# A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition Jeremy Rodgers · 12 November 2026 DOI: https://doi.org/10.5281/zenodo.24173681 Reflowed HTML prose and exact source-vector mathematical notation, with searchable linear PDF transcriptions. Vector expressions preserve the published geometry; their extracted alternatives are not semantic LaTeX or MathML. ## Source page 1 A Wilson Yang–Mills Construction A Natural-Language Mathematical Exposition Jeremy Rodgers Independent Researcher https://www.everythingequation.com DOI: 10.5281/zenodo.24173681 12 November 2026 Abstract This paper gives a self-contained natural-language exposition of the expanded Wilson Yang– Mills Construction: Mathematical Translation. For a given closed matrix subgroup of U(m) with nonabelian simple real Lie algebra, the construction supplies one calibrated four-dimensional Wilson family, full fixed-period mesh limits and a subsequential outer volume limit, a common hierarchy of local fields and phases, and a selected local Wightman theory with Poincaré covari- ance, forward-cone spectrum and a unique vacuum. We give the finite quadratic kernels and nonlinear geometry, marked integration, coupling reset and shooting histories, physical detec- tor and equation quotient, phase and boundary reconstruction, separated and nested operator products, contact extensions, and the finite-to-continuum stress and trace calculation with their hypotheses and derivations. The centered trace obeys lim r↓0 r8 log2(1/r)GΘ(r) = 3 dimR g π4 > 0. The associated exclusions retain their stated finite mass-shell and Maxwell comparison hypothe- ses. No positive mass gap is asserted. Two formula amendments are derived and documented with the conventions used throughout the argument. Attribution. The underlying Yang–Mills Lean work is credited to OpenAI. Jeremy Rodgers prepared this mathematical exposition with AI assistance. Funding. No external funding. ## Source page 2 A Wilson Yang–Mills construction J. Rodgers Contents 1 Introduction: the construction and its dependencies 1 2 Statement of the public conclusion 3 3 Notation and analytic conventions 5 4 Finite-lattice positivity 9 5 Quadratic kernels and their ultraviolet limit 11 6 Local nonlinear geometry and estimates 38 7 Marked integration and the coupling reset 111 8 Convergence of the interacting scale histories 149 9 Physical labels and their detection 238 10 Free shell charges and separation of the Ward kernel 254 11 The common continuum hierarchy 276 12 Phase decomposition and Wightman reconstruction 288 13 Separated and nested operator products 339 14 Extension across collision sets 360 15 The Gaussian short-distance comparison 372 16 The centered trace and its non-freeness test 385 17 Completion of the proof 451 18 Conclusion 453 A Editorial amendments and conventions 453 B Source, attribution and reproducibility 455 i ## Source page 3 A Wilson Yang–Mills construction J. Rodgers 1 Introduction: the construction and its dependencies The starting point is a finite probability law: group elements live on oriented lattice links, and the Wilson action measures each plaquette’s departure from the identity. The problem addressed by the supplied construction is to control one family of these laws while the mesh tends to zero and the physical period subsequently tends to infinity. Inserting local gauge-invariant functions into that same family is meant to produce a compatible hierarchy, rather than unrelated limits for different observables. The public theorem in Section 2 states the exact common-family and subsequence quantifiers. The derivation proceeds through a finite linear model, an actual nonlinear integration, and two successive continuum limits. The quadratic model provides local inverse kernels and stationary response maps. Nonlinear charts compare group links with that model, while positive energy bounds pay for exceptional configurations. Marked local activities keep products of insertions attached to the same represented density. An exact reset identity and finite coupling shooting select calibrated histories on one calendar. Detector estimates then separate physical label classes and identify the equation image before the common continuum hierarchy is formed. These constructions supply the estimates and compatibility statements used in the limits and reconstruction. Reflection positivity and slab bounds provide Hilbert-space control. The ground compression algebra supplies the phase space and its measure; tube continuation and tempered boundary limits supply fields on one common dense domain. The final part returns to actual finite current and stress functions. Their local equation estimates, coefficient calibration and Gaussian comparison determine a conserved physical stress and the centered trace. Positivity upgrades comparison along arbitrary retained scaling sequences to the full positive-distance trace limit. That limit supports the precise comparison exclusions stated in the public theorem. Source and the two-paper release. The mathematical source is the expanded project ex- traction, Mathematical Translation [1], prepared with AI assistance. It presents the Yang–Mills construction associated with the Lean work credited to OpenAI. This paper develops that source as a complete mathematical exposition. Its definitions, hypotheses, estimates and derivations are included here, together with the formula amendments in Appendix A. The accompanying flagship paper, Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction [2], applies Jeremy Rodgers’s Shadow Theory, source–readout and Source Lift programme to this construction [3]. Rodgers originated and directed that programme; AI assisted its execution and the preparation of the two expositions. The papers share the Zenodo release 10.5281/zenodo.24173681. Each contains its complete mathematical development: this pa- per presents the construction, and the flagship develops its source–readout realization and recovery. Their common subject does not require a reader of either paper to obtain an essential derivation from the other. The given gauge group, faithful representation, four-dimensional lattice, Wilson action and calibration conventions are physical and mathematical inputs. The detailed construction supplies stress Ward identities, nested expansions and contact extensions on its common witnesses; the public theorem retains its original list of clauses. How to read the argument. Read the finite setup and conventions first, especially the healthy- domain derivative and support budgets. The quadratic section explains the aliases, metric amend- ment and filled stationary predictors. The nonlinear and marked-integration sections use those objects to prove an actual density identity. The long scale-history section then buys every fixed fi- nite request in a specified order, and distinguishes a full Cauchy limit from an auxiliary extraction. 1 ## Source page 4 A Wilson Yang–Mills construction J. Rodgers The label, shell-charge and quotient sections are needed for reconstruction and stress detection. The OPE, contact, Gaussian and trace sections retain their own finite horizons, comparison devices and domains. The closing assembly returns to the theorem and identifies all its witnesses on one family and one refinement. Background. The finite law is Wilson’s lattice formulation [4]; compact group integration uses Haar measure [5]. Wick pairings [6], reflection positivity and Euclidean reconstruction [10, 11, 12], chessboard inequalities [13], positive-functional representations [7, 8], and the character-algebra approximation [9] form the mathematical background. Renormalization and cluster methods are described in [16, 17, 18, 14]; the field-theoretic framework is that of [15], and the conventional one-loop coefficient is associated with [19, 20]. These references credit the background; the actual hypotheses, constants and derivations used here are stated locally. Notation Role in this paper G ⊆U(m) Given closed group and faithful matrix representation; r = dimR g is the color dimension. hN = d−1 N , Lp Physical fine mesh and physical period in the public family. ai, Li, ti Intermediate spacing, integer block ratio, and ti = log(a−1 i ). bβi, βi Prescribed reference clock and actual represented coupling; their com- parison is proved in the running argument. θ, θ Healthy-field thresholds; the local text specifies which schedule or fixed chart is being used. Fd, Ed Finite physical label filtration and coefficient spaces; the relevant equa- tion quotient is specified before use. χ, Sχ, S Character phase, its separated hierarchy, and the unselected barycenter hierarchy. A scalar cutoffalso denoted χ is always a function with an explicit position argument. Tab, Θ Conserved physical stress and its selected-vacuum centered trace. Finite candidates are defined separately. 2 ## Source page 5 A Wilson Yang–Mills construction J. Rodgers 2 Statement of the public conclusion The assertion below is the supplied construction theorem in its public form. Its detailed proof occupies the subsequent sections. The Wilson family is fixed before a volume subsequence is requested, and the full mesh limit is obtained at each suitable fixed period before an outer volume subsequence is taken. An auxiliary compactness extraction is subsequently identified with that full inner limit. The concluding trace test concerns precisely the finite mass-shell class specified below. Let G be a closed subgroup of U(m) in its given faithful matrix representation. Its real Lie algebra is g = {X : exp(tX) ∈G for every t ∈R}. Assume that g is nonabelian and simple: every real linear subspace V ⊆g with [g, V ] ⊆V is either zero or all of g. No connectedness hypothesis is added. The common Wilson family. There is one Wilson family C for this G, with strictly increasing positive integer periods Lp and denominators dN. Its mesh is hN = d−1 N and its periodic lattice has side LpdN. Group elements are assigned to its oriented links. With normalized Haar measure ν and the independent link prior, the probability law is dμN,p(U) = Z−1 N,pe−βN,pS(U) Y e dν(Ue), S(U) = X x,i 0 and ε > 0, every fixed channel satisfies, for all sufficiently small r > 0, |Ca,j,λ(x; o)| ≤rj−|a|−ε whenever |xi −o| ≤Mr and |xi −xi′| ≥ηr for i ̸= i′. For a fixed spectator word b, c > 0 and D ≥0, the same cluster domain and the spectator separations |o −yl| ≥c and |yl −yl′| ≥c give Sχ(a(x), b(y)) − D X j=0 X λ Ca,j,λ(x; o)Sχ(Oj,λ(o), b(y)) ≤rD+1−|a|−ε. This holds for all sufficiently small r, uniformly in all phases and also for the unselected hierarchy. All terms belong to the same hierarchy. The threshold may depend on the fixed words and the stated parameters. The centered trace and its comparison class. There is a centered degree-four field Θ. Let GΘ(r) be its selected two-point correlation at spatial zero and Euclidean times −r/2, r/2. Then lim r↓0 r8 log2(1/r)GΘ(r) = 3 dimR g π4 > 0. It cannot agree at every sufficiently small radius with a finite complex linear combination of finite products of the specified mass-shell integrals. Each factor is of the form Z R3 Eμ(p)t−1 2 P(p)e−rEμ(p) d3p, Eμ(p) = q μ2 + |p|2, 4 ## Source page 7 A Wilson Yang–Mills construction J. Rodgers where μ ≥0, t ∈Z≥0 and P : R3 →R is continuous and positively homogeneous of a nonnegative integer degree. Here μ denotes the comparison mass called m in the source, to distinguish it from the representation size. This is the public non-freeness test with its precise comparison class. No positive physical mass gap is asserted. The public theorem is obtained by choosing blocking penalty b = 1 in the detailed construction, taking the refinement for the requested volume sequence, and projecting its single set of witnesses. This preserves the common hierarchy across all conclusions. The detailed construction also supplies stress Ward identities, nested expansions and contact extensions. They are developed below on its stronger common witnesses; they are neither extra assumptions nor additional clauses silently added to the public theorem. 3 Notation and analytic conventions The same letters occur in different mathematical roles in this long construction. In particular, Lp is a physical period, whereas Li is an integer block ratio; m is the dimension of the given matrix representation, whereas m(q) below is a Fourier multiplier. The context and arguments distinguish these uses. The reference coupling bβi is a prescribed clock, while an unadorned βi is the actual calibrated coupling. They agree only to the extent established by the running estimates. None of the estimates below permits substituting an arbitrary covariance for the particular positive covariance supplied by its construction. All lattices are four dimensional. The integer cover is Z4, and the periodic lattice of side n is (Z/nZ)4. A positive link is (x, i), joining x to x + ei; reading it in reverse gives the inverse group element. A plaquette in the positive plane i < j is the ordered product appearing in the theorem. Signed coordinate permutations act on links with this reversal convention. Lattice distance is the maximum of the centered coordinate distances. A support is a finite set of sites; its internal links have both endpoints in that set. The Lie algebra carries the physical inner product ⟨X, Y ⟩g = −1 m Re tr(XY ). It is positive definite on skew-Hermitian matrices and invariant under conjugation by G. Fix an orthonormal real color basis for this inner product, identifying the algebra with Rdim g. A color vector has matrix Frobenius norm √m times its color norm. The group chord is chord(g) = ∥g −1∥F/√m. Inversion and conjugation preserve it, and chord(gh) ≤chord(g) + chord(h). The Lie-algebra curvature of a holonomy g is the color projection of (g −g−1)/2. These normalizations are kept in the quadratic action, coupling slope and trace calculation. Write ai > 0 for the lattice spacing at scale i, and Li = ai/ai+1 for its integer block ratio. The construction chooses odd ratios at least three. The scale time is ti = log(a−1 i ), so ti+1 −ti = log Li. Block ratios can be increased stage by stage; subsequent lower bounds on the reference coupling are called delays. Such choices are always made before the terminal cutoffand physical period. The reference coupling is bβi = ̄β + sti, where s = 11CA 24π2 , CA = − 1 dim g X α tr(ad Tα)2. The simple nonabelian assumption gives CA > 0. Intermediate represented couplings are distinct from this reference coupling and are compared with it explicitly. For a finite support X, its radius-B halo is the union of the lattice balls of radius B about its points. The cover number covB(X) is the minimum number of such balls whose union is connected 5 ## Source page 8 A Wilson Yang–Mills construction J. Rodgers by unit lattice steps and contains X. The centers need not lie in X. This connected-union definition, including its behavior under periodic wrapping, is the one used in all support estimates. The numerical estimates often allow arbitrarily small power losses. For a fixed block ratio L, put βF (x, L) = x + s log L, B(x) = ⌈(log x)2⌉, temp(x) = x−1/2e(log log x)2. An envelope f has rate a if for every b > a, |f(x)| ≤βF (x, L)b for all sufficiently large x. The threshold may depend on the fixed stage and block ratio, but not on the period. Rate zero means subpower growth; it does not mean boundedness. The fine threshold is fiery(x) = temp(βF (x, L)). Finite sums preserve the larger rate, products add rates, and a strictly positive exponent gap absorbs any fixed finite multiplicity or subpower factor. For instance, if f has rate a and g has rate b, choose slightly larger exponents for their separate bounds; their product is then bounded by every exponent strictly greater than a + b. This is the meaning of adding rates, rather than an assertion of an exact leading power. Where the later operator-product argument uses rates along a scale path, that path and its limiting logarithmic exponents are specified separately. Weighted bounds on local terms include their derivative order, direction restrictions and expo- nential support weight. A bound on their sum always means an absolute bound, or equivalently a common bound on every finite partial sum of the nonnegative majorant. This convention permits countable dictionaries without assuming a finite number of terms. Terms such as a bill or a paid bound in the proof denote these explicit inequalities, not additional probabilistic assumptions. 3.1 Local activity domains, derivatives and absolute budgets An activity is a local contribution to the represented density. Its budget simultaneously controls size, the number of supports meeting a site, and the cost of their spatial extent. A flag records failure of a plaquette small-field test. A regular derivative is taken on its declared healthy domain; it does not differentiate the discontinuous indicator at that domain’s boundary. The distinction between an assigned gas support and the enlarged set of links actually inspected will be used in every locality claim involving flags. How the bounds are used The local estimates below have three separate pieces of data: the set of configurations on which they hold, the derivative order and directions allowed there, and the weight used when supports are summed. A bound on one healthy configuration is not a bound on a neighborhood of that configuration. Likewise, a support budget is a sum over an indexed dictionary, so two terms with the same support still count twice. The subsequent integration keeps these distinctions when it differentiates and sums the actual insertion gas. We fix the meaning of all activity estimates before using them. At a fixed finite torus Λn = (Z/nZ)4, let Cn = GΛn×{0,1,2,3} be the configuration space and An = gΛn×{0,1,2,3} its real space of left-link directions. A function is local on a site set X if its value depends only on links with both endpoints in X. Gauge invariance refers to Ux,i 7→kxUx,ik−1 x+ei. For a plaquette p, write V (p) for its four visited vertices and Up for its holonomy. At threshold 0 < θ ≤1 define Fθ(U) = {x : ∃p, x ∈V (p), chord(Up) > θ}, Hθ(X) = {U : X ∩Fθ(U) = ∅}. (AB1) Thus equality with the threshold is healthy. Each flag event is open, because it is a finite union of strict inequalities for continuous plaquette functions. Each healthy domain is closed and Borel. The mask is the literal function 1Hθ(X)F; its boundary is not differentiated in any regular budget. 6 ## Source page 9 A Wilson Yang–Mills construction J. Rodgers A coefficient which also reads the final flags in X is required to agree whenever both the links internal to X and X ∩Fθ(U) agree. Its actual link read set is bounded by Xins = X ∪ [ p:V (p)∩X̸=∅ V (p). (AB2) Agreement on internal links of this enlarged set fixes every plaquette incident to X, hence all flags inspected there. This is why flag-dependent payloads are assigned enlarged physical footprints even though compatibility in a gas is tested on the originally assigned supports. For A ∈An, set (inc(U, tA))e = exp(tAe)Ue, ∇AF(U) = d dtF(inc(U, tA)) t=0 . For an ordered list l= (A1, . . . , Aq), define recursively ∇∅F = F, ∇A::lF(U) = d dt(∇lF)(inc(U, tA)) t=0 . (AB3) The order matters; no commutation of left derivatives is assumed. These derivatives are required to exist where tested. A regular representative on X is a globally Borel, gauge-invariant local function F such that v 7→F(inc(U, v)) is Cj near zero for every U ∈Hθ(X), and its recursively specified left derivatives have the derivative identities in (AB3) through order j. In particular the smooth germ is required also at a healthy configuration lying on the threshold boundary. Off-domain values of the representative remain available for locality and measurability, but do not enter its regular derivative price. A budget point consists of integers p ≥2, j ≥6, B ≥1, support exponent s > 0, and threshold 0 < θ ≤1. We denote it by p = (p, j, s, B, θ). For anchor o, its allowed directions satisfy |Ae| ≤θ(1 + dist(e−, o))p for every link e. (AB4) Here e−is the initial vertex of the stored positive link, and the distance is the periodic maximum- coordinate distance. A nonnegative number b is a derivative envelope for F on (o, X) if |∇lF(U)| ≤b whenever U ∈Hθ(X), |l| ≤j, and every A ∈lsatisfies (AB4). (AB5) Norms replace absolute values for vector-valued coefficients. The support price is massp(X, b) = es covB(X)b. (AB6) The factor is at least one. Every sum of envelopes below means a nonnegative summable family together with its asserted bound on the sum. Equivalently, every finite partial sum obeys that bound; taking increasing finite partial sums proves summability and the same total bound for countable dictionaries. A regular address is (oi, Xi) with oi ∈Xi. A regular family Ei has anchored budget δ if it has envelopes bi ≥0 satisfying (AB5) and sup o∈Λn X i:oi=o es covB(Xi)bi ≤δ. (AB7) The representation used after the kinetic reset additionally has Ei(1) = 0. On a nonwrapping support its identity Hessian D2v 7→Ei(inc(1, v))  0[A, C] 7 ## Source page 10 A Wilson Yang–Mills construction J. Rodgers vanishes whenever A, C are affine link fields in the canonical lift about oi. Explicitly an affine field has Ax,a = aa + X b lift(x −oi)b dab. (AB8) The reset chapter constructs this Hessian cancellation; it is not imposed on arbitrary primitive functions before projection. A forced address consists of a support Yi, a nonempty inventory Ii ⊆Yi, and a fixed extra predicate Pi(U). Its coefficient Ki is globally Borel and gauge invariant and vanishes unless Yi ∩Fθ(U) = Ii, Pi(U). (AB9) The predicate is Borel and gauge invariant; both coefficient and predicate may read links and final flags on Yi, hence have the physical read enlargement (AB2). A forced budget κ consists of numbers ci ≥0 such that |Ki(U)| ≤ci on every configuration and sup x∈Λn X i:x∈Yi es covB(Yi)ci ≤κ. (AB10) This is a support-hit budget, in contrast to the single-anchor sum (AB7). Coefficients equal to zero still retain their addresses and eligibility predicates. For a finite set D of forced addresses, the gas term is zero unless their supports are pairwise disjoint and their inventories have union exactly Fθ(U); otherwise it is Q i∈D Ki(U). Summing these terms over finite D defines the forced gas. The regular exponent is the sum of 1Hθ(Xi)Ei. At a fixed finite torus these series are absolutely convergent: assign each regular term to its anchor and each forced term to one site of its nonempty support. Equations (AB7)–(AB10) bound their total envelope sums by |Λn|δ and |Λn|κ. The sum of absolute products over all finite subsets is at most Q i(1 + ci) ≤e P i ci. Restricting to compatible subsets only decreases it. If there are no flags, only the empty forced set can contribute, since each inventory is nonempty; the gas is then exactly one. Signed translations are implemented by fixed permutations of the whole regular and forced address lists, carrying anchors, supports, inventories, predicates and coefficient functions together. Thus comparing two activity values in a single chart always means comparing the same lists and symmetry permutations. It does not mean identifying two representations merely because their total densities agree. An early source budget uses s + 3, rather than s, in (AB7). This is the pre-reset reserve. The three additional support-exponent units pay the local splitting and conversion operations explicitly described in the reset and signed-expansion chapters. Reducing the exponent from s + 3 to s decreases every nonnegative summand. The rotation of derivative directions costs the finite factor turnFee(p) = (2 2p)j. (AB11) Indeed the anchor itself is transported exactly. A reversed link is stored at its other endpoint, at distance at most one from the transported base, so 1 + d + 1 ≤2(1 + d) costs 2p in (AB4). Its right increment can be converted to a left increment by the adjoint action of the background link, which preserves color norm. Expanding each pulled-back left derivative into its positive-link and negative-link contributions gives at most 2q ordered lists at order q; each has length q and direction inflation at most 2p. Multilinearity of the iterated derivatives bounds their sum by (2 2p)qb, which is at most (AB11) times b for q ≤j. This expansion follows by induction on the ordered list, using the exact left/right increment identity and the disjoint link supports of the positive and negative 8 ## Source page 11 A Wilson Yang–Mills construction J. Rodgers parts. Support enlargement is charged separately by its cover-number inequality; it is not included in this factor. At level i put xi = bβi and Mi = exp((log log xi)2). The normalizations used in (LCN1) are the calendar’s actual allowances bPi = x−11/10 i , kPi = exp(−M1/5 i ). (AB12) Both are strictly positive. Thus ∥A∥Pi ≤t means exactly that its regular and forced components obey (AB7) and (AB10) with budgets tbPi and tkPi, respectively, in their fixed address layout. The retained physical states have t ≤1/8. Triangle inequalities give the norm properties used later: sums of admissible nonnegative envelopes remain envelopes, derivatives are linear, and scalar multiplication multiplies all envelopes by the scalar modulus. Least envelopes, followed by finite partial-sum limits, give attainment of an infimum whenever it is used in (LCN1). The separated completion removes only differences of zero price in these tested germs and forced values. The source aliases RG/Masks, RG/Activities, Quotes, Investor and FullVolume/Sym/Shorn refer to the mask, activity, reset-budget and normalization definitions made explicit in (AB1)– (AB12). They will be used with the same domains and directions in every subsequent reference to a regular, anchored, early or forced-hit budget. 4 Finite-lattice positivity Reflection positivity starts at the finite Wilson law. Cutting a torus into two reflected halves rewrites each seam contribution as a positive kernel, so that a future observable paired with its reflected conjugate has nonnegative expectation. The finite-dimensional statement gives the null- space quotient used later to build a Hilbert space. It is also the input to the chessboard bounds: positivity converts a product of separated chamber observables into a product of individual rooted bounds. The details below retain the seam and orientation conventions needed for both uses. First the finite Wilson law is an actual probability measure. Every unitary matrix V has −m ≤Re tr V ≤m: each diagonal entry has modulus at most one. Thus every plaquette price lies in [0, 2]. If there are F plaquettes and β ≥0, its continuous density relative to the product Haar probability measure obeys e−2βF ≤e−βS(U) ≤1, e−2βF ≤Z ≤1. In particular normalization is legitimate and bounded measurable cylinders are integrable. A site gauge change multiplies a link on its two ends; by Haar invariance its change of variables preserves the product prior. Its plaquette holonomy is conjugated at the base site, so the trace and the action are unchanged. The normalized law is consequently gauge invariant. A signed lattice permutation merely reindexes plaquettes, possibly reversing their holonomies; inversion preserves their real traces. It too preserves both the prior and the law. The local representatives in (PF12), and their finite products, therefore have precisely the finite-lattice expectations used below. Reflection positivity is the nonnegativity of the expectation of a future observable times its reflected complex conjugate. It is a property of this finite law, before any quotient or completion. The Wilson measures used throughout have that property at every nonnegative coupling. We give the finite-dimensional argument, since it also supplies the rooted bounds used in extraction. First, the kernel K(g, h) = Re tr(gh∗) on G is positive semidefinite: it is the real Euclidean inner product of the real and imaginary matrix entries. Finite sums, pullbacks, nonnegative multiples and pointwise products of positive 9 ## Source page 12 A Wilson Yang–Mills construction J. Rodgers semidefinite kernels are positive semidefinite. For the product assertion, express each finite positive semidefinite matrix as a Gram matrix and tensor its Gram vectors. The exponential of such a kernel is positive semidefinite as well, by its power series with nonnegative coefficients and entrywise limits of the resulting finite matrices. Cut the periodic lattice perpendicular to a coordinate axis. A leaf X consists of the links tangent to one slice, and a thread consists of the links joining successive slices. With the thread fixed at the identity, the factor joining two leaves has the form Wβ(X, Y ; 1) = e−βq Eβ(X)Eβ(Y ) exp " β m X l Re tr(XlY ∗ l) # , (RP1) where q counts the crossing plaquettes and Eβ is the positive factor from the allocated half of the tangential action. Thus (RP1) is a continuous positive semidefinite kernel for β ≥0. For general threads, use their endpoint gauge action on the leaf links. Haar invariance gives the identity Z Wβ(X, Y ; u) du = ZZ Wβ(u · X, v · Y ; 1) du dv. (RP2) Both integrations use the product normalized Haar measure on the thread variables. The identity follows by replacing one thread variable by its left translate and combining the two endpoint gauge actions. The right side is positive semidefinite: apply the positive-kernel quadratic inequality to finite linear combinations of the averaged feature vectors, or approximate the integrals by simple functions in the same Gram identity. All integrands are continuous and bounded on compact spaces, so this passage and Fubini are justified. The averaged kernel in (RP2) is continuous by dominated convergence. Now split the lattice into the two halves of the reflection. For a reflection through sites, the two halves share the seam-site variables, and after fixing them the cross kernel is the constant kernel 1. For a reflection through links, there are two seam threads on the periodic lattice. They are distinct even at the shortest allowed half-length. Their joint Haar law is the product thread law. After integration, the cross kernel is the product of the two kernels (RP2), one with its leaf arguments reversed. It is again positive semidefinite. To make the splitting precise, list the seam variables, the two seam threads when present, and the open variables on the two halves. Assemble the original link configuration by their coordinate assignments and the endpoint gauge multiplications along the cut direction. Each change of coor- dinates multiplies a Haar link variable on the left or right by fixed already listed group elements, or inverts that variable. Iterated Haar invariance shows that this assembly preserves the product prior. Reading each plaquette shows that the action splits into the two half factors and the seam factor just described. A probe F supported strictly in one half reads only that half and the shared seam variables; its gauge invariance permits the endpoint gauge changes. Its reflected conjugate reads the other half. Consequently, conditional on the seam variables, the unnormalized reflected pairing has the form ZZ f(X)f(Y ) Kseam(X, Y ) dX dY, with Kseam positive semidefinite. This integral is nonnegative by the integrated Gram inequality. The half-action factors have been included in f. Bounded measurable probes are allowed: the Wilson density is bounded on the compact finite configuration space, and their reflected products are bounded and measurable, so all integrations are absolutely integrable. Integrating the conditional nonnegative quantities and dividing by the strictly positive Wilson partition function proves ⟨F F ◦θ⟩β ≥0. (RP3) 10 ## Source page 13 A Wilson Yang–Mills construction J. Rodgers Here θ is the actual geometric link reflection, including inverse links. In particular no crossing link has been replaced inside the probe. We next derive the chessboard inequality on an admissible rectangular tiling. Let v > 0 be its number of cells. For a function F in its strict reference chamber, write F (c) for its translated, reflected and, where prescribed, conjugated copy in cell c. Define R(F) = *Y c F (c) + β 1/v . For different chamber functions Fc, let V (F•) = |⟨Q c F (c) c ⟩β|. The Cauchy–Schwarz inequality from (RP3), applied across any translated cell seam, gives V (A)2 ≤V (A+)V (A−), (RP4) where A+ and A−are the assignments obtained by reflecting the corresponding half-assignment into the other half. For a finite alphabet of functions, choose a cell assignment maximizing V . Both folded values in (RP4) are at most this maximum. If the maximum is positive, (RP4) forces both folded values to equal it; if it is zero this is immediate. Hence the property of being a maximizing assignment is preserved by every fold. Such a fold-closed property has a constant assignment: in one coordinate start with a one-cell-wide strip of identical slices and fold at its preceding boundary. This extends the strip by one slice while preserving the property. Continue until a half-period is filled, then reflect it to fill that coordinate. A fold in another coordinate preserves equality along the coordinates already filled. Repeat in all four coordinates. This is a finite construction and needs no power-of- two restriction on the number of slices. For arbitrary Fc, apply this maximum principle to the finite alphabet eFc = Fc/(R(Fc) + ε), ε > 0. Every constant assignment of a normalized function has value at most 1, because its value is [R(Fc)/(R(Fc) + ε)]v. The maximum principle bounds their mixed assignment by 1. Rescale each cell and then let ε ↓0 to obtain *Y c F (c) c + β ≤ Y c R(Fc). (RP5) The same root is a seminorm on the chamber functions. Complex homogeneity follows because a copied scalar is either z or ̄z, and the absolute value of their full product is |z|v. To prove subadditivity, expand the full copied product of F + H into its 2v assignments of F, H. The triangle inequality and (RP5) bound its expectation by (R(F) + R(H))v. Take the positive v-th root. This proves all the rooted homogeneity, subadditivity and mixed-product estimates used below directly for the Wilson law. 5 Quadratic kernels and their ultraviolet limit The quadratic analysis constructs the linear response around the flat configuration. There are three separate issues. The evaluated quadratic form on link potentials must converge; its curvature representation must remain positive and exponentially local; and the stationary predictor must have a regular Fourier symbol at the apparent zero-momentum singularity. The auxiliary metric amendment changes only directions not detected by curls. Thus retaining the physical potential precision is a proved identity, not an inference from positivity. 11 ## Source page 14 A Wilson Yang–Mills construction J. Rodgers 5.1 Flat quadratic ultraviolet convergence Reading the finite linear records This opening convergence argument states the estimates required from a finite linear history before constructing that history below. A history, also called a horizon in the source notation, is a finite sequence of metrics and adjacent stationary response maps ending at the fixed bare quadratic form. Its price table records analytic widths, lower coercivity bounds and upper kernel bounds at each stage. A smoothing train is the finite chain of averaging patches defined in Section 5.8; a transition tower is the collection of auxiliary rows used to realize one adjacent response and its Gram identities. These names describe finite numerical and operator records. The later metric, tower and pinning calculations construct their required identities, so the opening conditional estimate is not an extra assumption of the public theorem. Write ai for the positive lattice spacing at level i, with ai tending to zero. For a reporting level k put T(k, i) = 2 + Z(k) + |ti −tk|, where Z is nonnegative. Let Qn,i denote the cover kernel of the quadratic action at level i constructed with finer cutoffn. The convention is the action Hessian convention; there is no additional factor one half. The kernel is obtained from the curvature kernel by composition on both sides with the lattice coboundary, so it describes the quadratic form on link potentials. It does not determine an arbitrary extension of the curvature metric away from curls. For s ≥0, let Rs(K) be the weighted row bound of a cover kernel K: for every output link and each fixed input orientation, sum the absolute kernel entries over input sites with weight es dist, then take the supremum over the output link and input orientation. This is the source’s RowCost at side one. Exponential weights control spatial locality as well as absolute summability. A finite quadratic history has, at each i ≤n, five real Laurent rows: an unamended two-form metric H, its inverse, an amended metric M, its inverse, and a residual Gram root F. Their weighted coefficient bounds have width α(J(i)) > 0 and price Ω(J(i)). At every torus period, HH−1 = MM−1 = I, F ∗F + cI = M, Md1 = Hd1, M(0) = I. Here c > 0 is the direct clip coefficient; the full Gram factor is the stack (√c I, F). Both metrics are symmetric with quadratic forms between η(J(i))I and Ω(J(i))I, where η > 0. The terminal unamended metric is I. Each transition has a fine potential response A0, curvature response T, and mixed, normal and image rows satisfying the stationary and Gram identities derived below. The mixed row has bound Ω/L2; the normal and image rows have bound Ω. A word of r carried fine differences of A0 or T has bound ΩL−1−r or ΩL−2−r, respectively, through order J(i + 1) −2. A carried difference increments the fine residue, shifts the coarse displacement when that increment crosses its centered seam, and subtracts the original entry. Thus it is the actual fine difference, including the carry. These all-period conditions are constructed by the metric, positive-margin, auxiliary-tower and pinning arguments below; they are not additional assumptions of the public theorem. Put yi = w(J(i)). The spacing hypotheses are 0 < w(j) ≤min{1, P.α(j)}, yi ≤min{1, P.α(J(i + 1)), P.α(J(i))} (i ≥e), one-step price at i ≤block(i)−θ, 2yi ≤block(i)yi+1, P.Ω(J(i)) ≤B T(k, i)l (k arbitrary, i ≥e), 12 ## Source page 15 A Wilson Yang–Mills construction J. Rodgers for θ > 0, B ≥0 and a nonnegative integer l. Here P.α and P.Ωare the width and upper-bound columns of the fixed price table. Each earlier history also uses the specified destination response T, the compatibility condition called Flat.meshed in the source. Under these hypotheses there exist A ≥0, an integer d ≥0, and c > 0 such that, for all k and e ≤i ≤N ≤n, Ryi(Qn,i −QN,i) ≤AT(k, i)d(aN/ai)c. (1) The source proves this through curvature packets. At the terminal comparison level N subtract the two saddle curvature kernels. Their strip bounds give a packet whose cost is at most topTab(N) = 2|Faces|P.Ω(J(N)). This packet represents the difference of the quadratic forms on every scheduled torus. Propagate it backwards from i+1 to i using the two histories’ fine response maps. The exact crossing identity preserves its representation of the quadratic difference; neutrality and translation covariance are retained. The weighted packet cost is multiplied by quote(i, yi+1). The width-crossing hypothesis licenses this propagation at width yi. In particular, this is an identity for the two actual histories rather than a comparison with a separately chosen limiting history. If Ci is the propagated cost, the recursive majorization is CN = topTab(N), Ci ≤block(i)−θCi+1. One may first define a nonnegative majorant using the reciprocal recursive weight, called Stair.weight in the source. Its recurrence is exact before the one-step factor is replaced by its upper bound. Applying the polynomial-age decay lemma of Section 5.2 gives Ci ≤AT(k, i)d(aN/ai)c. In this particular terminal-only recurrence the majorant can also be written without the auxil- iary weight notation. With tj = log(a−1 j ), it is bCi = topTab(N) N−1 Y j=i L−θ j = topTab(N)e−θ(tN−ti). The empty product gives its terminal value, and each backward step has exactly the required contraction. Set u = tN −ti ≥0. The triangle inequality gives T(k, N) ≤T(k, i) + u, whereas T(k, i) ≥2. The polynomial cap on the terminal price therefore gives bCi ≤CT(k, i)l(1 + u)le−θu ≤C′T(k, i)le−θu/2. The last constant is finite because the continuous function (1 + u)le−θu/2 is bounded on [0, ∞). Since e−u = aN/ai, this is the asserted estimate with c = θ/2. The more general recurrence lemma below also allows forcing at intermediate levels. Neither argument assumes continuum convergence. To recover the cover quadratic kernel, let D be the lattice coboundary and set W(K) = D∗KD, with K interpreted as its two-point kernel. Finite support of D gives a width-one row and column bound L ≥0 for this paragraph. Weighted convolution and the distance triangle inequality give Rs(W(K)) ≤|Faces|2L2C 13 ## Source page 16 A Wilson Yang–Mills construction J. Rodgers whenever the packet has width s ≤1 and cost C. This constant is independent of the torus size. The required uniqueness concerns periodization of absolutely summable cover kernels, not in- version of a finite matrix. In the source these operations are respectively named Flat.resolve and FreeResolvent.resolve. Suppose two summable cover kernels give the same matrices on tori whose side lengths tend to infinity. For a fixed pair of cover sites, express a periodized entry as the sum over sites with the same torus image. For each fixed summand, eventual injectivity of wrapping makes its indicator tend to the indicator of the selected cover site. The absolute kernel entries form a summable dominating family. Dominated convergence recovers that cover entry from the torus entries. Both kernels have the same limit and therefore agree entry by entry. Apply this argument to Qn,i −QN,i and W(Ki). Their finite-torus quadratic matrices agree by the propagated packet identity, and both have summable rows. Thus their cover kernels agree. The preceding convolution bound proves (1), absorbing |Faces|2L2 into A. For each fixed i ≥e, (1) makes every entry of Qn,i a Cauchy sequence, since aN tends to zero and c > 0. Let Fi be its entrywise limit. Fatou’s lemma for the nonnegative weighted absolute entries gives Ryi(QN,i −Fi) ≤AT(k, i)d(aN/ai)c. The horizon strip bound and the polynomial cap on the upper-bound column P.Ωsimilarly give Ryi(Fi) ≤B′T(k, i)l. The limits obtained with different reporting levels k coincide by uniqueness of entrywise limits. Consequently one family Fi works for every k. On levels before e it can be defined as zero because the conclusion imposes no requirement there. 5.2 Polynomial majorants and decay Let ti be increasing scale times, with ∆i = ti+1 −ti ≥log 3, and put τi = 2 + Z + |ti −tk|, where Z ≥0. Then τi ≥2 and τi+1 ≤τi + ∆i. Suppose fi ≤e−θ∆ifi+1 + gi (i < N), fN ≤gN, 0 ≤gi ≤Aτ d i , with fi ≥0, A ≥0, d ∈N, and θ > 0, uniformly in the schedules and terminal indices. Define K = 2(d + 1) θ , q = e−(θ/2) log 3 ∈(0, 1), B = A 1 −q. Since K + τi ≥K, the inequality 1 + u ≤eu gives K + τi+1 ≤(K + τi)e∆i/K. Thus d/K ≤θ/2 implies e−θ∆i(K + τi+1)d ≤e−(θ/2)∆i(K + τi)d ≤q(K + τi)d. Backward induction proves fi ≤B(K + τi)d. At N, use A ≤B. At an earlier index, fi ≤(Bq + A)(K + τi)d = B(K + τi)d. Finally K + τi ≤(K + 1)τi, so fi ≤B(K + 1)dτ d i . This includes d = 0 and A = 0. 14 ## Source page 17 A Wilson Yang–Mills construction J. Rodgers For the decaying version, suppose giec(tN−ti) has a uniform nonnegative polynomial majorant, with c > 0. Put μ = min(c, θ/2) > 0, and multiply the recurrence by eμ(tN−ti). The new unknown has contraction exponent θ −μ > 0; its forcing is polynomially bounded because μ ≤c. The preceding induction gives fi ≤A′τ d′ i e−μ(tN−ti) for uniform A′ ≥0, d′ ∈N. For the lattice schedule e−(tN−ti) = aN/ai. A forcing supported only at N automatically satisfies the weighted hypothesis, because its exponential factor is one on its support. This proves the decay estimate used in the flat overlap construction. The same argument extends to a well-founded collection of grades. At a grade j, permit polynomial forcing in finitely many strictly earlier grades, at the current and next scale. Sums, products, and integer powers preserve polynomial bounds. Next-scale ages are polynomially controlled when the block lengths are polynomially bounded in the current age, because τ i+1 ≤τ i + log(block(i)) ≤τ i + block(i). Induction on the grade therefore supplies the forcing bound needed for the scalar recursion. For differences, include the absolute bounds from both histories, including the current grade, in the forcing polynomial; permit difference bounds only at strictly earlier grades. A finite sum of exponentially decaying polynomial bounds retains such a bound with the minimum positive exponent, and multiplication by polynomial absolute bounds preserves it. The scalar weighted recursion then closes the induction. This recurrence is applied to the one-step estimates for the actual histories constructed in the nonlinear and history-convergence chapters. 5.3 Independence of the evaluated quadratic kernel Which quadratic object is independent The object compared here is the evaluated quadratic form on link potentials. The auxiliary two-form metric can change on directions not tested by the curl without changing that form. The word “shadow” in this particular calculation names the displayed modification of clip parameters in a second finite linear history. It does not introduce a source–readout framework into this companion paper. Equality is proved by the stationary matrix identities and backward induction from the common terminal form; no equality of the full nonlinear actions is inferred from it. The horizon’s terminal quadratic form is the same fixed bare quadratic form for every admissible horizon. Its one-step stationary action identity is, on each finite torus, Qcoarse = T TQfineT + b(I −probe T)T(I −probe T), where b > 0 is the blocking penalty and T is the extracted stationary response. Here is the exact comparison algebra. Write P for the common probe, Q,R for two symmetric fine precisions, A,B for their symmetric coarse precisions, and T,W for the responses. The primal and dual equations are QT = P TA, RW = P TB, A = b(I −PT), B = b(I −PW). Taking the transpose of RW = P TB gives W TR = BP. Consequently W T(R −Q)T = BPT −(PW)TA = B(I −A/b) −(I −B/b)A = B −A. The mixed terms cancel in their displayed order; no commutation of A and B is required. Thus equality of the fine quadratic forms implies equality of the coarse forms even if the auxiliary factor- izations differ. Backward induction from the common terminal form gives equality at every level 15 ## Source page 18 A Wilson Yang–Mills construction J. Rodgers i ≤N on every scheduled torus. The periodization uniqueness argument above then gives equality of the summable cover kernels. This proves the quadratic-history comparison. The harmonic and scaled-exterior sections construct the responses and derive their primal and dual equations. The comparison construction replaces each clip’s two numerical factors by γ(j) and γ(j)/32, with γ(j) > 0, retaining the relaxation, radius and slow selector. It retains the schedule, patches, smoothing indices and blocking penalty used by the block operator. The same terminal and back- ward comparison therefore applies to the original and modified horizons. Their evaluated quadratic cover kernels agree at every finite cutoff. This concerns the quadratic action on link potentials and does not identify higher-order horizon data. Here is the entrance argument, called entrance_gate in the source. First choose the retained- support linear construction for the fixed kernel choice; it supplies a positive clip curve γ and a length demand n. Apply the linear horizon construction to the numerical schedule with its rearranged clip profile. For its price choice X, set E(j) = max{tickets(X, j), n(j)}. The condition E(J(i+1)) ≤block(i) pays both demands, including auxiliary steps before the admitting tail. The delay at e supplies the spacing hypotheses proved next. Apply the flat convergence estimate to the modified horizon to obtain one family Fi, its polynomial absolute bound and its decaying difference bound. The exact comparison just proved, called same_shadow, identifies every original finite-cutoff quadratic output with its modified output. Equality of those sequences transfers their limits and difference bounds to every admissible original horizon. This proves the entrance statement once the retained-support construction is supplied. 5.4 Choosing the widths before the block lengths For the fixed price table P, choose w(j) = min{1, P.α(j), P.α(j + 1)} > 0. Let h(j, u) be the allotted bound for adjacent smoothing indices j, u, and put B(j, u) = |Faces|2[2/w(j)]h(j, u)2, EP (j) = l 2/w(j) + B(j, j)2 + B(j, j −1)2m . The smoothing index stays fixed or increases by one at each forward step. Thus, if j = J(i + 1) and u = J(i), then u = j or u = j −1. Requiring L = block(i) ≥EP (j) gives B(j, u)2 ≤L and L ≥2/w(j). The one-step price B(j, u)/L is at most L−1/2. Moreover 2w(u) ≤2 ≤Lw(j). These give contraction and width crossing with θ = 1/2. The same adjacency ensures that w(u) is no greater than either analytic width required at the crossing. All thresholds depend on the previously chosen prices, rather than on the subsequently chosen L. For the age bound, put Z(k) = 3 + log bβk, where bβk > 1, and let s > 0 be the coupling slope. Its affine dependence on scale gives bβi ≤bβk + s|ti −tk|. Divide by bβk > 1 and use log v ≤v −1 to obtain log( bβi) ≤log( bβk) + s|ti −tk| ≤(1 + s)[2 + Z(k) + |ti −tk|]. The delay condition P.Ω(J(i)) ≤log( bβi) on i ≥e now supplies a polynomial bound of degree one. Together with the previous inequalities this establishes every spacing condition used in the flat convergence argument. 16 ## Source page 19 A Wilson Yang–Mills construction J. Rodgers 5.5 Neutral packet propagation The predictor difference estimate used here follows directly from the one-step bound included in its Atlas data. Suppose its weighted forward difference in each fine direction has row bound c/Ld+1. Join x to z by a nearest-neighbor path of length a = |z −x|1 which never exceeds distance a from x. Such a path is obtained by changing each integer coordinate toward its target one unit at a time. The scaled-distance triangle inequality gives exp(σδL(x, y)) ≤exp(σa/L) exp(σδL(v, y)) for each intermediate site v. Thus each forward or backward difference along that path has weighted row sum, measured from x, at most exp(σa/L)c/Ld+1. Summing the a telescoping differ- ences and using the triangle inequality yields X y |U((z, i), (y, j)) −U((x, i), (y, j))| exp(σδL(x, y)) ≤a exp(σa/L)c/Ld+1. The centered estimate below is this bound with d = 2, followed by spending half the packet’s input exponential width on the factor a. The actual response and its one-step bound are constructed in the Atlas response and exterior-lifting arguments that follow. For a two-point curvature packet let K(x, v) be its kernel, where each argument includes a site and an orientation. Neutrality means P z K(x, (z, a)) = 0 for each fixed x and orientation a separately. If T, U are the response kernels used to carry the packet from the fine level to the coarse level, its pushed kernel is K′(y, z) = X x,v T(x, y)K(x, v)U(v, z). All rearrangements below use absolute convergence. It follows from the packet’s weighted row bound and the row and column bounds on T,U: first sum the nonnegative absolute majorant in the output variables, then in the input variables. The resulting finite product of bounds dominates the full multiple series. Thus both iterated sums and regrouping by the finite orientation sets are justified. Neutrality removes the spatially constant part of the right response. For fixed x, replace U(v, z) by U(v, z)−U((x.site, v.orientation), z). The removed sum vanishes separately for each orientation. If a = dist(x.site, v.site), the spatial difference estimate gains a/L relative to the ordinary row estimate. For positive input width τ and output width σ with 2σ ≤Lτ, a exp(σa/L) ≤(2/τ) exp(τa). Indeed σ/L ≤τ/2, and a ≤(2/τ) exp(τa/2), the latter following from u ≤exp(u) for u ≥0. The centered right row therefore costs |β|(2/τ)c/L3 for a degree-two predictor of cost c. The held left column costs |α|qL2. Multiplication, followed by the input packet bound p, gives foldedσ(K′) ≤[|α|q|β|(2/τ)c/L]p. This is the precise neutralCoin factor used in the flat recursion. The gain L−1 comes from centering before taking absolute values. The secured pilot and wire bounds in this chapter provide the predictor spatial difference estimate. 17 ## Source page 20 A Wilson Yang–Mills construction J. Rodgers To preserve neutrality after propagation, require that P z U((v, a), (z, b)) be independent of the input site v, for every pair of orientations a, b. Summing K′(y, (z, b)) over z gives a linear combina- tion of the zero orientation-wise sums of K. This is the source’s constant-sum condition travels. Translation covariance follows by shifting both fine summation variables by L times the coarse shift and using the covariance of K, T, U. Finally, periodization of K′ is exactly T T torusKtorusUtorus: absolute convergence permits splitting fine sites into residue classes, and covariance identifies each periodized factor. The mixed stationary identity then makes K′ represent the coarse quadratic difference. This proves the algebra, regrouping and price multiplication called toward, once the response bounds and constant-curvature transport have been constructed. 5.6 Positive three-form correction The linear flight construction uses a matrix symbol on the whole three-form coordinate space of the form A = V + lI −mS∗S, Here m ≥0 is the fine amendment parameter and l the coarse parameter. On real momentum the symbols are Hermitian. The analytic estimates give V ≥−a2CI and S∗S ≤a2BL−2I, where a is a fixed orientation-count constant, C, B > 0 depend only on the smoothing pair and penalty, and L is the block length. Choose l ≥a2C + 2 and require ma2B ≤L2. Then, for every vector x, ⟨x, Ax⟩≥[−a2C + l −ma2B/L2]∥x∥2 ≥∥x∥2. This is positivity on the full three-form space. No restriction to the image of the exterior derivative is used to obtain it. For the corrected form R = F ∗AF, a uniform inverse bound ∥x∥2 ≤W∥Fx∥2 yields R ≥W −1I. The source obtains W from a uniformly bounded right inverse of F, which in these square finite matrices is also a left inverse. An independent analytic row estimate supplies the upper bound. This explains why the coarse amendment threshold is chosen before the fine block length, while the subsequent length condition absorbs the already fixed fine amendment. The rebased correction identity has the following exact content. Let X,Y be the fine and coarse amended inverse metrics, B the coarse amended metric, T the installed inverse response, and d1, d2 the coarse exterior derivatives. Then B[Y −T TXT −b−1d1dT 1 ]B = dT 2 Rd2. To derive this, write Y = B0 + lRT 0 R0 and X = b0 + mF T 0 F0. The unamended defect is RT 0 V R0 and the intertwining identity is SR0 = F0T. Substitution makes the bracket RT 0 (V +lI −mSTS)R0. Since R0B = Fd2 and B is symmetric, the displayed formula follows with R = F TAF. The exterior lifts, regular symbols and analytic inverse estimates construct B0, b0, S, F with the bounds used here. 5.7 Roots for the bottom metric Let the amended metric M have real-axis lower bound b0I, and choose the clip coefficient 0 < c ≤ b0/2. Then M −cI ≥(b0/2)I. Subtracting the constant row cI preserves the analytic strip bound. The positive analytic-root construction gives a real Laurent row F with F TF = M −cI on every finite torus. Adjoin the constant block √c I. Its Gram matrix is exactly 18 ## Source page 21 A Wilson Yang–Mills construction J. Rodgers (√cI, F)T(√cI, F) = cI + F TF = M. The source chooses c’s allowed upper cap as half a uniform positive lower metric bound, inter- sects the finitely many positive analytic widths, and increases the common cost to cover the metric, inverse, diagonal, dual diagonal, and root rows. The resulting bed satisfies the required finite-torus floor identities and bounds for every permitted train. This reduces the remaining root step to the analytic square-root construction, rather than assuming a factorization with the desired decay. For completeness, the analytic square root can be constructed with one branch on a collar uniform over the entire family. Suppose a real Laurent matrix row Q has strip width η > 0 and entrywise weighted coefficient bound A, and its real symbols obey vI ≤Q(p) ≤AI with v > 0. Increase A if necessary so A ≥v. On the real axis, unitary diagonalization gives ∥A−1Q(p) −I∥≤1 −v/A. On the smaller strip η/2, differentiating the absolutely convergent Fourier series gives the en- trywise bound for symbol differences with constant D = A/(η −η/2). Let C be one plus the norm of the fixed identification from entrywise matrices to the matrix operator norm, and choose τ = min(η/2, v/(2CD)) > 0. For a complex momentum p in this collar, comparison with its real part gives ∥Q(p)−Q(Re p)∥≤ v/2. Consequently ∥A−1Q(p) −I∥≤1 −v/(2A) < 1 uniformly. The binomial series P(p) = √ A X n≥0 1/2 n ! A−1Q(p) −I n defines a bounded holomorphic periodic matrix function on the collar. On the real axis, diag- onalization reduces the series to the positive scalar square root, so P(p)∗= P(p), P(p)2 = Q(p) and √v I ≤P(p) ≤ √ A I. The Laurent coefficients of Q are real, and conjugation at opposite real momenta preserves the selected branch. Fourier inversion, taking real parts of the coefficients, therefore recovers the same real-axis function. Bounded holomorphy gives exponential Fourier decay by shifting each contour within the collar. Choose output width e = τ/3, leaving another width e for summing coefficient moduli. In four dimensions the weighted sum is bounded by K[2(1 + e)/e]4, where K bounds the entries of P on the collar. The constants depend on v, A, η and the fixed matrix size, not on the torus period. Discrete Fourier diagonalization gives P T torus = Ptorus and P T torusPtorus = Qtorus on every torus. The invertible discrete Fourier transform transfers the real-axis symbol identity to the real-coordinate matrices. Hence factorization and analytic decay hold simultaneously at all periods, as required for the bottom metric. 5.8 Smoothing histories and the actual block average A block average is an actual map on the fine lattice, with its carry across residue seams included. Its Fourier multiplier therefore has both a principal alias and nonzero aliases. The latter cannot be discarded before the stationary problem is solved. Their lower denominator bounds, proved next, are what make the filled principal formula and all the subsequent contour estimates uniform on the prescribed complex collar. 19 ## Source page 22 A Wilson Yang–Mills construction J. Rodgers Smoothing mass and physical mass The mass of a smoothing train below is a positive integer: the product of its block lengths. It fixes the rescaling in its Fourier multiplier. It is unrelated to the positive regulator masses in the later resolvent calculation, and to a spectral gap above the vacuum. Its useful property is multiplicative growth, which makes the inverse-square contributions of earlier smoothing stages summable independently of the depth of the history. For a positive odd integer M define the line average and its centered version by RM(z) = M−1 M−1 X u=0 exp(iuz), PM(z) = e−i⌊M/2⌋zRM(z). Both filters equal one at z = 0. Summing the geometric series gives MRM(z)(eiz −1) = eiMz −1, including the zeros of eiz −1. Writing an integer in {0, . . . , LM −1} as Mu + v yields RL(Mz)RM(z) = RLM(z). For odd L, M, M⌊L/2⌋+ ⌊M/2⌋= ⌊LM/2⌋, so PL(Mz)PM(z) = PLM(z). Reflection of the centered sum gives PM(−z) = PM(z), and conjuga- tion gives PM(z) = PM(−z). All these identities hold for entire functions. Set AM(p) = Q4 a=1 PM(pa). A train is a finite chain of patches whose smoothing indices stay fixed or decrease by one at each backward step. Its mass is the product of its odd block lengths. A terminal train of index j has mass and smoothing function both equal to one. If v has index J and a patch takes it to index j with j ≤J ≤j + 1 and block L, the new mass is L mass(v) and its smoothing function is smoothnew(p) = AL(mass(v)p)J smoothv(p). Equivalently, smoothv = Aj mass(v) extrav, extranew = AJ−j L mass(v) extrav . This separates the fixed-stage power from the extra smoothing introduced when the stage changes. The extra product is bounded independently of chain length. On | Im pa| < ζ/H, direct estima- tion of the finite exponential sum gives |AM(p)| ≤e4ζM/H. Successive masses increase by factors of at least three, and J −j is zero or one. Induction, equivalently summing the resulting geometric series, gives | extrav(p)| ≤e8ζ mass(v)/H, mass(v) ≤H. On the real principal region, |PM(pa/H)| ≥e−cM/H gives | extrav(p/H)| ≥e−8c mass(v)/H. Consequently | smoothv(p/ mass(v))| has a positive lower bound depending only on j, ζ, c. The normalized finite-sum proof is given in (QF1)–(QF4). At ql = (p + 2π lift(l))/ mass(v), the principal factor Amass(v) has inverse-linear decay in each centered integer alias coordinate, while the extra factor has the preceding upper bound. Hence | smoothv(ql)| ≤Cj Y a (1 + | lift(l)a|)−j. 20 ## Source page 23 A Wilson Yang–Mills construction J. Rodgers The equivalent source normalization uses up(n) = 1 + |n|. This is the decay used in the high-alias sums. The nonzero-alias reciprocal estimate supplies the required bound for RM. These are the actual patch weights. One centered box average of Fourier modes is AL; J independent box rolls give AJ L. Different rolls can have the same total displacement, and the patch weight retains their multiplicities. A positive line path in direction a has L edges at displacements 0, . . . , L −1, with Fourier sum LRL(qa). Thus its directional patch mean has symbol LAL(q)JRL(qa). Applying the defined mean to every discrete Fourier basis vector proves the source identity patch_service. The factor L remains because the path integral counts edges. Wrapped paths on a small torus still count every edge occurrence; no distinct-visit assumption is used. 5.9 Uniform reciprocals at the nonzero aliases A real lower bound is converted to a complex lower bound before inversion. Suppose fξ is holomor- phic with |fξ| ≤Cwξ on a common complex cube of width η, and |fξ| ≥cwξ on its real cube of width θ, where C, c > 0 and wξ > 0. Set r = min(η/4, θ, cη/(8C)). For p in the width-r cube, put y = Re p. The ball of radius η/2 about y stays in the original cube. Cauchy’s estimate along the line from y to p, using |fξ(z)−fξ(y)| ≤2Cwξ on that ball, gives |fξ(p) −fξ(y)| ≤4Cwξ/η|p −y| ≤(c/2)wξ. Thus |fξ(p)| ≥(c/2)wξ. Its reciprocal is holomorphic on this common cube and bounded by 2/(cwξ). The width r is independent of ξ. This is the required reciprocal estimate; real positivity alone would not prove complex nonvanishing. In particular, define λM(p) = X a [Mχ(pa/M)][Mχ(−pa/M)]. On real momenta this is P a |Mχ(pa/M)|2. The bounds for χ give c2 −|p|2 ≤λM(p) ≤c2 +|p|2 on the real cube, uniformly in M. For block L and alias l, put BM,L,l(p) = L2λM (p + 2π lift(l))/L . For l ̸= 0, choose a coordinate attaining grade(l) = 1 + maxa | lift(l)a|. The lower alias estimate gives BM,L,l ≥(c−ζ)2 grade(l)2 on the real cube. The holomorphic upper bound is C grade(l)2. Apply the reciprocal estimate with wl = grade(l)2 to obtain a fixed complex collar and |B−1 M,L,l| ≤ C′ grade(l)−2, uniformly in M, L, l ̸= 0. The principal alias instead satisfies the exact identity BM,L,0 = λLM and is not inverted on a collar containing zero. Define mix(l) = Y a (1 + | lift(l)a|), decayj(l) = mix(l)−j. Since all factors are at least one, grade(l) ≤mix(l). Therefore grade(l)u decayj+u(l) ≤decayj(l). Finally, 21 ## Source page 24 A Wilson Yang–Mills construction J. Rodgers X l mod L decay2(l) ≤  X n∈Z (1 + |n|)−2   4 < ∞, uniformly in L. The centered representatives are injective, so each one-dimensional finite sum is bounded by the full integer series. The finite four-dimensional sum factors into the product of those four finite sums. This proves the decay summation used in the earlier feed and high-alias estimates, without paying a factor equal to the number L4 of residues. For the residue-column estimates, a word u of fine differences contributes at most CL−|u| grade(l)|u|. The scale factor contributes L−e. The residue phase is uniformly bounded because residue coordinates are of size at most L and | Im qa| ≤ζ/L. The symbol contributes decayj(l). Absorption and summation therefore bound the residue sum by C′L−(e+|u|) whenever j −|u| ≥2. Reciprocal shifts permute aliases, giving periodicity; conjugation and reflection give real Fourier coefficients. Fourier estimates on a narrower collar yield the priced fine-difference rows. The coefficient and symmetry arguments below verify these symbol bounds for the actual potential and curvature responses, called pilot and wire in the source. The scalar bounds use χ(z) = eiz −1. Choose c−, c+ > 0 so that c−|z| ≤|χ(z)| ≤c+|z| on |z| ≤3π/2. Such constants exist because χ(z)/z extends continuously with nonzero value i at zero, and has no zero on that disk: eiz = 1 requires a real multiple of 2π. Compactness bounds this nonvanishing quotient above and below. If M|z| ≤3π/2, the geometric-sum identity gives |RM(z)| ≥c−/c+ for z ̸= 0, and RM(0) = 1. Put a0 = min{1, c−/c+}. Centering leaves the modulus unchanged for real z, so |PM(z)| ≥a0. If also H|z| ≤3π/2 and M ≤H, put x = M/H. Estimating each centered monomial minus one gives |PM(z) −1| ≤kx, where k = c+(3π/2). For y = |PM(z)| we have y ≥a0 and 1 ≤y + kx. Since ekx/a0 ≥1 + kx/a0, y exp(kx/a0) ≥y + kx ≥1. It follows that |PM(z)| ≥e−(k/a0)M/H. The coefficient in front of this exponential is exactly one, which is essential when the estimate is composed along arbitrarily long smoothing trains. For a complex alias q = (p + 2πl)/L with centered nonzero integer l, the selected cube gives L|q| ≥ζ up(l) and L| Im q| ≤ζ, where ζ = π/4 and up(l) = 1 + |l|. These follow by separating 2πl from | Re p| ≤π + ζ. Also |q| ≤3π/2 for odd L ≥3. The geometric-sum identity and χ(p + 2πl) = χ(p) give L|RL(q)|c−|q| ≤|χ(p)| ≤c+(3π/2), so |RL(q)| ≤C/ up(l). For l = 0, the finite-sum estimate gives |RL(q)| ≤eζ. Centering changes this by at most eζ, yielding the same inverse-alias bound for PL. Coordinate products give the four- dimensional estimate above. The remaining alias estimates concern the Laplacian denominator and the sum of the decay weights. To install a response kernel, split a fine site into its centered residue r modulo L and a coarse displacement. Advancing by ea changes the residue to r + ea and adds a coarse displacement ea exactly when lift(r)a = ⌊L/2⌋. Denote that added displacement by ca(r). Then lift(r + ea) + Lca(r) = lift(r) + ea. 22 ## Source page 25 A Wilson Yang–Mills construction J. Rodgers In the residue Fourier sum this identity gives a factor exp(ip · ca(r)) on the advanced residue. Subtracting the original residue value is exactly multiplication of each alias term by exp(iqa)-1. Repeating this operation gives a product of those factors for a word of fine differences. Fourier inversion therefore identifies the actual carried differences of the cross row with the separately priced difference symbols. This identity follows from the real sums, rather than treating a residue wrap as an ordinary shift within the same copy. For a fine-difference word u, the symbol estimates give a kernel bound CL−(e+|u|), with a positive width and C uniform over all transitions at fixed smoothing indices, whenever |u| ≤j −2. There are finitely many such direction words. Intersect their widths and enlarge their costs to obtain one table for the row and all required differences. The coefficient estimates and the inverse-third-power estimate below supply the analytic symbol bound; the residue calculation identifies the actual carried differences to which it applies. 5.10 Laurent exterior division Write qs(p) = eips −1 in the lattice Fourier convention, and let δr(p) be exterior multiplication by q on degree-r forms. We divide a cycle without assigning a singular quotient at qs = 0. For a lattice monomial ex, x ∈Z4, let Es replace its s-coordinate by zero. Define Jsex by summing monomials with s-coordinates 0, . . . , xs −1 when xs ≥0, and minus the sum over xs, . . . , −1 when xs < 0, keeping other coordinates fixed. Telescoping gives qs(p)(Jsex)(p) = ex(p) −ex(p|ps=0). If P x |ax|eη|x|1 ≤C, then Es preserves that bound. The sum defining Js has |xs| terms, each of length at most |x|1. For θ = η/2, |xs| exp(θ|x|1) ≤θ−1 exp(η|x|1). Therefore Js maps the width-η bound C to the width-η/2 bound 2C/η. This proves absolute convergence of the transformed rows and licenses termwise telescoping. In particular qsJsA = A −EsA as an identity of real-momentum symbols, including the hyperplane qs = 0. Suppose A maps degree-(r + 1) forms to a fixed coordinate space and Aδr = 0. Let Lr(s) be exterior multiplication by the s-th coordinate vector. Define the relief row As = JsALr+1(s)T. Obtain the residual Rs by evaluating A at ps = 0 and deleting columns whose orientation contains s. Equivalently, Rs = (EsA)[I −Lr(s)Lr(s)T]. The exterior contraction identity Lr+1(s)Tδr+1 = qsI −δrLr(s)T and δr(p) = δr(p|ps=0) + qsLr(s) imply Asδr+1 + Rs = A. First prove this where qs ̸= 0 by multiplying by qs, using qsJsA = A −EsA, and cancelling that nonzero scalar. The symbols are continuous, so perturbing ps through nonzero real values proves the equality also at ps = 0. Furthermore Rsδr = 0 because δr+1δr = 0. Thus the residual remains a cycle and one may repeat the operation. 23 ## Source page 26 A Wilson Yang–Mills construction J. Rodgers Apply the four axes successively to the preceding residual and add the relief rows. Once an orientation contains a processed axis its column is zero and stays zero. In positive degree every orien- tation contains an axis. After four operations the residual is zero and the sum B satisfies Bδr+1 = A. The explicit finite width losses preserve a positive analytic width, so B retains exponential coeffi- cient decay uniformly for a common input bound. Fourier evaluation yields Btorusdr+1 = Atorus at every period. Conversely, Atorusdr = 0 at all periods implies the symbol cycle identity by Fourier uniqueness. That reverse implication uses all periods. Transposing gives the corresponding left lift. A two-sided lift requires a correction because an arbitrary left lift need not remain a right cycle. Suppose A is a two-form row with Ad1 = 0 and dT 1 A = 0 on all periods. Choose W by a left lift so dT 2 W = A. Put F = Wd1; then dT 2 F = 0. Lift again on the left to obtain a degree-four row T with dT 3 T = F. Now Td0 = 0: its left derivative vanishes since dT 3 Td0 = Wd1d0 = 0, and a left cycle in top degree four vanishes by the same lifting theorem, whose next form space in dimension four is zero. Right-lift T to obtain X with Xd1 = T. Define K = W −dT 3 X. Then Kd1 = 0 and dT 2 K = A, using d3d2 = 0. Right-lift K to obtain B with Bd2 = K. Thus dT 2 Bd2 = A. Every step is a finite composition of the preceding analytic lifts, exterior derivatives, sums and transposes, so the same familywise exponential decay statement holds. This proves the exterior factorization used for the fund; the refined block-length power in the sale estimate is additional quantitative information. 5.11 The regular two-form symbols Fix a smoothing train v with scale parameter M. Set za(p) = seedQM(p, a) seedQM(−p, a) and λM(p) = P a za(p). These za are nonnegative at real momentum. The positive smoothing factors are h0 = halo(v) and la = lid(v, a). Let ra = drain(b, v, a) be the regular remainder and put ga = la + λMra, h = h−1 0 − X a z(1) a ra laga . Here z(1) a is the analogous factor for M = 1. The exact identity z(1) a h0 = zala, after taking common denominators, gives λMh = X a z(1) a /ga. (2) The two regular symbols are diagonal on the oriented two-forms. At orientation {a, c}, the symbol called vault has entry hgagc and the symbol called pulse has its reciprocal; off-diagonal entries vanish. At p = 0, every ga and h equals one. These are the values of the same analytic expressions at zero, with no separately substituted harmonic value. The positivity proof uses la > 0 and ra ≥0 on the real cube, so ga ≥la > 0. For p ̸= 0, (2) gives h ≥0. More quantitatively, if ga ≤C and the comparison constants for λM and λ1 with |p|2 are c2 −and c2 +, then (2) gives h ≥c2 − Cc2 + . At p = 0, h = 1, giving the joint lower bound min(1, c2 −/(Cc2 +)). Hence h never vanishes. Pulse and vault are therefore positive, real diagonal mutual inverses throughout the real cube. Uniform bounds and positive lower bounds for ga, h give analytic narrower-collar bounds for their reciprocals 24 ## Source page 27 A Wilson Yang–Mills construction J. Rodgers and their finite products. Taking Fourier coefficients gives the exponentially decaying metric and inverse rows used above. Away from zero, set Da = ga/λM and ρ = P a z(1) a /Da. Equation (2) gives ρ = λ2 Mh. Thus vault = ρ mask2(Da), pulse = ρ−1 mask2(D−1 a ), where mask2 takes the pairwise products of diagonal entries on two-form orientations. Periodicity of Da and ρ proves agreement under reciprocal-lattice shifts on overlapping real cubes. The regular ga, h expressions give the values through the apparent pole. At the terminal train, Da = λ−1 1 and ρ = λ2 1, so both two-form symbols are the identity, as required for the bare Wilson quadratic form. The centered-average, nonzero-alias and cumulative-filter estimates in this chapter provide the smoothing bounds, regular remainder bounds and scale-uniform comparison of λM used in (2). The depth-independent analytic estimate works as follows. Each earlier train contributes with weight mass−2. A hop of block length L multiplies its mass by L, dividing the sum of old weights by L2; the new list adds one train of mass one. Hence, if t(v) is the sum of inverse-square masses over that list, t(top) = 0, t(hop(v)) = 1 + L−2t(v). Since L ≥3, induction gives t(v) ≤2, uniformly in depth. Each train’s actual contribution has analytic bound C mass−2, so the finite list has bound 2C. This controls the function called levelNoise; an unweighted contribution is not accumulated at every renormalization step. At high aliases the smoothing factor has decay indexed by j, while the inverse nonzero-alias divisor costs an inverse squared grade. Their product is dominated by the summable reference decay of order two when j ≥4. Its sum is independent of the alias side length. The same argument gives the inverse-square-mass bound for each contribution. Combining the high-alias sum with b−1 levelNoise bounds drain. The alias lower bounds, inverse-divisor bound and reference decay sum used here were proved above; their scalar assembly is also displayed in the cumulative-filter construction. The factor lid is the product of its smoothing expression at p/M and −p/M. A uniform positive smoothing lower bound, together with the positive lower line-filter bound, makes lid uniformly positive. The same holds for halo. The bound on drain and the bound on λM bound every ga above. Equation (2) then bounds h below. Only halo, lid and ga occur in the rational denominators defining h. Their positive lower bounds permit a common smaller holomorphic collar and analytic bounds for h. No bound for the singular quotient 1/λM at zero is required. 5.12 Amendment and preservation of the potential precision Let H be the unamended two-form metric and B = H−1. Both are symmetric positive matrices on each finite torus. Put X = d2B and, for l ≥0, define A = B + lXTX, G = A−1, R = I + ld2BdT 2 , D = R−1. The source names these rows pledge, amend, bail and treble, respectively. Positivity gives A > 0 and R ≥I, so their analytic inverses exist on uniform narrower strips. Since d2d1 = 0, we have XHd1 = d2BHd1 = 0. Hence AHd1 = d1 and Gd1 = Hd1, so dT 1 Gd1 = dT 1 Hd1. 25 ## Source page 28 A Wilson Yang–Mills construction J. Rodgers The amendment changes the two-form extension while preserving the potential precision exactly. Also RX = d2A; multiply by D on the left and G on the right to obtain XG = Dd2. These are the rebasing identities used above. At zero momentum d2 = 0, and B(0) = I gives G(0) = I. Uniform inversion uses the fixed orientation dimension, rather than the torus site count. If the real symbols satisfy vI ≤A(p) ≤CI, then diagonalization gives | det A(p)| ≥vs in dimension s and C−1I ≤A(p)−1 ≤v−1I. On a common narrower collar the determinant stays separated from zero: analytic coefficient bounds control its change from the real slice, and the collar can be reduced to lose at most half the positive bound. Adjugate divided by determinant gives uniformly bounded holomorphic inverse symbols. Periodicity and real-coefficient symmetry survive. Fourier coefficients decay exponentially, and discrete Fourier evaluation gives the inverse matrices at every period. This proves the inversion needed for G and D. The same Fourier calculation transfers all real-axis spectral inequalities to finite tori. The discrete Fourier matrix has Gram matrix n4I; that positive scalar multiplies both the quadratic form and the squared norm and cancels. Thus a bound uniform in momentum gives the identical bound on every finite torus, including side one. 5.13 Constant-curvature transport For a translation-invariant Laurent row, its total spatial sum is its zero-momentum symbol and is independent of the starting site. For an installed cross row, the starting site also has a residue modulo the block length. The required independence is therefore a separate assertion. For the specified duo response, its zero-momentum entry at each residue z is toll(L, 2) times the identity in face orientations. In the alias formula all nonzero aliases vanish at the origin; the zero alias equals the normalized identity. Consequently the spatial sum of each orientation column is independent of that residue as well. The property survives convolution. If the spatial sums of S, T are the orientation matrices s, t, absolute convergence gives total sum st for their convolution. The fine inverse and coarse metric are translation-invariant rows, and the intermediate curvature response has the constant- residue property just proved. Their sandwich therefore has constant spatial sums. This is the condition called travels_route, required to preserve packet neutrality. The regular curvature- response formulas and harmonic values establish the alias vanishing and normalization used here. 5.14 Exterior factorization of the unamended defect Exterior factorization separates the part of a two-form kernel seen by curls from its complementary directions. The calculation is performed in Laurent symbols before evaluating on a torus, so the same algebraic identity holds at all permitted periods. The zero Fourier mode is then restored by its explicit harmonic equation. This order avoids claiming that division on the punctured momentum domain alone determines the harmonic fibre. From symbols to a finite history The exterior factorization supplies the auxiliary rows needed by the adjacent stationary problem. Those rows are then evaluated at every discrete torus momen- tum. Invertibility of the finite Fourier transform turns the symbol identities into actual matrix identities, including the separately checked zero mode. The subsequent assembly labels such as “Present” and “H.Suffices” record precisely this collection of identities and purchased bounds for a scheduled horizon; they are not invocations of an additional theorem in an unavailable file. On a coarse torus let H1 be the original metric and B1 = H−1 1 its inverse, with fine counterparts H2, B2. For the installed response T and penalty b > 0, define 26 ## Source page 29 A Wilson Yang–Mills construction J. Rodgers V = B1 −T TB2T −b−1d1dT 1 , M = H1V H1. The stationary identities of the transition are TH1d1 = H2e1A0, T Te1 = d1Q and QA0 = I −b−1dT 1 H1d1. Consequently, T TB2TH1d1 = d1[I −b−1dT 1 H1d1], so V H1d1 = 0 and Md1 = 0. Symmetry gives dT 1 M = 0. The two-sided exterior lift therefore gives a three-form row F with dT 2 Fd2 = M on every torus. Replace F by (F +F T)/2; this preserves the factorization and makes its real symbols Hermitian. Multiplying by B1 on both sides gives V = (d2B1)TF(d2B1). This is the fund used in the positive correction. Analytic bounds are retained through the two exterior lifts, their symmetrization and the surrounding bounded rows. The exterior-division construction below supplies the analytic lift and its bounds. The fine curvature charge e2B2TH1 annihilates d1, because TH1d1 = H2e1A0 and e2e1 = 0. A right exterior lift produces S with Sd2 = e2B2TH1. Multiplying by B1 gives Sd2B1 = e2B2T, the rebasing intertwining identity. The refined estimate takes a fine difference before lifting and keeps L−3 per residue; the residue sum then gives the L−2 bound on S∗S used in three-form absorption. The inverse-third-power argument below proves this quantitative bound. The retained-support construction makes stagewise choices before the eventual block lengths. For a legal pair (j, t), where j, t ≥4 and t = j or t + 1 = j, the linear transition estimate supplies a positive coarse-floor threshold δ(j, t). Choose l(i) = max(δ(i, i), δ(i + 1, i)). This pays the coarse threshold for both possible incoming transitions at stage i. With fine and coarse parameters l(j),l(t), choose the positive analytic width, positive coercivity parameter, cost, and integer block threshold supplied by each legal pair. At each stage also choose the positive clip cap and the bottom metric estimates. Take the minimum of the finitely many widths and coercivity parameters for the bottom and two outgoing possibilities, the maximum of their costs, and the maximum of their length thresholds. Weakening an analytic width or lower coercivity bound and increasing an upper cost preserves each estimate. Thus one table works for every allowed adjacent transition. This assembles the common price table from the metric, positive-margin and flight estimates established in this chapter. For a fixed level schedule A, construct the linear trajectory backwards. Start with the top train at level N, and at each coarser level take the landing train of the flight determined by the block and adjacent smoothing indices. Denote the resulting train at i by RN,i. At every pair (N,i), the bottom estimate chooses a metric bed defined by that train and l(J(i)). Choose as its transition tower the flight’s specified duo with the fixed penalty. The identity RN,i = land(destination(N, i)) aligns the bottom metric with this tower. The flight’s tangent equations give the primal and dual stationary identities on each scheduled finite torus; its rebased identity supplies the associated inverse and error data. The collected bottom bounds, transition bounds, and terminal normalization form the required linear horizon. The pinning step retains these metric beds and the indicated tower components while arranging the additional horizon conditions for the given sheet profile. Since retention preserves the beds and the relevant components, their defined train metrics and destination response equalities survive. 27 ## Source page 30 A Wilson Yang–Mills construction J. Rodgers This supplies Present for the pinned horizon as well as the admissibility statement H.Suffices. The bottom metric, flight and pinning estimates used in this assembly are proved in the harmonic, positive-margin and auxiliary-tower arguments in this chapter. 5.15 Harmonic values and finite-torus stationary equations At p = 0 every exterior signal vanishes; the regular factors bulb and bend equal one, and the principal patch multipliers are identities. Thus the filled potential response (pilot) and curvature response (wire) have principal value I. At a nonprincipal zero alias choose a coordinate with nonzero residue. Its line multiplier vanishes because LRL(qa)χ(qa) = χ(0) = 0 and χ(qa) ̸= 0. The box mass filter vanishes, since its smoothing exponent is positive. Every nonprincipal pilot and wire value is zero. The pilot’s leading term contains S = 0 and its correction contains the zero coarse longitudinal row. The wire’s first term contains the zero coarse adjoint derivative, and its second contains the vanishing flux. These explicit values establish the stationary symbol equations on the harmonic fibre; no harmonic projection is deleted. For clarity, the finite Fourier transfer is as follows. Suppose a real matrix A has representation AG = Fa, with Fourier matrices satisfying FF ∗= F ∗F = vI and GG∗= G∗G = uI. Taking adjoints and multiplying by G, F yields ATF = G[(v/u)a∗]. Two real matrices with the same symbol and Fourier matrices agree, since their difference annihilates the invertible matrix G. Products and differences of represented matrices become products and differences of symbols when the intervening Fourier matrices coincide. The coarse Fourier Gram scalar is c4 and the fine one is f4, with f = cL. The transpose of the block mean therefore has alias column L−3V1(ql): the mean carries L and c4/f4 = L−4. The installed A0 and T columns are L−1 pilotl and L−2 wirel. The fine exterior derivative is L−1dL on the scaled alias, and its precision is L−2Kl. Restoring these factors gives the exact finite real-matrix identities QfA0 = meanT Qc, mean A0 = I −b−1Qc, T Tdf = dc mean, THcdc = HfdfA0. The installed cross rows have these symbol representations because their residue Fourier sum is the alias Fourier matrix times the stacked alias column. Evaluation at every coarse discrete fre- quency then gives the represented identities above. Invertibility of the finite Fourier matrices proves the real matrix identities at every period. Together with the exact amended-metric preservation and pinning arguments, this supplies the finite linear horizons used in the flat entrance estimate. The cumulative-filter section gives the scalar covariance feedback formulas and the train identities used in this construction. 5.16 Scaled exterior identities for the stationary columns For scale S, set qS a (p) = S(eipa −1), dS r = qS∧, δS r (p) = dS r (−p)T. On real momentum, δS r is the Hermitian adjoint. The patch identities are Fr+1(ql)dL r (ql) = dr(p)Fr(ql), RL(ql,a)qL a (ql) = qa(p). 28 ## Source page 31 A Wilson Yang–Mills construction J. Rodgers The second proves the first coordinatewise. Transpose at −p to obtain the dual transport identity. Also dS r+1dS r = 0 identically. At regular momentum let D = diag(S−2Da), ID = D−1 and α = S4ρ. The two-form metric is α−1 mask2((ID)aa), with the pairwise-product convention above. The weighted exterior contraction identity makes its potential precision K = ID −α−1IDdS 0 δS 0 ID. Indeed expanding the two-form wedge product yields the diagonal weighted scalar P a qS a (−p)(ID)aaqS a (p) = α; the off-diagonal terms are exactly the rank-one product subtracted above. Thus, with P = α−1δS 0 ID and C = α−1IDdS 0 , K = ID −IDdS 0 P, KD = I −CδS 0 , DK = I −dS 0 P. Also KdS 0 = 0 and δS 0 K = 0 directly from the exterior chain identities. These are the rotor identities used in matching pilot and wire. Write Dl for the fine diagonal covariance at ql, Hl for its two-form pulse and Kl for its potential precision. Use Dc, Hc, Kc for the coarse quantities, Fr, Vr for the patch multiplier and its dual, and Pc for the coarse longitudinal row. Define El = m(−ql) tr(Dl) Z , Z = X l |m(ql)|2 tr(Dl). On the punctured real cube all summands in Z are nonnegative and the principal one is positive. Hence Z ̸= 0 and P l Elm(ql) = 1. Covariance recursion gives X l F1(ql)DlV1(ql) = Dc −b−1I. The regular potential response is Al = DlV1(ql)Kc + EldL 0 (ql)Pc. Multiply by F1(ql) and sum over l. The patch identity turns the second term into P l Elm(ql)d0(p)Pc. The preceding identities and DcKc = I −d0Pc therefore give X l F1(ql)Al = I −b−1Kc. Similarly KlAl = V1(ql)Kc. The longitudinal term is killed by KldL 0 = 0, and the remaining correction in KlDl is killed by δL 0 V1δ1 = 0. This proves the coarse averaging and primal stationary equations on the punctured cube. Set C = Hcd1Dcδ1. The weighted identities give δ1(I −C) = 0 and (I −C)Hcd1 = 0. The regular curvature response is Tl = HldL 1 DlV1δ1 + ylV2(I −C), where yl is the original chain scalar and all fine factors are evaluated at ql. Left multiplication by δL 1 uses KlDlV1δ1 = V1δ1 and δL 1 V2 = V1δ1, giving δL 1 Tl = V1δ1. Right multiplication by Hcd1 kills the complementary term. The potential response’s longitudinal term vanishes under dL 1 dL 0 = 0. Hence 29 ## Source page 32 A Wilson Yang–Mills construction J. Rodgers TlHcd1 = HldL 1 Al. The filled pilot and wire equal these expressions away from the puncture. Their continuity extends the identities to p = 0 once the Fourier and scale normalization factors are restored. The finite Fourier matrices use the coarse volume and fine volume separately when taking adjoints; their ratio is L−4. Accordingly the installed columns use toll L−1 for Al and toll L−2 for Tl, while the adjoint averaging column carries toll L−3. With these factors the equations are exactly the primal, dual and transport equations on every compatible pair of tori. The Fourier transfer in “Harmonic values and finite-torus stationary equations” restores these factors and proves the asserted real matrix identities at every period. 5.17 Extending the three-form positive margin Suppose the analytic Hermitian three-form symbol Q obeys ⟨d2(p)a, Q(p)d2(p)a⟩≥v∥d2(p)a∥2 for every real p, with v > 0. First this controls every vector at zero. For a complex three-form x in four dimensions, the real-linear map n 7→(in)∧x from R4 to the complex one-dimensional top-form space has a nonzero real kernel vector n. The exterior contraction identity gives x = (in) ∧a for some two-form a. Since q(tn)/t →in, the vectors d2(tn)(a/t) tend to x. Apply the image bound and continuity of Q to obtain Q(0) ≥vI. Let C ≥0 be its analytic coefficient bound and N the fixed orientation-count constant. When all |qa(p)| ≤δ, the symbol departure estimate bounds every entry of Q(p)−Q(0) by δC/η. Increase C to a positive value and choose δ = ηv/(2N2C). Then Q(p) ≥(v/2)I in that region. Elsewhere choose k with |qk(p)| > δ. Cartan decomposition gives x = y + z, with y = d2(p) qk(p)−1ιkx , z = qk(p)−1ιkd3(p)x. Thus ∥z∥2 ≤Nδ−2∥d3(p)x∥2. Put M = N2C. The quadratic-form and cross-term bounds imply ⟨x, Qx⟩≥v∥y∥2 −2M∥y∥∥z∥−M∥z∥2 ≥(v/2)∥y∥2 −(M + 2M2/v)∥z∥2. Since ∥x∥2 ≤2∥y∥2 + 2∥z∥2, it follows that ⟨x, Qx⟩+ [M + 2M2/v + v/2]Nδ−2∥d3(p)x∥2 ≥(v/4)∥x∥2. Adding r d∗ 3d3, where r = [M + 2M2/v + v/2]Nδ−2, makes Q uniformly positive. Near zero it adds only a nonnegative form. This correction has finite-support Laurent coefficients and a common analytic bound. It preserves the physical curvature contraction because d3d2 = 0. Enlarging the upper cost therefore gives a uniformly positive full-space row with exactly the original dT 2 Qd2 at every period. This is the positive extension used in pinning. 5.18 The auxiliary tower Gram identity Weighted convolution gives common analytic bounds at width γ = min{σf, σc, ν} > 0, where ν is the three-form-root width. The full Gram row J stacks √clip.direct I with the bottom root row, and is bounded by max{√clip.direct, rootCost}. Each exterior derivative has finite support: a bound 30 ## Source page 33 A Wilson Yang–Mills construction J. Rodgers valid at all periods is the finite sum of eγ|z|1 times absolute stencil coefficients over orientations and sites. Transposition preserves that bound. Composing rows with an intermediate orientation space of size n costs at most n times their entrywise weighted bounds, by the distance triangle inequality and absolute convergence. Apply this first to JR, then to composition with the fine response T, then to multiplication by J′T. The bound for T retains its factor L−2 throughout, so the mixed row costs CfL−2. The normal row s−1d′TJ′T has an L-independent bound, as do I −J′R′J′T and PeR′J′T. Stacking takes their maximum. Choose one upper price covering these and the original input prices. Keep A0, T unchanged; every carried fine difference keeps its L−1−r or L−2−r bound at the reduced width and enlarged price. This proves the estimates named F08; positivity and the Gram identity are separate algebraic statements proved here. On a fixed pair of tori put W = JTJ and W ′ = J′TJ′, the fine and coarse metrics from the full Gram rows. Let R = W −1 and R′ = W ′−1. Retain the inverse response T; let d, d′ be the fine and coarse first exterior derivatives, e the coarse second derivative, and s = √ b. The factorized inverse defect is W ′ZW ′ = eTP TPe, Z = R′ −T TRT −b−1d′d′T. Define Q = J′R′J′T and X = PeR′J′T. Then QT = Q, Q2 = Q and QJ′ = J′. For the mixed, normal and image rows M = JRTJ′T, N = s−1d′TJ′T, O = stack(I −Q, X), direct multiplication gives MTM = J′T TRTJ′T, NTN = J′b−1d′d′TJ′T, OTO = I −Q + J′ZJ′T. Substitution of Z, Q makes their sum I. This holds on the entire source of the stack, including harmonic coordinates. The transport identities TW ′d′ = WdA0, T Td = d′ mean give MJ′d′ = JdA0, M TJd + NT(−s mean) = 0. Also sNT = J′d′, NJ′d′ = s−1d′TW ′d′ and OJ′d′ = 0. The last follows from (I −Q)J′ = 0 and ed′ = 0. These are the exact identities called F09 in the source. They use the same A0, T and defect, without an additional spectral estimate. 5.19 Pinning the auxiliary tower without changing the quadratic history The remaining assembly changes only auxiliary tower rows. Suppose a transition has metric factors Hf, Hc with the stated analytic and positive lower bounds, response rows A0, T with their carried fine-difference bounds, and inverse defect dT 2 Cd2, where C is analytic and positive on im d2. Extend C to a uniformly positive full three-form row without changing dT 2 Cd2, take its analytic square root 31 ## Source page 34 A Wilson Yang–Mills construction J. Rodgers P, construct the auxiliary rows below, and apply their Gram identity. The preceding positive- margin, root and Gram arguments justify each operation. Let Ff, Fc be the full two-form Gram factors and Bf, Bc the inverse metrics. Keep A0, T unchanged. On the finite torus use mixed row FfBfTF T c , normal row b−1/2dT 1 F T c , and the image row obtained by stacking I −FcBcF T c Pd2BcF T c . These formulas are supplied by common cover rows at all torus periods. The analytic auxiliary- row construction gives a positive width and common costs for them, including the required fine differences. Its inputs are fixed analytic estimates, clip profiles, and penalty; its constants are chosen before a particular block length. The exact Gram identity, using the bed floors and defect factorization, proves the required tower roll equations. For each smoothing index first apply the positive extension and root operations, then choose the auxiliary-row prices for each of the two adjacent-stage possibilities. Intersect the finitely many widths with the original width and increase the cost to their maximum. Choosing at each tran- sition the corresponding constructed tower gives a horizon with the identical beds and first two response rows and with the full horizon conditions at those new prices. Terminal normalization, crossing equations and bed bounds are inherited because these components were retained. The positive extension, auxiliary-row estimates and exact Gram identity therefore complete the pinning construction used in the retained horizon. 5.20 Completing the regular coefficient bounds and symmetries All potential- and curvature-response bounds are made on one output momentum cube before choosing L. The mass filter is bounded by C decayj(l) and its principal value has a positive real lower bound from the centered-average estimate. The line-filter factors are uniformly bounded, with positive principal lower bounds. At l ̸= 0, Dl,a and R−1 l have respective bounds C grade(l)−2 and C grade(l)−4, by the common-collar reciprocal estimate and the bulb and bend bounds. The seed expression bounds S uniformly. The sums H, Ta therefore have common analytic bounds. Their two smoothing factors con- tribute decay2j, and each D contributes inverse squared grade. For j ≥4 these are bounded by the uniformly summable decay2. Also U = |m(q0)|2 P a fa + SH is bounded above. Every real summand is nonnegative, and the principal filter and one fa have positive lower bounds. Thus U has an L-independent positive lower bound. The reciprocal estimate bounds U−1 on a narrower collar. These observations bound all factors in the regular correction coefficients of the potential response; every matrix has a fixed face-orientation dimension. For the curvature response, Λ is bounded because its only inverse factors are the principal mass filter and line filters. The other principal factors B, M, y are bounded products and sums of controlled quantities. At l ̸= 0, the first branch has a signal of grade order one, Dl of order grade(l)−2, and a smoothing flux with decayj(l). The second has two smoothing factors, the inverse-fourth-grade bound on Rl, a bounded coarse factor and a further smoothing flux. Both products are bounded by C decayj(l), with constants fixed before L. For real overlap identities, first use the regular stationary expressions. Every covariance Da and its scalar sum ρ is periodic, and exterior signals and patch multipliers have the corresponding reciprocal periodicity. Shifting coarse momentum by 2πx therefore permutes l to l + wrapL(x) in both responses. If x ̸= 0 and both momenta lie in the enlarged real cube, neither is zero, so the 32 ## Source page 35 A Wilson Yang–Mills construction J. Rodgers regular matching applies to both. For x = 0 the identity is immediate. At nonzero momentum, scalar real symbols are even, Fourier rows conjugate at −p, and exterior matrices transform by entrywise conjugation. Substitution gives pilot(p, l) = pilot(−p, −l), wire(p, l) = wire(−p, −l). At zero, the principal formulas have their normalized values and all other branches vanish. Re- flection therefore holds there too. These shift and reflection identities permit the common-collar Fourier export of the residue sums. 5.21 The inverse third power before residue summation The response kernel called buckle is obtained from the symbol vault(wing, ql) wire(p, l) with scale factor L−2. Vault and its overlap identities preserve the common analytic estimates, so one actual fine difference of buckle has coefficient bound CL−3. Its fine curl e2 sums the carried forward differences over four directions and the fixed two-form orientations, with incidence signs of modulus at most one. Hence its bound is 4|Faces2|CL−3. Only orientation counts enter; no residue count is paid at this step. Multiplying the curl by the bounded coarse metric gives the charge kernel with the same L−3 scale. Apply the analytic right exterior lift on the fixed coarse orientation columns; it retains the prefactor, so the lifted kernel called sale is bounded by C′L−3 per fine residue. Its Gram row sums over L4 residues and the fixed fine orientations. The bound is therefore C′′L4(L−3)2 = C′′L−2. This is the inverse-square scale used in three-form absorption. Estimating before taking the fine difference would lose the power; the actual proof takes the fine curl before lifting. 5.22 Filling the stationary predictor at the principal alias Predictor convention and amendment A1. The regular principal coefficient uses the paren- thesized expression below. The extraction placed m(q0)−1 on its first term only. To see the required scope of that multiplier, retain the directional rail in the covariance identity. Put m+ = m(q0), m−= m(−q0) and, for this calculation only, a = | pullLinea(q0)|2fa/(m+m−), F = P c fc, g = m+m−a + ST and U = m+m−F + SH. Thus a is a reduced principal coefficient, rather than an unannounced replacement of the source’s directional multiplier by m+m−. The corresponding weights are E0 = m−F/U, r0 = m−a/g, and E0 −r0 = m−S(FT −aH) Ug = S m+ T g −H U  . In the real even reduced convention m+ = m−= m this is the shorter m2 calculation; on the complex collar the two factors retain the source’s opposite-momentum convention. This is the cancellation needed before extracting Fourier coefficients. The amendment changes the parentheses, and retains all nonprincipal branches, the collar hypotheses and the harmonic restoration. For one transition, use ql = p + 2π lift(l) L , q0 = p/L, S = λmass(land)(p). Put ga = bulb(land, p, a), h = bend(land, p), m(q) = massFilter(q) and fa = bulb(wing, q0, a). The land and wing are the coarse and fine smoothing trains of this transition. For l ̸= 0, define 33 ## Source page 36 A Wilson Yang–Mills construction J. Rodgers Dl,a = bulb(wing, ql, a) blow(mass(wing), p, l). Set Ta = P l̸=0 | pullLinea(ql)|2Dl,a+b−1. On real momentum the notation is an absolute square; on complex momentum it means the product at ql and −ql. The alias covariance recursion gives ga = | pullLinea(q0)|2fa + STa. (3) Put H = P l̸=0 |m(ql)|2 P a Dl,a and U = |m(q0)|2 X a fa + SH. Away from p = 0 this equals S times the original sum of weighted covariance entries. The first term has a positive real lower bound; the other term is nonnegative, so U has a positive lower bound. The earlier alias estimates bound H, Ta and U on a common collar, and the reciprocal lemma bounds U−1 there. Thus U is the regular denominator used in place of the original singular covariance sum. Define the regular correction coefficients G0,a = m(q0)−1(Ta/ga −H/U), Gl,a = m(−ql)[( X c Dl,c)/U −Dl,a|RL(ql,a)|2/ga], l ̸= 0. Every denominator is regular: ga, m(q0), U and the nonzero-alias divisor. The estimates above bound G0 uniformly and Gl by C decayj(l) grade(l)−2 for l ̸= 0. The principal formula follows from (3). For the original normalized covariance weights El and directional rebates rl,a, the identity is El −rl,a = SGl,a. The original longitudinal factor is S−1 times the regular row h−1d0(p)∗diag(g−1 a ). Thus S cancels before Fourier coefficients are extracted. The pilot is its regular transverse leading term plus diag(Gl,a)dL 0 (ql) h−1d0(p)∗diag(g−1 a ). The transverse leading terms are l = 0 : diag(fa) flux1(q0) diag(g−1 a ), l ̸= 0 : S diag(Dl,a) flux1(ql) diag(g−1 a ). Here flux1 is the transpose of the directional one-form patch multiplier, and dL 0 has the specified scaled normalization. The scalar cancellation and these regular formulas supply the value at zero. This is the filled potential response; the original expression with an isolated longitudinal denomi- nator is not separately asserted to extend holomorphically. The potential-response bound follows term by term. Its leading term has decayj(l). Its cor- rection has decayj(l) grade(l)−2 times a signal bounded by a constant times grade(l); every other factor is bounded. Since the grade is at least one, the same decay bounds the correction. Sec- tion 5.24 proves the covariance recursion; Section 5.16 gives the stationary equations; Section 5.15 34 ## Source page 37 A Wilson Yang–Mills construction J. Rodgers restores the finite-torus normalizations; and Section 5.20 gives reciprocal-shift and reflection com- patibility. Together these prove the analytic predictor and its actual stationary response, including the harmonic fibre. 5.23 The regular dual predictor on two-forms Continue the filled-potential-response notation. Let Hf(q) and Hc(p) be the fine and coarse pulse metrics, d = d1(p) and δ = dL 1 (q0). The adjoint d∗on real momentum is continued analytically using −p. Write Fr(q) = hingesr(q) for the patch multiplier on r-forms, and Vr(q) = fluxr(q) for its transposed symbol at −q. Define Λ = m(q0)−1 mask2 RL(q0,a)−1, y = |m(q0)|2hc/hf(q0), xa = Ta/ga, B = Hf(q0) diagpairs(xa + xc) −S mask2(x) , M = diag(ga). Here diagpairs sums the two coordinate entries in each two-form orientation. The principal wire is the regular expression W0 = yV2(q0) + BΛdM −Hf(q0)Λd diag(Ta) d∗. For a vector x = (xa), let mask2(x) have diagonal entries xaxc at orientation {a, c}, and let diagpairs(xa + xc) have entries xa + xc. The entrywise identity I −mask2(1 −Sx) = S diagpairs(xa + xc) −S mask2(x)  and covariance identity (3) prove SB = Hf(q0) −yV2(q0)Hc(p)F2(q0), by substitution in each two-form diagonal entry. Together with F2(q0)Λ = I and F2(q0)δ = dF1(q0), this identifies W0 with the original stationary dual expression for p ̸= 0. Every factor of the displayed regular expression is uniformly analytic, so it extends the response through zero. For l ̸= 0, set Rl = blow2 l hf(ql) and yl = |m(ql)|2hc/Rl, and put N = S2I −SHc(p)dMd∗. The remaining wire branches are Wl = Hf(ql)dL 1 (ql) diag(Dl,a)V1(ql)d∗ +ylV2(ql)N. These are the nonprincipal dual branches, called stake in the source. The identities portD = scaleD, portR = scaleR and N = S2(I −cloud) on the regular set identify them with the original dual response. Their nonzero-alias divisor is separated, so no principal denominator is inverted. The lower bound gives R−1 l = O(grade(l)−4) and Dl = O(grade(l)−2) with the common analytic bounds. Signal and smoothing-flux factors then give C decayj(l), as for the bounded principal branch. Section 5.20 gives the reciprocal-shift and reflection identities needed for Fourier export. The algebra here identifies the regular symbols themselves. 35 ## Source page 38 A Wilson Yang–Mills construction J. Rodgers 5.24 Cumulative filters and propagation of the quadratic covariance The cumulative-filter calculation closes the quadratic argument. Each one-step identity is composed along the same scale history, and the weighted alias sums remain absolutely convergent. The resulting covariance is therefore attached to the constructed history, rather than to an independently chosen continuum kernel. These identities are later used as the fixed linear comparison in the nonlinear predictor and in the free hierarchy. A finite linear history begins at smoothing order j. It is either empty, or consists of a patch with odd block side L ≥3, new order J with j ≤J ≤j + 1, followed by a history beginning at J. Let Mv be the product of its block sides, with M∅= 1. For the centered one-dimensional box filter pL, write AL(p) = 3 Y ν=0 pL(pν). Its exact composition identity for odd sides is ALM(p) = AL(Mp)AM(p). Define an extra filter recursively by E∅= 1 and E(L,J;v)(p) = ALMv(p)J−jEv(p), Sv(p) = AMv(p)jEv(p). The composition identity gives S(L,J;v)(p) = AL(Mvp)JSv(p). (QF1) The exponent J −j is zero or one. Thus extra smoothing factors enter at the successive cumulative lengths, while the principal filter has exactly the original exponent j. The filters are entire, periodic under reciprocal integer translations, invariant under signed coordinate permutations, equal to one at zero, and satisfy Sv(p) = Sv(−p). These statements follow first for the finite box sums, then for AL, Ev, Sv by finite products and induction. There is a bound independent of both the number of steps and the total dilation. On the strip | Im pν| ≤z/L0, a box filter satisfies |AM(p)| ≤e4zM/L0. When M ≤L0, at real principal momenta p/L0 in the fixed domain one also has |AM(p/L0)| ≥e−4cboxM/L0. For a history of total mass Mv ≤L0, induction using each block side at least three gives |Ev(p)| ≤e8zMv/L0, |Ev(p/L0)| ≥e−8cboxMv/L0. (QF2) To check the induction, a new extra factor costs 4zMv/L0 in the upper exponent, and the preceding bound costs at most 8zMv/(3L0). Their sum is at most 8zMv/L0. If no extra factor occurs, monotonicity of the mass suffices. The lower bound has exactly the same calculation with negative exponents and cbox in place of z. At an alias indexed by l, the principal box estimate is |AM(aliasl(p))| ≤B4 mix(l)−1. 36 ## Source page 39 A Wilson Yang–Mills construction J. Rodgers Combining this with (QF2), with L0 = Mv, proves |Sv(aliasl(p))| ≤B4je8z mix(l)−j. (QF3) At the real principal alias the lower bound is |Sv(p/Mv)| ≥e−4jcbox−8cbox > 0. (QF4) These bounds are uniform over all finite histories, with constants depending on the initial order only. In particular all alias-filter functions have a common holomorphic neighborhood and the indicated uniform alias decay. Include now the longitudinal edge box filter rM(pa), and set Dv,a(p) = Sv(p)rMv(pa), PL,J,a(p) = AL(p)JrL(pa), Y (f)(p) = f(p)f(−p). The exact rail composition and (QF1) imply D(L,J;v),a(p) = PL,J,a(Mvp)Dv,a(p). (QF5) On real momenta Dv,a(−p) = Dv,a(p), so Y (Dv,a) = |Dv,a|2 ≥0. For any scalar symbol f, define its propagated alias sum by Fv,af(p) = M−2 v X l∈(Z/MvZ)4 Y (Dv,a)(aliasl(p))f(aliasl(p)). (QF6) The normalization is M−2 v . Define LL,J,a by the same formula for one patch. The radix bijection between aliases of LMv and pairs of aliases of L and Mv, together with (QF5), gives the exact identity F(L,J;v),af = LL,J,a(Fv,af). (QF7) Indeed substitute the nested alias formula in the sum, factor out the outer Y (PL,J,a) by periodicity, and use (LMv)−2 = L−2M−2 v . The empty history has F∅,af = f. Every renormalization step adds an independent local covariance k−1. Let Nv be the sum of the covariances of these noises propagated through their respective prefixes. More explicitly the empty history has no prefixes; extending a history introduces the empty prefix and extends each old prefix by the new patch. Thus N∅= 0, N(L,J;v),a = 1 + LL,J,aNv,a. If λ(p) is the initial Wilson quadratic divisor, the cumulative diagonal covariance is Dv,a = Fv,a(λ−1) + k−1Nv,a. Linearity and (QF7) prove the feedback equation D(L,J;v),a = LL,J,aDv,a + k−1. (QF8) This formula accounts for every earlier noise; retaining only the final noise would give a different covariance. All sums are finite. Real nonnegativity follows term by term when k > 0, and re- ciprocal periodicity, evenness and covariance under signed coordinate permutations follow by the corresponding permutation of aliases. 37 ## Source page 40 A Wilson Yang–Mills construction J. Rodgers Separating the principal alias in (QF6) yields the pole decomposition Dv,a(p) = Y (Dv,a)(p/Mv) λMv(p) + Hv,a(p) + k−1Nv,a(p), (QF9) where λMv includes the M2 v scaling of the principal divisor and Hv,a(p) = X l̸=0 Y (Dv,a)(aliasl(p)) M2v λ(aliasl(p)) . This is an algebraic identity on the regular set of the denominators. The auxiliary divided symbols are not asserted to be continuous extensions at harmonic momenta. Their regular parts and their harmonic reconstruction are handled by the pole-removal construction already given above. Both H and the propagated noise are nonnegative on real regular momenta, and (QF4) supplies the uniform nonvanishing principal numerator required there. The one-dimensional filters in this construction are literal normalized finite sums: rM(z) = 1 M M−1 X u=0 eiuz, pM(z) = 1 M M−1 X u=0 ei(u−⌊M/2⌋)z. Partitioning the indices 0, . . . , LM −1 as Mu + v proves rL(Mz)rM(z) = rLM(z). For odd sides the centers satisfy M⌊L/2⌋+ ⌊M/2⌋= ⌊LM/2⌋, proving the corresponding identity for p. The principal lower bound used in (QF2) has coefficient one in front of its exponential, which is essential for arbitrarily long histories. To derive it, choose a disk on which c−|z| ≤|eiz −1| ≤c+|z|. Such positive constants exist because the ratio extends continuously with nonzero value at zero, and the disk excludes the other zeros. The geometric-sum identity MrM(z)(eiz −1) = eiMz −1 gives |pM(z)| = |rM(z)| ≥c0 for real z with M|z| in that disk, where c0 = min(1, c−/c+) > 0. If M ≤L0 and L0|z| ≤R0, the centered finite sum also gives |pM(z) −1| ≤c+R0 M/L0. Put x = M/L0, k0 = c+R0, and u = |pM(z)|. Then u ≥c0 and u + k0x ≥1. Since ey ≥1 + y, uek0x/c0 ≥u + uk0x/c0 ≥u + k0x ≥1. Thus u ≥e−(k0/c0)x. This proves the stated principal estimate with cbox = k0/c0, without multiply- ing a fixed prefactor less than one at every step. The complex-strip upper bound follows simply by bounding every exponential in the centered average by eM| Im z|. 6 Local nonlinear geometry and estimates The nonlinear argument replaces a flat Lie-algebra field by actual group-valued links while control- ling the difference from the quadratic model. It first constructs the chart and its Haar density, then obtains local inverse and predictor estimates, and finally pays for the exceptional configurations through packet and support budgets. Each estimate records whether it is a pointwise bound, a derivative bound on a healthy domain, or a summable bound over supports. These three forms of control have different uses in the primitive integration and cannot be substituted for one another. 38 ## Source page 41 A Wilson Yang–Mills construction J. Rodgers 6.1 Constructing the exponential chart and its Haar ratio The chart is constructed from the closed matrix image of the compact gauge group. Its real Lie algebra consists of matrices X such that exp(tX) belongs to that image for every real t. Let N be the orthogonal complement of this Lie algebra in the ambient real matrix space. First suppose Xj ̸= 0, Xj →0, exp(Xj) ∈G, and Xj/∥Xj∥→u. For t ≥0, put kj = ⌊t/∥Xj∥⌋. Then kj∥Xj∥→t and exp(kjXj) = (exp Xj)kj ∈G, exp(kjXj) −→exp(tu). Closedness gives exp(tu) ∈G; negative t follow by inversion. Thus u belongs to the Lie algebra. Consequently, in some neighborhood of zero in N, the condition exp(n) ∈G forces n = 0. Otherwise choose nonzero such nj →0 and extract a convergent subsequence of their unit directions. Its limit has norm one, belongs to N, and by the preceding argument belongs to the Lie algebra, a contradiction. Identify the color space linearly with the Lie algebra by C, and write expG(a) = exp(C(a)). The map (a, n) 7−→exp(C(a)) exp(n) has derivative (a, n) 7→C(a) + n at zero, an isomorphism by the orthogonal decomposition. Its local smooth inverse has a color component logG. In particular logG(exp(C(a))) = a for small a. If a nearby g ∈G has inverse coordinates (a, n), then exp(n) = exp(−C(a))g ∈G. The isolation just proved gives n = 0, so exp(C(logG g)) = g. Shrinking the color ball, its exponential image is open in G: continuity of the inverse sends a group neighborhood of each image point back into the ball. No global logarithm or connectedness assumption is used. For u ∈G, define Fu(a) = logG(u exp(C(a))) where the translated point lies in the chart. Put A(a) = DFexp(−C(a))(a), J(a) = det A(a). Smoothness of the ambient inverse makes J smooth, and F1 is the identity germ, so J(0) = 1. If b = Fu(a), the identity of germs near a Fexp(−C(a)) = Fexp(−C(b)) ◦Fu gives, by differentiation and determinants, J(b) det DFu(a) = J(a). Shrink the ball S so 1/2 < J < 2 there and put U = expG(S). Push 1SJ(a) da forward by expG, obtaining a finite nonzero measure v supported on U. For measurable A ⊆U, the change-of- variables formula and the overlap law give v(u−1A) = v(A ∩uU). Indeed Fu is injective on the relevant chart pieces, and positivity of J makes its density identity | det DFu(a)|J(Fu(a)) = J(a). Average over normalized Haar measure. Tonelli and right invariance make the left integral v(G)μG(A). For fixed g ∈A, the set of u with g ∈uU has measure μG(U), by inversion and right translation, so the right integral is μG(U)v(A). Both constants are positive and finite: U is nonempty and open, and J is bounded above and below on the finite-radius ball. Hence c = μG(U)/v(G) > 0 satisfies (expG)∗(1ScJ(a) da) = μG|U. Intersecting an arbitrary measurable test set with U proves the full local measure identity. This gives the required normalization, positivity, smoothness and value J(0) = 1. 39 ## Source page 42 A Wilson Yang–Mills construction J. Rodgers 6.2 Local Haar charts in the primitive integral Active and dormant coordinates; amendment A2. The chart integral contains rh active real coordinates and dv dormant Gaussian coordinates. Integration of the latter gives one, including when dv = 0; it does not authorize dropping their normalization. Put K = √ 2π and mg = c(Kg)r. The f −h frozen free links cancel as mf ge−(f−h) log mg = mh g. Consequently K−(rh+dv)mh g = K−(rh+dv)chKrhgrh = K−dvchgrh. The extracted exponent −rh + dv would instead produce Kdvchgrh, an extra factor K2dv. We use the reconciled exponent −(rh + dv) below. The determinant of the reference form remains a separate factor. The local change of variables in the primitive expansion uses the chart ratio of the same compact gauge group. If Bρ is its chart ball and the local Haar ratio is cJ(a), then a 7−→expG(a)V sends 1Bρ(a)cJ(a) da to Haar measure restricted to expG(Bρ)V , for every fixed V ∈G. This follows by the defining chart measure identity for the exponential followed by right Haar invariance. On the finite set of active links, take the product of these maps. Product measure and multiplication of densities give the exact joint density Q i cJ(ai) on the product chart ball, with the product of the restricted Haar measures as its image. The inactive links are held fixed. The active chart is sufficient because the sector selectors force every active link of a nonzero integrand to lie in that chart relative to its replica background. More precisely those links have the form expG(ai)Vi with |ai| ≤a0t inside the chart ball. Consequently restricting integration to the product chart does not discard a nonzero contribution. The transformed integrand’s powers of c and its logarithmic Jacobian terms come from this product density. Before this change, the tree-gauge configuration measure decomposes into active-link and exterior-link coordinates by a measure-preserving splitting. Fubini permits integration first over the active coordinates and then the exterior coordinates, even after the team’s bounded local ingredients are inserted. Regrouping the integrand after this substitution produces its quadratic Gaussian density and the remaining explicit vertex factors. The chart identity and product-measure operations are exact. The local Haar-chart construction, grafted-signal argument and Gaussian exponent rearrangement in this chapter supply the chart ratio, selector implication and vertex decomposition used here. The selector-to-chart implication has the following numerical form. On an active link let z = Ui V −1 i , with Vi its replica background. The actual sector and tree-slice conditions give chord(z) ≤Kt, where K is the selected shell width and t the small-field scale. The local logarithm estimate gives ∥logG z∥≤γ chord(z) and expG(logG z) = z on the wider shell chord(z) ≤2Kt. Choose a with γK ≤a and at < ρ, the chart radius. Then v = logG z satisfies ∥v∥≤at < ρ and Ui = expG(v)Vi. This is the chart representation of every active link of a nonzero sector integrand. Its substantive geometric input is the shell-access estimate deriving chord(z) ≤Kt from the actual sector conditions. The inserted cutofffunctions can be handled at all color arguments. Write χa(v) for the inner- share cutoffat radius at. It lies between zero and one, equals one on ∥v∥≤at, and can be nonzero only on ∥v∥≤2at. Therefore χ2a = 1 wherever χa is nonzero. Define the extended Jacobian logarithm by ga(v) = χ2a(v) log J(v). On the support of χa, positivity of J in the chart gives exp(ga(v)) = J(v), provided 2at < ρ. This identity yields the exact product Jacobian in 40 ## Source page 43 A Wilson Yang–Mills construction J. Rodgers the integral. Outside the chart the cutoffmakes the transformed integrand vanish, so no exterior logarithm value is used in the change of variables. Since J(0) = 1 and J is smooth and positive near zero, log J is locally Lipschitz with | log J(v)| ≤ C∥v∥. Wherever χ2a is nonzero, ∥v∥≤4at. Hence |ga(v)| ≤4Cat for all v, after making 4at smaller than the fixed local radius. The zero-cutoffcase is immediate. This proves the uniform small Jacobian-log bound used in the primitive vertex envelopes. The later rescaling from active chart variables to fluctuation variables is likewise exact. Let g > 0 be the coupling scale, p the real number of active coordinates, and separate fluctuation coordinates into active and vacant parts. The active affine dilation used in the source sends gp times Lebesgue measure to active-chart Lebesgue measure. Its coordinate reassembly is a permutation and preserves measure. The integrand depends on the active part through the original chart expression; the vacant part contributes a separate normalized Gaussian comfort factor with integral one. Thus, in these coordinates, its integral factors as Z Y (x)gpC(y)dxdy = gp Z Y (x)dx, Z C(y)dy = 1. The dormant quadratic energy identity gives precisely this factorization of the transformed ribbons, cutoffand Jacobian terms. Fubini is licensed by the primitive integrable envelopes. This accounts for the active-coordinate Jacobian and the unused Gaussian coordinates in the exact fluctuation integral, rather than discarding them by a dimension convention. The affine dilation, dormant energy and normalization are derived next; the packet and non-packet estimates below supply the vertex envelopes. The coordinate map in that rescaling is simply affine dilation followed by reindexing. Index active real coordinates by pairs (active link, color coordinate), let p be their cardinality, and put D(x) = μ+gx. Its derivative is g times the identity, so det(DxD) = gp > 0. The finite-dimensional change-of-variables formula says D pushes gp dx to Lebesgue measure. Reindex the pairs into one color vector per active link; this is a coordinate permutation followed by regrouping product coordinates and preserves Lebesgue measure on measurable rectangles, hence on all measurable sets. The active coordinate of the original lifted fluctuation μ + gz is exactly this D(x). The configuration, active cutoffproduct and Jacobian-log sum depend only on these active coordinates. On the vacant coordinates the source’s dormant matrix is the identity and on active coordinates it is zero. Its quadratic energy is therefore one half the sum of the squared vacant real coordinates. If dv is their number, its comfort density is C(y) = ( √ 2π)−dv exp(−∥y∥2/2). The product of the one-dimensional Gaussian integrals gives R C = 1, also for dv = 0. This proves the dormant factor and its normalization used in the previous integral identity. The remaining volume constants can be checked by actual finite set partitions. If r is the color dimension, h is the number of active free links, and f is the total number of free tree-gauge links, then p = rh, the number of dormant coordinates is dv, and the ambient fluctuation dimension is p + dv. The number of frozen free links is f-h. Define mg = c( √ 2π g)r, where c is the Haar chart scale. The chart/rescaling prefactor satisfies the exact identity ( √ 2π)−dvgrhch = ( √ 2π)−(rh+dv) mf g exp(−(f −h) log mg). 41 ## Source page 44 A Wilson Yang–Mills construction J. Rodgers It follows by cancelling mf−h g and substituting its definition. The determinant of the reference blank quadratic form contributes the separate factor exp(−1 2 log det Qblank); it is not included in a dimension count. Together these are the deducted Gaussian normalization used by the local integral. Its reference covariance has density ( √ 2π)−d exp(−log det Q/2 −zTQ−1z/2) in ambient dimension d, with positive definite Q. Thus all determinant and Gaussian-volume factors are explicit before the remainder vertices are estimated. There is also a useful exact cancellation in the raw density before this transformation. Splitting the action into a retained crown and its reduced part gives β cutR(U) + ER(U) = vR|Λ| −reducedAction(U) −additiveActivity(U). The block-kernel correction has whole gap cutR(U)-coarseCrown(V ). Its exponential cancels the retained crown already moved into the raw density, since exp(−β coarseCrown) exp(β cutR) exp(−β(cutR −coarseCrown)) = 1. Consequently the sector sum of the raw marked density equals the tagged block kernel times the original marked density pointwise. A summable envelope for the original activity gas and bounded sector factors licenses integration and summation. This is the initial action cancellation in the primitive Mark formula. The local vertex decomposition and primitive-integral estimates in the marked-integration chapter complete this formula. 6.3 The quadratic floor before choosing the local covariance The reference precision is a Gram matrix with an identity completion on dormant coordinates. Let L u be the active part of a link direction, let D u be the physical stacked motion, and let Pd be the diagonal projection onto dormant real coordinates. The plate is Q = F TF + Pd, where F is the scaled motion matrix. The scale σ satisfies uTF TFu = σ2 X |Du|2. To prove a volume-uniform positive floor, the source first obtains at each link the local inequality |Lui| ≤ZRi(u) + MCi(u). Here Ri is the square root of the sum of squared physical motions within a fixed native radius d of i, while Ci is the square root of the sum of squared active link directions within radius 3B. Their definitions use actual torus max-distance and the actual link census. Summing squared inequalities and charging how often any term occurs gives W ≤2Z2NdV + 2M2N3BW, where W = P |L ui|2, V = P |D u|2 and Nr = (2r + 1)4 Lblock4 |Axis|. The local geometric driving estimate supplies Z fixed at the chosen block and stage, while M has reference-coupling order −1/2 up to the displayed width factors. Since N3B has only polynomial width growth, eventually 2M2N3B ≤1/2. Absorption gives W ≤4Z2Nd V. Enlarge the constant to a > 0 if needed. 42 ## Source page 45 A Wilson Yang–Mills construction J. Rodgers Every real direction splits orthogonally into its active and dormant components: ∥u∥2 = ∥Lu∥2 + uTPdu, uTPdu ≥0. Taking k = min(σ2/a, 1) > 0 proves uTQu ≥k∥u∥2. This floor uses the physical motion estimate, not an assumed lower bound for the full matrix. The local driving and flattening arguments are developed explicitly in (NF1)–(NF4). The localized inverse-column construction also treats dormant coordinates exactly. The plate has identity row and column on each dormant coordinate, and its linear source there is zero. The same is true in every principal restriction. Its inverse therefore has identity row and column at those coordinates, by multiplying the inverse equation by that row. Padding a chosen principal inverse column into the full space preserves this property; offits footprint it is zero, and the diagonal belongs to the footprint. Symmetrizing these padded columns makes the reference covariance identity on the dormant rows and columns. The mean, defined as minus the transpose of the padded inverse times the linear source, is zero on dormant coordinates. These identities justify the dormant Gaussian factor used earlier. Positivity and the uniform upper bound of the full symmetrized padded covariance additionally require the localized inverse-window estimates. 6.4 Exponentially local inverses and a positive finite-window covariance The covariance has its own construction An exponentially decaying inverse and a positive covariance on a finite window are separate objects. Truncating an inverse kernel need not preserve positivity. The argument below therefore constructs the window covariance with a positive margin and retains the comparison error in the norm used by the Gaussian integral. The later coordinates, mean and determinant refer to this particular covariance. They do not permit substitution of an arbitrary matrix with similar entrywise decay. Let A be a real symmetric matrix indexed by coordinates placed at sites ai. Suppose A ≥rI, r > 0, and its weighted absolute row and column sums at width η ≥0 are at most X > 0. All principal restrictions inherit these statements: extend a vector by zero for the quadratic floor, and restrict the nonnegative row sums for the price. Choose 0 ≤θ ≤1 with θX ≤r/2. For u ≥0 put Au = A + uI. It is positive definite and has an actual two-sided inverse. Fix a column j, put si = exp(θη dist(ai, aj)), and set y = A−1 u ej, wi = si yi. Conjugation by the diagonal s gives (Au + E)w = ej, Eik = (Au)ik(si/sk −1). The triangle inequality for distance gives |si/sk −1| ≤θ exp(η dist(ai, ak)). For completeness, the logarithm of si/sk lies between −θD and θD, where D = η dist(ai, ak) ≥0. Convexity gives eθD −1 ≤θ(eD −1) ≤θeD. Also 1 −ev ≤e−v −1 when v ≤0. These two bounds prove the displayed inequality. The diagonal contribution of uI vanishes from E, so its absolute row and column sums are at most θX ≤r/2. The elementary Schur quadratic estimate is |vTEv| ≤(r/2)∥v∥2: expand the double sum and use 2|vivk| ≤v2 i + v2 k, then apply the two sum bounds. Taking the inner product of the conjugated equation with w yields 43 ## Source page 46 A Wilson Yang–Mills construction J. Rodgers (r/2 + u)∥w∥2 ≤wj. With c = min(1, r/2), this implies c(1 + u)∥w∥2 ≤wj ≤∥w∥, hence ∥w∥≤[c(1 + u)]−1; the zero-vector case is immediate. Therefore exp(θη dist(ai, aj))|(A + uI)−1 ij | ≤[min(1, r/2)(1 + u)]−1. This is uniform in matrix dimension and in all principal restrictions. For the positive-width conclusion now take η > 0, set θ = min(1, r/(2X)), b = θη and k = b/3 > 0. The preceding entrywise estimate also holds when η = 0; the following exponential site sum uses k > 0. If at most m coordinate labels occupy each site, the entrywise inverse estimate converts to row and column estimates at width 2k. Indeed the sum of exp(−(b −2k) dist) over torus sites is bounded by the corresponding absolutely convergent lattice sum, independently of the period. Multiply that sum by m[min(1, r/2)]−1; call the resulting price P. Thus both the full inverse and every zero-padded principal inverse column have price at most P at width 2k. For each column j choose a principal window Wj containing all coordinates within distance B of aj and contained within distance B+2. Its padded inverse column is Kij =    (A|Wj)−1 ij, i ∈Wj, 0, i /∈Wj. The error E = AK −I vanishes exactly when i ∈Wj, by the principal inverse equation. In particular it vanishes for dist(ai, aj) ≤B. Product price bounds at width 2k give price(E) ≤XP +1 there. On its remaining entries, losing one width unit k supplies exp(−kB), so at width k its price is at most εB = exp(−kB)(XP + 1). The resolvent identity K −A−1 = A−1E then gives width-k price at most PεB. These are bounds on actual matrices, not a declaration that the padded inverse is the full inverse. Symmetrize C = (K + KT)/2. Since A−1 is symmetric, averaging the two error bounds gives price(C −A−1) ≤PεB, and price(C) ≤P. The Schur bound turns the error price into |vT(C − A−1)v| ≤PεB∥v∥2. Also A ≤XI implies A−1 ≥X−1I. For example, put w = A−1v and use nonnegativity of (w −X−1v)TA(w −X−1v), together with A ≤XI. Expanding and using Aw = v gives 0 ≤(w −X−1v)TA(w −X−1v) ≤vTA−1v −X−1∥v∥2, which proves vTA−1v ≥X−1∥v∥2. Consequently, once PεB ≤1/(2X), (2X)−1I ≤C, |vTCv| ≤P∥v∥2. Thus C is the positive covariance required for the reference Gaussian, with the uniform upper bound M = P used in the warning estimate. Its entries vanish beyond distance B+2. The residual AC−I has width-k price at most XPεB, by multiplying A(C −A−1). All these choices are uniform in the torus volume and preserve the original endpoint-defined windows. For the physical plate, the quadratic floor proved above supplies r, while the stencil bounds supply its width-η row price X and the coordinate census supplies m. At fixed block and stage, increasing B makes the exponential error small enough to meet the displayed requirement. The physical operator bounds and local driving estimate of this chapter establish these inputs for the plate. 44 ## Source page 47 A Wilson Yang–Mills construction J. Rodgers 6.5 Axial gauge and bounded logarithmic extension Axial gauge is used locally to make a controlled logarithm available on a clean hull. Its chosen tree is an auxiliary coordinate device. The proof must therefore bound the resulting logarithm and preserve the observables when it extends links outside the hull. Subsequent comparison estimates depend on the size of the fixed inspected hull, not on a simultaneous global gauge choice for every query. For a finite subset X of the integer cover, let eX(U) be the sum of the chords of its internal positive links and fX(U) the sum of the chords of its wholly internal elementary plaquettes. Bound- ary links and incomplete plaquettes contribute zero. Both sums are nonnegative, and fX is gauge invariant. An ordered list l of distinct coordinate axes defines a slab: coordinates in l range over the integers −r,. . . ,r, while the other coordinates are pinned at o. From a point p in the slab, travel first in the first listed axis to the corresponding coordinate of x, then in the next axis, and so on. Let Hp(x) be the ordered product of links along this route. Its endpoint is x because the remaining coordinates are pinned. For l followed by a new axis j, the gauge Hp makes every internal j-link equal to 1. Indeed the routes to x and x+ej have the same initial part; their final straight j-segments differ by precisely the link (x,j), so the gauge-transformed link cancels. When these j-links equal 1, the plaquette relation gives | chord U(x + ej, i) −chord U(x, i)| ≤chord holonomy(square(i, j), x). This follows from the triangle inequality applied to a b−1 and its inverse, where a,b are the two horizontal links. For each site the right side is at most the sum of its internal plaquette chords; reversing the ordering of the two axes only inverts the holonomy and leaves its chord unchanged. Summing over the four positive axes and over a slice gives a bound of four times the plaquette sum in the adjacent layer. Telescoping between two slices therefore yields |eslice(z)U −eslice(u)U| ≤4fslab(U). All layers used by the telescoping sum lie inside the slab; adding the unused nonnegative layers proves this bound even when the two slices are far apart. We prove by induction on k = length(l) that some route root p satisfies eslab(gauge HpU) ≤4k(2r + 1)fslab(U). For k = 0 the slab is a single site with no internal links. For the induction step choose, among the 2r+1 slices normal to the new axis, one whose internal face sum is minimal. Its face sum is at most fslab(U)/(2r + 1), because the slice face sets are disjoint subsets of the full slab face set. Use the induction hypothesis to choose a root on that slice, then extend its routes in the new axis. The edge sum on the chosen slice is unchanged, the new-axis links are 1, and the preceding slice comparison bounds every other slice by the chosen slice edge sum plus 4 fslab(U). Summing all 2r+1 slices gives the required bound with k+1. This also proves existence of the root; no curvature smallness was required. For all four axes the slab is the box [−r, r]4, so the bound is 16(2r+1) fbox(U). If every plaquette chord in a periodic radius-r patch is at most q, then fbox ≤(2r +1)4 |PlaqAxes|q. Lift the patch to that box and apply the preceding construction. When 2r < n, the lift is injective on the box, so Hp descends to a gauge on the corresponding torus sites, extended arbitrarily elsewhere. If d + 1 ≤r, 45 ## Source page 48 A Wilson Yang–Mills construction J. Rodgers both endpoints of every radius-d bond lie in the box. Each such link chord is at most the full nonnegative edge sum, giving chord(gauge zU)b ≤16(2r + 1)5|PlaqAxes|q. The logarithmic extension used after this gauge choice has an especially simple construction. Analyticity of the local logarithm at the identity, log(1) = 0, gives ∥log M∥≤D∥M −1∥on a small matrix ball. Shrink the ball so that exp(log U) = U for group elements in it. Since ∥mat(U) −1∥= colorScale · chord(U), there are C, δ > 0 with exp(log U) = U and ∥log U∥≤C chord(U) whenever chord(U) ≤δ. Define the logarithmic field linkwise by log U on the prescribed interior patch and by zero outside it. This field is globally bounded by Ch and exponentiates to U on that patch whenever its links have chords at most h ≤δ. No continuity across the boundary of the finite patch is needed. We will also use two consequences of the same analytic Taylor argument. For an analytic map f at 0 with directional derivative L, the analytic series identifies L with the bounded derivative Df(0). Its first two terms and the uniform series remainder give ∥La∥≤K∥a∥, ∥f(a) −f(0) −La∥≤K∥a∥2 on one fixed small ball. Applied to the matrix exponential this bounds chord(exp a) by K∥a∥. Applied to the curvature of exp a0 exp a1 exp(−a2) exp(−a3), its derivative is a0 + a1 −a2 −a3. Hence the difference between the actual plaquette curvature of exp a and its discrete curl is at most K∥a∥2, uniformly in the plaquette and lattice size. Repeated edges cause no change: each of the four coordinates is bounded by the same field norm. Finally, ∥Ad(U)a −a∥≤2 colorScale · chord(U)∥a∥. In matrix coordinates write UXU∗−X = (U −1)XU∗+X(U∗−1), use unitary invariance of the Frobenius norm, and use the chord identity above together with ∥colorMat(a)∥= colorScale ∥a∥. Induction along a walk likewise gives chord(travel(U, w)) ≤length(w)h if every link read by the walk has chord at most h; inverses have the same chord. 6.6 Quantitative nonlinear and flat row comparison Put widen(R, h, τ) = (1 + R)h + exp(−τR). All cutoffs in this paragraph use actual path length, including winding, rather than the distance between the two torus endpoints. First consider the plotted curvature field. A path slot consists of a walk from its reading site to a plaquette, followed by adjoint transport of that plaquette curvature. If ∥a∥≤h is in the chart ball, the preceding estimates give link chords ≤Bch h, curl norm ≤Ach h, and curvature-minus-curl norm ≤Ach h2. For a retained slot of physical length at most R, transport changes the curl by at most 2 colorScale RBchh · Achh. Together with the quadratic curvature error this bounds slot-minus-curl by Jh widen(R, h, τ), J = Ach(1 + 2 colorScale Bch). 46 ## Source page 49 A Wilson Yang–Mills construction J. Rodgers For an omitted slot of length lwith dl> R and stage width σ > 0, 1 ≤exp(σl) exp(−σR/d) ≤exp(σl) widen(R, h, σ/d). This pays its whole linear curl with the exponentially weighted path budget. If the path coefficients ui obey P i |ui| exp(σli) ≤paths, absolute summation of retained errors and omitted linear terms proves ∥xlin(a)j∥≤Ch, ∥plotted(exp a)j −xlin(a)j∥≤Ch widen(R, h, σ/d), where C = √ directAch + paths J. The direct-curvature coordinate uses just the quadratic face estimate and h ≤widen. The path coordinate uses the same summable envelope |ui| exp(σli); thus subtraction and summation are justified by absolute convergence, including countably many slots. Constants and the small chart radius depend on the fixed stage and d, not on torus size or R. For the kernel rows, let Kab(u) be a cover coefficient with X u exp(η|u|)|Kab(u)| ≤A for each pair of finite residue types a,b. A routed term is its scalar multiple of adjoint transport along its designated walk, retained when length ·d ≤R, and included only in the endpoint congru- ence class. Suppose length ·d ≤D(1 + |u|). The full flat term is Kab(u) times the identity on color space, without the cutoff. Unitary adjoint action bounds the operator norm of each routed term by |Kab(u)|. If link chords are ≤Cch h, a retained transport differs from the identity by at most 2 colorScale Cch R h. For an omitted term the length inequality implies ηR/D ≤η + η|u|, 1 ≤exp η · exp(η|u|) exp(−ηR/D). Thus both kinds of difference are bounded by J widen(R, h, η/D) exp(η|u|)|Kab(u)|, J = 1 + 2 colorScale Cch + exp η. Sum first on the cover. Each u contributes to exactly its endpoint congruence class; summing over endpoints therefore costs only the finite residue census, even on short tori with many windings. Absolute convergence allows the order of these nonnegative bounds to be interchanged. If there are na source types and nb target types, the convenient common row and column bound is CK = (na + nb)AJ. Both the cut nonlinear matrix and the full flat matrix have row and column sums at most CK, and their difference has row and column sums at most CK widen(R, h, η/D). For the same-level lanes, the ordered coordinate route to u has length |u| and d = L. For the fine-to-coarse row, its route goes from a residue lift to Lu; its length is at most L|u|+4L. Both cases satisfy the preceding inequality with D = 5L. The replica background has either identity links or original coarse links, so it preserves the required chord bound. Apply this calculation to the F, I and N rows with A = max(max(0, E), E/L2), τ = η/(5L). 47 ## Source page 50 A Wilson Yang–Mills construction J. Rodgers Take the maximum of 1 and the three corresponding CK values, and the common exponential- chart radius. This proves all the kernel-chart estimates used below, uniformly in the period and cutoff, with their row and transpose-row bounds. All constants are chosen after the block length L has been fixed. 6.7 Clean-hull guide comparison The guide is the pair of coarse response fields Mv,Mw, while its linear comparison is obtained by applying the flat F and I rows to the linearized field xlin(a). There is also an N row. Its linear comparison vanishes identically: the supplied stationary identity is flatN ·Dfull · d = 0, and xlin has precisely the coordinate form Dfull · d a. This includes harmonic coordinates; no scalar-mode deletion is used. On a clean radius-B/4 hull agreeing with exp a, the clipped vision field equals the unclipped plotted field of exp a at every coordinate read by a guide row. The row’s reach is at most R = till(x); the plotted field needs only R+4 further sites, still within the same hull by the room inequalities. Cleanliness makes the retained indicator equal to one and makes the clipping factor min(1, temp / toll) equal to one. The row itself can also be replaced by its value at exp a by its local dependence on this hull. Here is the resulting quantitative calculation. Suppose the nonlinear row Q has row bound X and differs from its flat row D by row bound XW. Suppose also ∥xlin(a)j∥≤Z, ∥plotted(exp a)j −xlin(a)j∥≤U. Writing B = Q(exp a), f = plotted(exp a), and g = xlin(a), the exact decomposition Bf −Dg = B(f −g) + (B −D)g and the row-sum inequality give ∥Bf −Dg∥≤XU + XWZ. All equalities before this bound are restricted to the same clean hull. In the source the uniform kernel-chart and plotted-chart estimates supply positive constants X, Z0, s, t and M(x) = Z0H(x), U(x) = M(x) widen(till(x), H(x), t), e(x) = XU(x) + X widen(till(x), H(x), s)M(x). Here t is the lower-stage width divided by the positive block length. If H has rate −1/2 and is eventually nonnegative, broadening by widen preserves that rate: its bound is a radius factor times H plus an exponentially small cutoffterm. Products in the displayed formula therefore have rate −1. The same e bounds the two guide differences and the nonlinear N guide itself, since its linear part is zero. The kernel-chart and plotted-field estimates of this chapter give these constants. 6.8 Stationary cancellation in every signed orientation The stationary comparison uses a single supplied pair of quadratic beds and their common diagonal and roll identities. Their attached path realizations, positive floors, and invariance under signed coordinate changes are chosen together. Let W be the installed stationary map, U its extracted fine-field matrix for the chosen forest, and D0 = I −probe ·U. The diagonal action identity reads Kcoarse = UTKfineU + penalty · DT 0 D0. 48 ## Source page 51 A Wilson Yang–Mills construction J. Rodgers For coarse logarithmic data a, the shifted fine field has color-coordinate vectors U a, while its residual has vectors D0 a. Pairing this matrix identity with each color coordinate and summing gives fineEnergy(shifteda) + penalty∥residual(a, shifteda)∥2 = coarseEnergy(a). This is an identity of quadratic forms, including zero harmonic modes. The same identity holds after any signed coordinate change. To justify the noise part, apply the exact signed covariance of the actual nonlinear predictor and noise energy to the exponentiated fields exp(t a), exp(t shifted a), divide by t2, and pass to their already established quadratic limits. The invariant fine and coarse bed metrics give the other two terms. Thus no isotropy of individual matrix entries is assumed. Write w for the coarse linear moment field of the transformed input, u for the fine linear moment field of its transformed shifted field, and v for its linear noise field. The attached path realization and floor identify moment-square sums with the corresponding bed energies. The preceding energy identity is therefore ∥u∥2 + ∥v∥2 = ∥w∥2, where the norms here are finite counting sums of squared color norms. Let F,I,N be the full flat rows, and put U1 = Fw, V1 = Iw. The N row annihilates w. The diagonal roll and cross roll identities then give w = F TU1 + ITV1, w = F Tu + ITv. The transpose here means the adjoint for the counting inner products. Pairing the first equality with w gives ∥U1∥2 + ∥V1∥2 = ∥w∥2. Pairing the second with w gives ⟨U1, u⟩+ ⟨V1, v⟩= ∥w∥2. Expand squared differences and use these three equalities to obtain ∥U1 −u∥2 + ∥V1 −v∥2 = 0. Every summand is nonnegative, so u = U1 and v = V1 at every row. This proves equality of linearCost and linearGuide in every signed orientation. Subtracting the linear copy then gives precisely the uncut spring cancellation used below. 6.9 Nearest group points and the analytic focus All matrix norms in this paragraph are Frobenius norms, with the real inner product. Compactness of the closed unitary matrix group gives a nearest group element to every matrix X. Left or right multiplication by a group element preserves this minimization property, by unitary invariance of the norm. If 1 is nearest to X, differentiate the squared distance ∥exp(tY ) −X∥2 at its minimum t = 0 for every Lie algebra matrix Y. The derivative is zero, so X−1 is orthogonal to the Lie algebra. In general, if u is nearest to X, left multiplication by u−1 gives u ∗X −1 ∈lie(G)⊥. Here u* denotes the adjoint matrix of u. Consider the tubular map 49 ## Source page 52 A Wilson Yang–Mills construction J. Rodgers T(a, n) = exp(colorMat(a))(1 + n), n ∈lie(G)⊥. It is analytic and its derivative at (0,0) is colorMat(a)+n. This is an isomorphism by the orthogonal tangent/normal decomposition. The inverse function theorem therefore gives a local analytic inverse near X = 1. Define focus(X) as the exponential of the tangent coordinate of that inverse. Every nearest point must agree with focus sufficiently near 1. To prove this without assuming continuity of a minimizing selection, choose any nearest b(X). Its minimizing property gives ∥b(X)− X∥≤∥1 −X∥, hence ∥b(X) −1∥≤2∥X −1∥. Thus b(X) →1. Its normal coordinate n(X) = b(X)*X−1 tends to zero and is orthogonal to the Lie algebra by the preceding variational identity. Its tangent coordinate a(X) = log b(X) also tends to zero. Moreover T(a(X), n(X)) = X exactly. Both coordinates therefore lie in the inverse-function neighborhood and equal its unique inverse coordinates, proving b(X) = focus(X). If there were exceptional nearest points arbitrarily near 1, choosing one at each exceptional X would contradict this conclusion. Thus every nearest point equals focus there. This proves local uniqueness simultaneously over all components of the group. Translating X by a nearby group element extends the same uniqueness radius to a tube about the whole group. Differentiating the tubular inverse gives D(mat ◦focus)(1) = orthogonalProjectionlie(G). Indeed the inverse derivative sends colorMat(a)+n to (a,n), and differentiating the exponential discards n. Analyticity makes this a strict derivative. Consequently, for every ε > 0, after shrinking the neighborhood of 1, nearest points u of X and v of Y satisfy ∥mat(u) −mat(v) −projectionlie(G)(X −Y )∥≤ε∥X −Y ∥. These prove the focus properties used in the frame and predictor derivatives from the local group chart and the ordinary inverse function theorem. 6.10 Analyticity of the normalized frame This is a fixed-period assertion; no volume-independent analytic radius is claimed here. Let v be the logarithmic input and Hp its actual frame matrices at the patch points, with Hp0 = 1 at the distinguished center. At v = 0 all path products equal 1. The identity frame minimizes the nonnegative frame score with score zero; uniqueness of the minimizer identifies the selected frame with it. The minimum branch has an open neighborhood of this input because its cutoffis positive. Continuity of the unique selected minimizer therefore gives continuity of all Hp at zero. On that neighborhood all frame tags select this same frame. The frame self-consistency condition says that Hp is a nearest group element to cloudp(v, H) = averageq averageo Hq · travelv(p →qinordero) ∗. This finite expression is analytic in v and the matrix array H: its path products are finite products of exponentials and adjoints. At (v, H) = (0, 1), every cloud is 1. Continuity then puts every cloud in the neighborhood where its nearest group element equals the analytic focus map. Thus the actual frame solves the analytic equations Fp(v, H) = Hp0 −1 if p = p0, Fp(v, H) = Hp −focus(cloudp(v, H))otherwise. 50 ## Source page 53 A Wilson Yang–Mills construction J. Rodgers Here focus is viewed in matrix coordinates. To apply the implicit function theorem, compute the derivative in the H variable at (0,1). If an array h belongs to its kernel, then hp0 = 0 and hp = Aforp ̸= p0, A = projection(averageq hq). Let N be the positive number of patch points. Averaging gives average h = (N −1)A/N. Since projection(A) = A, its defining equation yields A = (N −1)A/N, hence A = 0 and h = 0. The derivative is an injective endomorphism of a finite-dimensional matrix-array space, so it is invertible. For precision, an analytic equation F(v, H(v)) = 0 and a continuous solution H(v) with invert- ible H-derivative identify that solution with the analytic implicit branch: apply the inverse function theorem to (v, H) 7→(v, F(v, H)). Continuity puts the given solution inside its local inverse do- main, where uniqueness forces equality. This proves analyticity of the actual frame. It supplies the analytic frame input used in the following predictor calculation. The nearest-point, self-consistency and selected-minimizer arguments in this chapter establish the geometric properties used here. 6.11 First derivative of the actual rooted predictor For coarse root y and patch points p,q, let feey(p, q, o; a) be the tangent link sum along the actual ordered route from p to q in order o, embedded at y. Define tapy(p; a) = −averageq averageo feey(p, q, o; a). Each color coordinate of tap is the corresponding scalar path functional τyp applied to the color-coordinate input. The derivative of a route product at the identity is colorMat of its link sum, by differentiating its finite product of exponentials; reversed links contribute with a minus sign. Let Hy(p; a) be the actual normalized frame matrix near zero, and let dp be its derivative in direction a. The local frame equation says that Hy(p) is the nearest-group focus of its averaged cloud. Differentiating that cloud gives averageq dq + colorMat(tapy(p; a)). The derivative of focus at 1 is the orthogonal projection onto the matrix Lie algebra. Hence dp = projection(averageq dq) + colorMat(tapy(p; a)). The frame at the distinguished patch center p0 is identically 1, so dp0 = 0. Subtracting its equation eliminates the common projected average: dp = colorMat(tapy(p; a) −tapy(p0; a)). The local analytic frame equation and its normalization are the inputs to this differentiation; the subtraction shows explicitly why the derivative is independent of the frame tag. For a coarse bond (y,i), a connector consists of the left frame, the actual length-L bond route, and the inverse right frame. Its derivative is therefore colorMat of billp(a) = tapy(p; a) −tapy(p0; a) +linkSum(bondRoutefromp; a) −tap(y+ei)(p; a) + tap(y+ei)(p0; a). Near zero every connector is near 1. The best-predictor branch is open and is used at zero because its cost there is zero and its cutoffis positive. Thus the actual predictor near zero is the nearest-group focus of the weighted connector average, irrespective of its fallback tag. The connector average equals 1 at zero. It is analytic because the frame and finite route maps are analytic, and focus is analytic near 1; consequently the actual predictor is analytic near zero. 51 ## Source page 54 A Wilson Yang–Mills construction J. Rodgers Differentiating the weighted average gives colorMat(P p weightp billp(a)). This lies in the Lie algebra, so the focus derivative leaves it unchanged. Expanding the definitions of tap, the rooted path functionals, and the probe matrix shows coordinate by coordinate that X p weightp billp(a) = probe(a). The patch weights sum to one; this retains both frame-center subtraction terms. Thus the predictor equals 1 at zero and its literal first derivative is colorMat(probe(a)), exactly the derivative used in the noise Taylor estimate. The remaining foundation of this calculation is the local analytic frame construction and focus derivative, rather than any limiting or averaged predictor. 6.12 Uniform predictor linearization and entry bounds For a fixed patch and kernel, the predictor at one coarse bond reads a finite cover stencil. Include all vertices on every ordered route between patch points, the translated copy of this set at the other bond endpoint, and every straight length-L connector from a patch point. Call their union Spred. If two cover fields agree on links rooted in Spred, every route product, axial frame, via-frame value and score agrees. Their minimizing frame also agrees by the supplied frame uniqueness; in the fallback branch their axial frame agrees directly. Thus their actual predictor values agree with the same endpoint tags and connector tag. Taking the maximum route distance of these finitely many roots defines predictionReach. Agreement on that parent-distance hull therefore suffices for predictor equality. At fixed volume and fixed tags define Zk(a, b) = p penalty · curvature(exp b · prediction(exp a, k)−1) at the specified coarse bond. This is analytic near zero by the local analytic predictor theorem, matrix multiplication and adjoint, and the linear matrix formula for curvature. At zero the predictor is 1 and Zk = 0. Its derivative is zlin(a, b) = p penalty · (b −probe(a)) in the physical color coordinates: differentiate the predictor to obtain colorMat(probe(a)), dif- ferentiate exp b, and use skew adjointness to differentiate the inverse. The curvature derivative recovers the difference. Taylor’s theorem gives a linear bound and a quadratic remainder. Since there are finitely many tags and bond positions on a fixed torus, a maximum of their constants and a positive minimum of their radii makes the bounds simultaneous there. To remove volume dependence, choose one reference torus with coarse side c0 > 1 so large that every point of Spred lifts uniquely from its fine torus. One explicit choice is c0 = 2M + 2, where M is the largest absolute coordinate of a stencil point. For any input torus and any bond rooted at x, fill the reference fine logarithmic field by pulling back the actual input values under y 7→wrapf(lift(y −embed(0))) + embed(x). Fill its coarse logarithmic field constantly with the actual coarse value at the bond being read. Neither operation increases the supremum norm. Put the original left and right frame tags at 0 and ei of the reference coarse torus; these sites are distinct because c0 > 1. Put the original connector tag at the reference bond. All predictor stencil reads then agree exactly for every scalar multiple t(a, b), even if the original torus identifies different stencil points. Thus the nonlinear Z values agree 52 ## Source page 55 A Wilson Yang–Mills construction J. Rodgers at t = 1 and their linear terms agree by uniqueness of the derivative at t = 0. The reference-torus Taylor bounds consequently give constants Cz, rz independent of the input volume and all its tags: ∥zlin(a, b)∥≤Cz max(∥a∥, ∥b∥), ∥Zk(a, b) −zlin(a, b)∥≤Cz max(∥a∥, ∥b∥)2 when the maximum is at most rz. The local analytic predictor theorem and its derivative formula are the inputs to this Taylor argument; the transplantation adds no new branch choice. The entry field is the direct sum of the fine plotted moment and Z. Its linear counterpart is the direct sum of xlin and zlin. For logarithmic inputs bounded by H, combine the plotted comparison already proved with the last inequality. If Cx is its plotted constant, set M = (Cx + Cz)H, W = CxH widen(R, H, widthtop) + CzH2. Then each linear entry has norm at most M and each entry-minus-linear-entry has norm at most W, simultaneously in the two kinds of row. For R = till(x) and H of rate −1/2 these have rates −1/2 and −1. The entry is gauge covariant by predictor covariance and curvature conjugation. It is local on the protected hull: the plotted component uses its cutoffstencil and the Z component uses predictionReach together with the coarse link at the row root. These establish the entry-near and entry-locality inputs used next. For completeness, the full and cutofflinear rows have uniform stencil bounds at every cutoff, not merely at till(x). A discrete curl has norm at most 4∥v∥. Summing a path row against its weighted absolute coefficient budget gives ∥ellLine(v)∥≤4 paths∥v∥. Terms omitted beyond R cost at most 4 paths∥v∥exp(−widthtopR). The direct-curvature term has bound 4 √ direct∥v∥; the noise term has bound Cz∥v∥. Thus, writing Ust = 4(paths + √ direct) + Cz, Kst = Ust + 4paths, both full and cutoffrow norms are at most Kst∥v∥and their difference is at most Kst∥v∥exp(−widthtopR). The cutoffbound follows by comparing to the full row and using exp(−widthtopR) ≤1. Absolute summability justifies every linear path sum and its subtraction. These bounds also give the spatial impulse estimate used in the plate prices. If a single-link input of color a is at parent distance g from a row root, the cutoffrow at s is zero whenever s + predictionReach +4 < g. For g > predictionReach +5 choose s = g −predictionReach −5. Compare both cutoffs R and s to the full row and apply their tail bounds; if R is too short the row is already zero. The result, also covering the nearby case by the uniform cutoffbound, is exp(widthtopg)∥barFieldR(singlelinka)∥ ≤2Kst exp(widthtop(predictionReach +6))∥a∥. 6.13 From a shell comparison to the tree-slice link estimate For a differentiable group-valued curve U(t) with U(0) = 1 and matrix derivative colorMat(a), the norm of its difference quotient gives chord(U(t))/|t| →∥a∥ast →0, t ̸= 0. 53 ## Source page 56 A Wilson Yang–Mills construction J. Rodgers Indeed chord is the Frobenius distance from 1 divided by colorScale, and ∥colorMat(a)∥= colorScale ∥a∥. This preserves the physical normalization of the color norm. Let w be a fine logarithmic field vanishing on the trunks of the given forest. Then exp(tw) lies on that exact tree slice for every t. The actual frame minimum branch is active near t = 0 at every coarse site; its frame mass is positive, so the chosen tags are usable there. Apply the nonlinear shell link comparison at the link e to U(t) = exp(tw), coarse comparison field 1, and its actual predictor Q(t). It gives chord(U(t)e) ≤C port(parent(e), U(t)) + chord Q(t)(parent(e),axis(e)) . This upper bound is valid in the internal-block case as well, where adding the last nonnegative term is harmless. The port sum is bounded by the full macro-box sum of plaquette chords, including repetitions on a periodic lattice. Divide by |t| and pass to the limit. Each plaquette derivative is its discrete curl, and the actual predictor derivative was computed above. The finite macro sum commutes with this limit. We obtain ∥we∥≤C X z∈box(macroR) X p ∥curl(w)(wrap z+embed(parent(e)),p)∥ +∥probe(w)(parent(e),axis(e))∥. This proves the local link estimate from the nonlinear shell comparison without a separate linear gauge choice. The same calculation gives a volume-independent counting bound. Put N = | box(macroR)|, m = |PlaqAxes| and q = L4. For any plaquette function f, Cauchy–Schwarz on the macro sum costs Nm. For each fixed offset z, the map (coarse site,plaquette type) 7→(wrap z+embed(site),plaquette type) is injective, so summing its squares costs at most the full fine plaquette sum. Summing all N offsets adds the factor N. Each coarse site has q fine children and four outgoing fine-axis choices. Thus squaring the local link inequality with (a + b)2 ≤2a2 + 2b2 gives X e ∥we∥2 ≤β X p ∥curl(w)p∥2 + γ X b ∥probe(w)b∥2, β = 2C2 · q · 4 · N2m, γ = 2q. This counting uses injectivity in the coarse site for each fixed offset, not injectivity of different offsets; repeated periodic plaquettes remain allowed. If the fine quadratic bed has curl floor a > 0 and the predictor penalty is κ > 0, its vertical Hessian dominates a P ∥curl w∥2 + κ P ∥probe w∥2. The restriction to the vertical tree chart has the same Hessian as the full germ on directions (0,v), by the linear inclusion chain rule; the collected vertical field is v offtrunks and zero on trunks. Therefore, with η = min(a, κ)/(1 + β + γ) > 0, the vertical Hessian is at least η P free ∥ve∥2, since ηβ ≤a and ηγ ≤κ. This also unfolds the fixed-block counting-coordinate Hessian floor. The shell comparison supplying C is described next. 6.14 Uniform branch cutoffs and continuity of selected minima The finite-pair uniqueness result gives a positive d depending only on G. Choose the frame cutoff at most d/colorScale. For any patch satisfying the all-minimum criterion, use its center-based axial 54 ## Source page 57 A Wilson Yang–Mills construction J. Rodgers frame a and define pair transports Wxyk = ax travel(x →y, k)a−1 y . The criterion bounds their chords by the frame cutoff, so their matrix distances from 1 are at most d. For an arbitrary frame b, the transformed array bx a−1 x has pair cost exactly the original frame score. A normalized global minimizer of the frame score is a global minimizer of this pair cost: every unnormalized competitor can be multiplied on the left to fix its center, without changing its score. Pair uniqueness then gives uniqueness up to a left multiple, and the center value 1 makes that multiple equal to 1. Existence follows by compactness of the finite product of G with its center fixed. This proves the uniform frame cutoffindependently of patch order and size. The predictor cutofffollows from a related weighted-centroid calculation. For real weights wi summing to one and group points fi, set m = P wi mat(fi). Expansion of the real inner product gives 2repDim X i wi price(fiu−1) = X i wi∥mat(fi)∥2 −∥m∥2 + ∥m −mat(u)∥2. Thus minimizing the weighted price is equivalent to being nearest to m. For nonnegative weights, the squared norm inequality for a weighted average shows ∥m −mat(u)∥2 ≤2repDim X i wi price(fiu−1). If the right side is below the square of the nearest-point tube radius r, m lies within r of the group and has a unique nearest point. Choosing predictor cutoffat most r2/(2 repDim) proves the uniform predictor uniqueness condition. Both cutoffchoices depend only on G. The selected minimizers are continuous wherever their defining fibers are unique. Here is the compactness argument including arbitrary selections away from that set. Let M ⊆X × Y be the closed minimizer relation, with Y compact. For an open neighborhood V of the unique minimizing value at x0, project M∩(X×V c) to X. This projection is closed because Y is compact, and it omits x0 by uniqueness. Its complement is therefore a neighborhood of x0 on which every selected minimizer lies in V. This is continuity at x0. For frames the relation is closed because it is the intersection of the closed center-normalization condition and the score inequalities against all normalized competitors; scores are continuous finite path expressions. The predictor minimizer relation is closed by the corresponding continuous cost inequalities. Applying the compactness argument proves continuity on the respective minimum branches. Their strict, finite cutofftests make those branches open. These complete the uniqueness and continuity inputs used in the analytic-frame construction. 6.15 Minimizer self-consistency and uniqueness For a finite pair family define term(h; x, y, k) = price(hxWxykh−1 y ), and cost as its average over all three indices. Reversal of the pair exchanges x,y and replaces k by its mirror permutation, preserv- ing the summed term. The diagonal terms vanish because Wxxk = 1. Consequently, replacing only hx by a competitor b changes the total cost by twice the change in the off-diagonal x-row. Global minimality implies that this off-diagonal row is minimal. Adding the frozen self term price(bh−1 x ), which is nonnegative and is zero at b = hx, proves the stronger comparison with the whole row in which every hy, including hx, is held fixed. The normalized whole-row price is precisely the weighted squared-distance problem to the ma- trices hy W −1 xyk, up to a positive constant and a term independent of b. Expanding the squared 55 ## Source page 58 A Wilson Yang–Mills construction J. Rodgers Frobenius norms, or completing the square about their average, identifies its minimizers with near- est group points to feedback(h)x. Thus hx is nearest to its feedback. This proves the self-consistency assertion used in both alignment arguments. A uniform small-transport condition gives uniqueness modulo a common left multiple, indepen- dently of the numbers of sites or pair labels. Suppose ∥mat(Wxyk) −1∥≤d. Comparison with the constant array bounds the average squared pair separation of any minimizing h by 4d2. If 1 is nearest to its centroid, averaging this bound at a suitable index gives ∥centroid(h) −1∥≤2d. The feedback differs from the centroid by at most d, so every feedback is within 3d of 1. Choose δ from the focus two-point estimate with coefficient 1/4, and d = min(δ/4, 1/4). Given two minimizers, multiply each on the left by the inverse of a nearest point to its own centroid. This preserves minimization and makes 1 nearest to both centroids. Let u = maxx ∥mat(hx)−mat(kx)∥ and take an index attaining it. Write B = feedback(h)x −feedback(k)x. Unitary invariance gives ∥B∥≤u. Subtracting the centroid difference before estimating gives the sharper bound ∥B −(centroid(h) −centroid(k))∥≤ud, because each summand factors as (mat(hy)−mat(ky))(mat(Wxyk)−1 −I). Both centroid-minus- 1 matrices are normal to the Lie algebra by the nearest-point normal equation. Their difference has tangent projection zero. Thus ∥projection(B)∥≤ud. Self-consistency and the focus estimate, applicable since 3d < δ, now imply u ≤∥projection(B)∥+ 1 4∥B∥≤ud + u/4 ≤u/2. Therefore u = 0. Undoing the two centroid normalizations shows that the original minimizers differ by one common left group element. A fixed center value removes this last freedom. 6.16 Finite-pair alignment and the route census The synchronization estimates compare overlapping local predictor charts. They cannot identify all gauges by declaration: each query has its own route, and every route contributes to the finite multiplicity count. The common macro gauge is obtained only after these comparison errors have been estimated. This supplies the coordinate compatibility used by the affine rows and protected mean in the following steps. Let Wxyk be a finite family of group-valued pair transports with its reversal symmetry, and let h minimize the averaged pair price of hx Wxyk h−1 y . Put income(x) = averagey,k chord(Wxyk) and m = averagex income(x). Comparison with the identity array and chord2 ≤2 chord show that its minimum price is at most m. Factoring hx h−1 y through the pair discrepancy gives averagex,y ∥mat(hx) −mat(hy)∥2 ≤8 colorScale2 m. Indeed the squared triangle inequality bounds each chord(hxh−1 y )2 by 2 times the squared dis- crepancy plus 4 chord(Wxyk). Averaging and using the minimizing comparison proves the factor 8. Let centroid(h) = averagex mat(hx) and feedback(h)x = averagey,k mat(hyW −1 xyk). Direct sub- traction gives ∥feedback(h)x −centroid(h)∥≤colorScale · income(x). A minimizer is nearest to its feedback at each coordinate; this coordinatewise variational asser- tion is the finite-pair self-consistency input. Choose a nearest group point b to the centroid and 56 ## Source page 59 A Wilson Yang–Mills construction J. Rodgers replace h by b−1h. The centroid now has nearest point 1. By choosing one index whose averaged squared pair distance is at most the global average and applying the squared norm inequality for averages, its distance from 1 is at most √ 8 colorScale2 m. Take δ > 0 so that the nearest-point comparison from the focus derivative holds with error coefficient 1. Put σ = (δ/2)2/(8 colorScale2) and a = δ/(2 colorScale). If m < σ, the centroid is within δ/2 of 1. At any index with income(x) < a, its feedback is within δ of 1 as well. Apply that comparison to the nearest points hx and 1 of feedbackx and centroid. The projection has norm at most one; therefore ∥mat(hx) −1∥≤2∥feedbackx −centroid ∥, chord(hx) ≤2 income(x). At the remaining indices chord(hx) ≤2 ≤(2/a) income(x). Averaging proves the alignment estimate with max(2, 2/a). If m ≥σ, simply choose b = 1 and use average chord(hx) ≤2 ≤(2/σ)m. Thus the single positive constant Calign = max(2/σ, 2, 2/a) gives averagex chord(hxb−1) ≤Calign m for every finite pair family, with no smallness hypothesis on its individual transports. Here is the route census used to bound m. For a radius-r box, replace selected coordinates of an endpoint x by the corresponding coordinates of y; call the resulting site dp(x, y). Swapping those coordinates between x and y is an involution of endpoint pairs. Therefore every site function f satisfies X x,y f(dp(x, y)) = N X x f(x). Likewise swapping a site’s j coordinate with an independent height in −r,. . . ,r is an involution of (site,height), giving X x,height f(updatej(x, height)) = (2r + 1) X x f(x). A straight segment in direction j has chord at most the sum of the internal positive j-link chords on its whole coordinate line. For a positive segment this follows by the product triangle inequality and then enlargement to the full line; for a negative segment reverse the walk and use inversion invariance. Append one new axis j to an ordered route. Its initial part ends at dp(x, y), and its last segment is bounded by that site’s full j-line sum. The two involution identities bound the sum of this added cost over endpoint pairs by N(2r + 1) edgeSum(U). Induction over the distinct axes proves the route bound length(p)N(2r + 1) edgeSum(U); four axes give the constant 4. This establishes the route estimate and the finite-pair alignment used in synchronization. The preceding minimizer self-consistency argument gives the coordinatewise nearest-feedback condition used here. 6.17 Predictor query and synchronization estimates Let Yp be connector values with nonnegative patch weights summing to one. For arbitrary group elements x,y put q = xy−1 and τ = X p weightp chord(x−1Ypy). 57 ## Source page 60 A Wilson Yang–Mills construction J. Rodgers Cyclic invariance of chord identifies this with P weightp chord(Ypq−1). The weighted matrix mean m = P weightp mat(Yp) satisfies ∥m−mat(q)∥≤colorScale ·τ. In the best-predictor branch its value b is a nearest group point to m. Hence ∥mat(b) −mat(q)∥≤∥mat(b) −m∥+ ∥m −mat(q)∥ ≤2 colorScale ·τ. In the other branch the minimum connector price is at least predictorCut. Since price(g) = chord(g)2/2 ≤chord(g) for 0 ≤chord(g) ≤2, comparison with q gives τ ≥predictorCut. The universal chord bound 2 then gives the same common estimate in both branches: chord(x−1predict(Y )y) ≤2 max(1, predictorCut−1)τ. Thus the query estimate uses the actual fallback outcome as well as the nearest-point outcome. We apply the preceding finite-pair alignment estimate to synchronization. The ordered route sum on a radius-r box obeys X p,q chord(travel(p →qinordero)) ≤4N(2r + 1) edgeSum(U), N = numberofpatchpoints. It follows by charging each coordinate segment to its internal positive edges; the precise route- count argument is developed with the finite-pair input below. Averaging this inequality over orders and endpoints gives N · meanPairChord ≤4(2r + 1) edgeSum(U). In the minimizing frame branch, gauge by H and set bp = framep H−1 p . This array minimizes the corresponding finite pair energy: gauge covariance of route products preserves that energy, and any competing array can be normalized at the center by a common left multiplication without changing it. The finite-pair cluster inequality gives c with averagep chord(bpc−1) ≤C meanPairChord. Therefore X p chord(c−1framepH−1 p ) ≤4C(2r + 1) edgeSum(gauge HU). The fallback branch is elementary and does not use that alignment inequality. Its positive frame mass makes its axial tag eligible, so its normalized route gauge d satisfies edgeSum(gauge(d, U)) ≤ axialFactor(2r + 1) faceSum(U). Put g = dH−1. Edgewise factorization and the chord triangle inequality give edgeSum(gauge g1) ≤edgeSum(gauge dU) + edgeSum(gauge HU). For this pure gauge, route transport from p to q is gp g−1 q . Choose q whose sum over p is at most its average over q, and apply the ordered route sum above. Taking c = gq yields X p chord(c−1framepH−1 p ) ≤4(2r + 1)[edgeSum(gauge HU) +axialFactor(2r + 1)faceSum(U)]. With A = max(4C, 4), both branches prove the synchronization estimate below. The finite-pair cluster inequality and route census proved in this chapter supply its two inputs. 58 ## Source page 61 A Wilson Yang–Mills construction J. Rodgers The patch weight bound used in the macro predictor estimate also has a direct counting proof. Its weight is the distribution of a sum of J independent uniform small-box rolls. Separate the first roll from the other J−1 rolls. For every fixed choice of those remaining rolls, at most one first roll can give a specified sum. Since the small box has L4 points (L is odd), that sum has probability at most L−4. Thus L4 weightp ≤1, with no assumption that different roll histories give different sums. 6.18 Common macro gauge and predictor price The planting argument uses the axial estimate already proved and a frame synchronization inequal- ity. The latter states that for a usable frame on a patch of radius r and any gauge H there is a constant group element p with X patch chord(p−1frame(q)H(q)−1) ≤A(2r + 1)[edgeSum(gauge HU) +axialFactor(2r + 1)faceSum(U)]. The preceding synchronization proof establishes this inequality. Its constant A depends only on the group; the block-length factors are displayed explicitly. Let j be the smoothing index, set l0 = 2(j +7)+1, D = 16l0 and C = Al0(D+axialFactor·l0). The macro radius is the patch radius plus 4L; consequently 2 macroR +1 ≤l0 L. Apply the four-axis axial construction on this whole macro box to obtain a single H with edge sum at most D L·port. Every patch translated by centerRoot(y) = Ly, y ∈[−2, 2]4, lies in this macro box. Its internal edge and face sums are bounded by the full macro sums by nonnegativity and translation. Apply synchronization separately on each such translated patch using the same H. The displayed bounds give an element py with frame error sum at most CL2 port. This proves the planting estimate used below and preserves one common macro gauge for all neighboring roots. To price a predictor connector between adjacent roots y and y + ei, factor it as the left frame error, the straight length-L transport in gauge H, and the inverse right frame error. Its chord is bounded by the sum of these three chords. Let the two frame-error sums be at most KL2 port and the macro edge sum at most TL port. The sum of the straight-path chord over all patch starting points is at most L times the macro edge sum: each step’s starting-point translation is injective, its links remain in the macro box, and the nonnegative edge census bounds that partial sum. This bound counts all L steps even if they repeat a torus link. The patch weight bound is L4 weightp ≤1. Therefore the weighted connector chord sum τ satisfies L4τ ≤L2(2K + T) port . The predictor query inequality bounds its chord, in this left and right frame, by 2 max(1, predictorCut−1)τ. Multiplying by L2 and using L > 0 proves L2 chord(p−1 y Q(y,i)p(y+ei)) ≤2 max(1, predictorCut−1)(2K + T) port . This proves the indicated-predictor estimate from synchronization and the predictor query in- equality. It also supplies the coarse curvature comparison. Four such gauged predictors around an elemen- tary coarse plaquette have chord at most α port /L2 each, where α = 2 max(1, predictorCut−1)(2C+ 59 ## Source page 62 A Wilson Yang–Mills construction J. Rodgers D). Their endpoint p factors cancel in the plaquette product. Triangle and conjugation invariance yield chord(face Q) ≤4α port /L2. For any V, compare its four links to Q. The two-factor inequality chord(ab(cd)−1) ≤chord(ac−1) + chord(bd−1) follows by factoring the product into one discrepancy and a conjugate of the other. Iteration, with inversion invariance for the reversed links, gives chord(face V ) ≤chord(face Q) + fuel(Q, V ), where fuel is the sum of its four wire discrepancies. This is the local curvature comparison used by the clean-patch and shell arguments. 6.19 The nonlinear shell link comparison Fix the forest and its actual slice. To compare a fine link (x,i), close it with the reverse tree hike from its first coarse root to x and the tree hike from x + ei to its second coarse root. On the slice both hike transports are 1, so the resulting walk has transport U(x, i), endpoints the two embedded coarse parents, and length at most 8L+1. Lift the whole walk to the cover starting at 0 over its first root. Every point lies in the macro box. The planting estimate supplies on this same macro box a gauge H and comparison elements py with totalEdgeChord(gauge HU) ≤DL · port, X patch chord(p−1 y framey(q)H(q + centerRoot(y))−1) ≤CL2 · port . The second inequality is read at the distinguished patch point, whose frame is 1, to give chord(H(centerRoot(y))py) ≤CL2 port. The indicated-predictor estimate on adjacent roots gives chord(p−1 0 Qpei) ≤αL−2 port, α = 2 max(1, predictorCut−1)(2C + D). These planting and indicated-predictor estimates are inputs to this step. The triangle inequality along the lifted forest walk bounds its gauged transport by u · port, u = (8L + 1)DL. The period assumption 100 macroR ≤fineSide makes the cover wrapping injective on the macro box. If the two coarse parents are equal, the lifted endpoint is therefore 0; conjugation invariance bounds chord(U(x, i)) by u port and its replica comparison link is 1. If they differ, the centered parent-hop identity makes the second parent the positive coarse neighbor, and injectivity identifies the lifted endpoint with centerRoot(ei), even across a periodic seam. Factoring the gauged predictor through p0 and pei gives its chord bound v · port, v = 2CL2 + α/L2. Compare the fine link and predictor in that same macro gauge and use the triangle inequal- ity to get chord(U(x, i)Q−1) ≤(u + v) port. Comparing Q with the desired coarse link V then adds precisely its wire discrepancy, with coefficient one. This proves the shell comparison with constant u+v uniformly in the period and in the given forest. The room hypotheses supply its size assumptions, so the same constant is available in the local driving inequality. 60 ## Source page 63 A Wilson Yang–Mills construction J. Rodgers 6.20 The local driving inequality Let lens(u) be the active part of the fine tangent field. At a coarse site o define current(o, u) as the square root of the sum of ∥lens(u)e∥2 over the radius-3B fine-link neighborhood, and response(o, u) as the square root of the sum of squared selected motion rows over radius macroR+2. A summand’s norm is bounded by the corresponding square root because all squared norms are nonnegative. Fix a hot fine link e and put o = parent(e). Suppose a gauge z makes the background links in radius 2B about o have chords at most h. Every row root within macroR+2 of o is selected in the identity orientation: hotness puts e far enough from the seed set, and the room radius leaves this entire neighborhood clear. Its oriented motion norm is bounded by response(o, u). Apply the transported-to-flat row comparison to the gauged background and the gauged lens. The required tangent values lie in the radius-3B neighborhood of o, so their norms are bounded by current(o, u). Gauge covariance and norm preservation give, at each such row, ∥barFieldR(Ad(z) lens(u))∥≤Eo, Eo = response(o, u) + Kst · 2 colorScale ·(B + 1)h · current(o, u). The local link estimate for a tree-slice tangent field w with zero trunk values states ∥we∥≤K X qinmacro box X plaquettesp ∥curl(w)(q,p)∥ +∥predictorLinearBill(w)(o,axis(e))∥. Here K is the common local comparison constant for the actual predictor tags; the gauge- flattening and local-driving arguments in this chapter supply it. Apply this estimate to w = Ad(z) lens(u). It vanishes on trunks, since these are not hot. The direct-curvature rows in the preceding bound imply ∥curl(w)∥ ≤ Eo/ √ direct, and the noise row implies ∥predictorLinearBill(w)∥≤Eo/√penalty. All macro-box plaquette roots belong to the radius macroR+2 neighborhood. Thus ∥lens(u)e∥≤ZEo, Z = K| box(macroR)||PlaqAxes|/ √ direct + 1/ p penalty. This proves the local driving estimate used in the plate floor. A clean-patch gauge provides h of rate −1/2 with its radius factor already included in the flattening bound; hence the coefficient of current tends to zero fast enough for the squared-neighborhood absorption previously described. The tree-slice link estimate in this chapter gives the constant uniformly in the tags. 6.21 Transported affine rows For coarse sites x,y choose the coordinate-order route from 0 to the centered lift of y−x and start it at x. Its endpoint is y and its length is at most 4 separation(x, y), since each of its four coordinate lengths is bounded by that separation. Let cx(V ; y) be its actual parallel transport in V. For a fine tangent field u, spin each link value at its coarse parent y by Ad(cx(V ; y)), then apply the cutoff flat linear row rooted at x. This defines the straightened row. The signed oriented row first changes the input coordinates by the actual tangent transformation associated with the replica of V, then applies this straightened row in the signed background. These constructions are linear in u. The tangent change at a reversed link includes its back- ground adjoint factor and a minus sign; it is not a mere permutation of color coordinates. Both this 61 ## Source page 64 A Wilson Yang–Mills construction J. Rodgers tangent change and the carrier adjoint preserve the norm of each color vector. Thus the straight- ened and oriented rows have the same Kst∥u∥bound as the cutoffflat row. At V = 1, every carrier is 1 and the straightened row is exactly the cutoffflat row. Gauge covariance is exact. The carrier transforms as z(x)cx(V ; y)z(y)−1, because its endpoint is y. Combining its adjoint with the adjoint on u cancels (z(y))−1z(y), leaving the common factor Ad(z(x)) at the row root. The flat row commutes with this common color action. Signed tangent transformation commutes with the gauge change after pulling the gauge to the signed sites; hence the oriented row transforms by the adjoint at its signed root as well. The locality and comparison bounds can be proved without controlling the background outside the queried hull. A carrier to y depends only on the radius-d hull about x if 4 separation(x, y) ≤d. If link chords there are at most h ≥0, replace all links outside the hull by 1. The carrier is unchanged and its chord is at most length ·h ≤dh by the walk estimate. Similarly, if two tangent inputs differ by at most M on the radius R+predictionReach+4 input stencil, truncate their difference to zero outside this stencil. The resulting global field has norm at most M, and linearity and locality give a row difference at most Kst M. It follows that, when 4(R + predictionReach +4) ≤d and the input is bounded by M on the needed stencil, ∥straightenedR(V, u) −barFieldR(u)∥ ≤Kst · 2 colorScale ·dhM. Indeed each carrier changes the input by at most 2 colorScale·dh M; apply the preceding row comparison. The signed case has one additional error: at a reversed stored link the tangent trans- formation at V differs from that at 1 by at most 2 colorScale h M. A signed link root moves by at most one site, so radius d+1 bounds in the original orientation supply the required radius-d bounds after turning. Adding the carrier and reversal errors gives ∥orientedRowR(V, u) −orientedRowR(1, u)∥ ≤Kst · 2 colorScale ·(d + 1)hM. All fields in this inequality refer to the same signed row and input; no independent branch outcome is selected. For a single-link input of color a, the signed tangent transformation and the carrier spin still produce a single-link input with color norm ∥a∥. Its parent distance from the row root differs from the original signed distance by at most one. The flat impulse estimate above therefore yields exp(widthtop · distAt(link, signedRoot)) ∥orientedRowR(V, singlelinka)∥ ≤2Kst exp(widthtop(predictionReach +7))∥a∥. This bound requires no smallness of V. Locality at R = till(x) follows from the same carrier reach and the room inequality, using a half-sized hull before the signed change to absorb its one-site root shift. 6.22 Affine accuracy on the protected stencil The local hypothesis here records both actual field agreement and a tangent bound. Say that U is grafted to V by u on a radius-d hull if it agrees there with exp(u) replica(V ) and ∥ui∥≤H on the same hull. This condition is preserved by a coarse gauge transformation with u changed by its adjoint action. Under a signed coordinate change it is preserved on radius d−1, because the stored 62 ## Source page 65 A Wilson Yang–Mills construction J. Rodgers positive branch of a reversed link may shift the root by one site. Restriction to a smaller hull and enlargement of H also preserve it. Fix a row p, its root i, d = B/4 and R = till(x). Suppose V agrees with exp b on that hull, ∥b∥≤H, and U is grafted by u there. Replace u outside this hull by zero to obtain v with ∥v∥≤H, and set a = linearCopy(b), so ∥a∥≤H. The logarithm-product estimate supplies w with exp w = exp v exp a, ∥w −v −a∥≤JH2, ∥w∥≤(1 + J)H for eventually small H. The equality is linkwise and uses the exact replica exponential iden- tity. Locality of the entry row now identifies its actual value C with entry(exp w,exp b), and its background value Y with entry(exp a,exp b). Let E0 bound the error between these two logarithmic entries and their corresponding full linear costs; let M0 bound the full linear cost for w. These are the uniform entry-near estimates at the enlarged envelope (1+J)H. Then ∥C∥≤M0 + E0. To estimate the affine remainder write z = linearCost(w, b), z0 = linearCost(a, b). Linearity gives the exact decomposition C −Y −straightened(V, u) = (C −z) + uncut(w −v −a) + (z0 −Y ) +(uncut(v) −truncated(v)) +(truncated(v) −straightened(V, u)). The five terms have bounds E0, q1, E0, q2, q3 respectively, where, for the common stencil constant K and exponential-chart constant A, q1 = KJH2, q2 = KH exp(−width · R), q3 = K · 2 colorScale ·B · AH2. The first uses the full stencil bound on w−v−a. The second uses its exponential tail bound. For the third, V’s links in the hull have chords ≤AH; the adjoint transport comparison bounds the background-to-flat row difference by the displayed radius-times-quadratic error. Locality first replaces u by v, so the bound never uses u outside the hull. The room inequality 4(R + predictionReach +4) ≤d ensures that every needed input lies inside it. Consequently the common affine error can be chosen as Eaff= 2E0 + q1 + q2 + q3. It also bounds ∥Y −linearCost(a, b)∥. If H has rate −1/2, the entry-near estimate gives E0 rate −1 and M0 rate −1/2; the quadratic, cutoff, and radius-weighted quadratic terms all have rate −1. Thus M0 + E0 has rate −1/2 and Eaff has rate −1. The predictor entry-near estimate and the uniform full and tail stencil bounds of this chapter supply the constants used here. 6.23 The protected mean and its stationary comparison The protected mean is the stationary comparison variable retained when vertical fluctuations are integrated. Its affine accuracy is tested on the prescribed stencil. The proof first estimates the discrepancy of the actual nonlinear rows from their flat counterparts, then inserts those bounds in the stationary identity. The cancellation concerns these actual rows; it does not assert that every nonlinear minimizer is equal to the flat one. 63 ## Source page 66 A Wilson Yang–Mills construction J. Rodgers Here the scale, geometry, selected rows, and gauge forest are held fixed throughout. Write A for the counting-coordinate plate, D for the selected linear form, b for its linear bill, and S for the diagonal dormant-coordinate operator. For every fine tangent field v the pricing identity gives Av + b = DTpriced(v) + Sv. Coordinates are expressed in the fixed orthonormal color basis. All norms of fields in this argument are supremum norms; a color-vector norm is bounded by the sum of the absolute values of its color coordinates. Let Y bound both the row sums of D and the row sums of DT. Its matrix entries vanish beyond distance B/4. We first produce a comparison field in the same gauge as the actual background. A clean background on a ball of radius d+1, with d ≤32B, admits an axial gauge in which each link in the radius-d ball has chord at most u(x) = 16(2(32B(x) + 1) + 1)5|PlaqAxes| · 2 fiery(G, x, L). The axial-patch estimate supplies this bound using the clean plaquette toll and the room temper- ature comparison. The local logarithm extension then gives a globally defined bounded logarithmic field a agreeing with that gauged background on the smaller ball, with ∥a∥≤C u(x). Thus the bound w = C u has rate −1/2. The axial-gauge and bounded logarithmic-extension argument of this chapter proves this assertion. For clarity, the logarithm-product estimate used by the subsequent comparisons is a genuine group-chart estimate. Set h(p, q) = log(exp p exp q). Differentiating at (0,0), using the derivative of the matrix exponential and the inverse-chart identity, gives Dh(0)(p, q) = p + q. Analytic Taylor expansion on a sufficiently small ball therefore gives, with fixed positive K,r, exp h(p, q) = exp p exp q, ∥h(p, q) −p −q∥≤K max(∥p∥, ∥q∥)2, ∥h(p, q)∥≤K max(∥p∥, ∥q∥). Applying this link by link proves the same uniform bounds for the logarithm of two successive increments. No identification of a group-valued density with a tangent vector is made here. Let spring(a) be the stationary response with zero values on the chosen forest trunks. It is linear in a. The stationary response estimate bounds each color coordinate for ∥a∥≤1 by C0 = Response.total(j, η, C) + exp η. Consequently its field norm is at most Cs∥a∥, where Cs = max(1, max(0, C0) dim(Color)). Indeed this is the coordinate bound summed over colors for unit input; rescaling a by ∥a∥−1 proves the general case, and linearity handles a = 0. Signed coordinate changes preserve the field bound and commute with the linear copy. The latter assertion follows either from the copy formula or by differentiating the exact identity for its exponentiated replicas. If z is the axial gauge transformation, put v = Ad(z−1) spring(a). Then v vanishes on trunks and ∥v∥≤Cs w. Gauge covariance of the selected path, away from the seed region, identifies its value at v with request(gauge(z, V )) + orientedRow(gauge(z, V ), spring(a)). 64 ## Source page 67 A Wilson Yang–Mills construction J. Rodgers The source uses one common clean hull for all rows and columns subsequently read. On this hull the uncut stationary identity is exactly linearCost(linearCopy(a), a) −linearGuide(a) +uncutField(spring(a)) = 0. To see this, the stationary shifted field is the sum of the linear copy and spring(a). Linearity of the uncut field and the defining stationary equation identify its image with linearGuide(a). This proves the displayed cancellation before any estimate is applied. Subtract this identity from the gauged real path. There are four errors: the base-stroke linear approximation, the guide linear approximation, replacement of the background-oriented row by the flat truncated row, and replacement of the truncated row by the uncut row. Denote the first two envelopes by A1 and A2; the guide and local affine comparison estimates give each rate −1. If Cch bounds chord(exp a) by Cch∥a∥and Kst bounds the stationary stencils, the last two errors are bounded respectively by U1x = Kst · 2 colorScale ·(B(x) + 1) · Cchw(x) · Csw(x), U2x = KstCsw(x) exp(−width · till(x)). The first has rate −1 because it is quadratic in w with only the radius factor; the second has rate −1 by the cutoffdecay. Hence A0 = A1 + A2 + U1 + U2 has rate −1 and bounds every path of v whose root lies outside the 20B seed neighborhood and whose radius-2B background hull agrees with exp a. These bounds use the previously specified guide, affine-row, stencil, and stationary identities on that same hull. Now let P be the nonsymmetric inverse-column pad, E = AP −I, and μ = −P Tb. Symmetry of A gives ET = P TA −I, whence the exact identity μ −v = ETv −P T(Av + b). For a coordinate i, only the column window of P at i is read by the second term. If |(Av +b)j| ≤ M on that window and ∥v∥≤W, the inverse-window estimates already proved imply |μi −vi| ≤payE · M + scraps(B) · W. This uses only the elementary inequality | P j Mij tj| ≤K P j |Mij| when |tj| ≤K on the nonzero entries of the row. It requires no bound on t outside that row support. For coordinates outside the 11B seed neighborhood, S v = 0: on a trunk v = 0, and any other such coordinate is active. The pricing identity and the transpose row bound therefore show that path bounds A0 within distance B imply |(Av + b)i| ≤Y · scale · A0. For a selected path p, locality and the row bound of D also imply ∥realPathp(u) −realPathp(v)∥ ≤dim(Color)scale−1Y U whenever |ui −vi| ≤U on the radius-B coordinate neighborhood of its root. 65 ## Source page 68 A Wilson Yang–Mills construction J. Rodgers Take a path root outside the 30B seed neighborhood. Its clean hull has radius 8B+1. Flatten on radius 8B and use the resulting single comparison field v for every coordinate needed below. Its paths within distance 6B satisfy the A0 bound. Each column window read at distance B from the root lies within distance 3B, so the residual bound above applies throughout that window. All these rows remain outside the required 11B seed neighborhood. Thus, with W = Csw, M = Y scaleA0, U = payE · M + scraps(B) · W, Emean = A0 + dim(Color)scale−1Y U, we obtain ∥realPathp(μ)∥≤Emean. All envelopes are eventually nonnegative. The scraps bound at any prescribed negative power gives rate −1 for U and hence for Emean. The root is selected by the same seed-distance condition. This proves the protected-mean estimate used in the watch argument, with one common choice of gauge and comparison field for its entire finite stencil. The axial patch estimate follows by lifting the torus patch to its radius-r cover box. The axial gauge construction bounds the total edge chord by 16(2r+1) times the face-chord sum. If every face chord is at most q, the sum is at most (2r + 1)4 |PlaqAxes| q. Descend the gauge to the torus using injectivity on the box when 2r < n. Both endpoints of every radius-d bond lie in that box when d + 1 ≤r, and each individual edge chord is bounded by the total nonnegative sum. Thus the bound is 16(2r + 1)5 |PlaqAxes| q, as used above. 6.24 Local normalization differences The condensation letter at a fine link e is condensation(e) = [joint(Ablank, window; e) −joint(A, window; e)]/2. The blank geometry has the same forest, scale, windows and tags, but background 1 and empty seeds. Its hot links are exactly the nontrunks, and all its rows are selected. No second forest or tag choice is made. We first compare joint diagnostics on the same finite window U. For positive plates A,Z let fA = AU −I and gA(s) = I + s(AU −I), with 0 ≤s ≤1, and similarly for Z. The window intensity is fA gA(s)−1, evaluated at its distinguished diagonal coordinate. Suppose all entries of A−Z on U have absolute value at most e0 ≥0. The census bounds |U| ≤∆= ballCost(B + 2)m, so the row sum of fA −fZ and of gA −gZ is at most d = ∆e0. Let Cinv = payE bound both inverse row sums uniformly along this interpolation, and let Q+1 bound the row sum of fZ. The resolvent identity gives g−1 Z −g−1 A = g−1 Z (gA −gZ)g−1 A , so its row sum is at most C2 invd. Decomposing the difference of the two intensities into (fA − fZ)g−1 A −fZ(g−1 Z −g−1 A ) yields the diagonal bound pulseHead(e0) = [Cinv + (Q + 1)C2 inv] · ∆e0. All inverse matrices here are the principal-window inverses at that same s. Integrating from 0 to 1 and summing over the color coordinates of e gives |joint(A; e) −joint(Z; e)| ≤dim(Color)pulseHead(e0). 66 ## Source page 69 A Wilson Yang–Mills construction J. Rodgers Continuity in s, supplied by positivity of the interpolated principal matrices, justifies subtraction of these integrals. The same calculation without taking a difference bounds each individual joint diagnostic by dim(Color)(Q + 1)Cinv. To apply the local difference bound, take e outside the 30B seed neighborhood and flatten its background in a single radius-6B patch. If the resulting link chords are at most h0, compare each actual gauged form entry to the blank one by testing on a single coordinate impulse. Trunk columns are zero in both forms. A nontrunk column in the needed neighborhood is hot in both geometries. Rows farther than B/4 from that column vanish in both; all other rows are selected by the seed margin. The transported-row comparison already proved therefore gives |Dgauged(p, i) −Dblank(p, i)| ≤swell, swell = scale · Kst · 2 colorScale ·(B + 1)h0. The dormant diagonal terms of their plates agree in this neighborhood. Expand DTD entrywise using ab −cd = (a −c)b + (b −d)c. The two transpose row sums are at most Y, so |Agauged(i, j) −Ablank(i, j)| ≤2swellY =: repair throughout every window read at e. This is an entrywise comparison only on the patch, with the census factor retained in pulseHead. Gauge invariance of joint follows by conjugating the plate by the block diagonal orthogonal color action, which preserves every link window. The same conjugation preserves its principal inverse, interpolation intensity and color trace at e, hence its integrated diagnostic. Thus the gauged plate can replace the actual plate in the condensation difference. Flattening gives h0 = Zch w(x) with w of rate −1/2. Hence repair has rate −1/2, including its radius factor. The window census ∆has rate zero. The envelope Econd(x) = dim(Color)pulseHead(repair(x)) therefore has rate −1/2 and bounds the absolute condensation outside the 30B seed neighbor- hood; dropping the factor 1/2 only enlarges the bound. This derives the remote normalization estimate used below from the local form comparison and the already established inverse-window bounds. 6.25 Exact locality of the cut gaps Every walk of length at most R reads only the radius-R hull of its starting point, in any projection for which neighboring sites have projected separation at most one. This includes stored negative links: their positive-branch root is one step behind the current point, still paid by the step being read. Induction on the walk proves equality of its transport for fields agreeing on that hull. A retained plotted slot has its transport length at most R, and its final plaquette needs four more steps. Thus plotted fields agree under radius R+4 agreement. The cutoffF,I,N kernel rows are banded at radius R and their entries read only the radius-R coarse hull about the row root. For the fine F row this follows by lifting its designated route in the replica; its endpoint projects to the coarse column, and the route-length cutoffbounds that projected separation. The lane rows use the same argument on their coarse route. For a banded row F(V ), its transpose at output j reads input rows within distance R. Agreement of V,V′ on radius 2R about j makes each such row entry equal; input values need agree only at those rows. Likewise F(V )TFV w is unchanged by this background replacement on that hull, with 67 ## Source page 70 A Wilson Yang–Mills construction J. Rodgers the entire arbitrary array w held fixed. The inner row coefficients agree individually, so no locality assumption on w is needed. Apply this to lGap. The fine plotted inputs at relevant rows require at most R further travel plus their four plaquette steps; the noise input additionally queries its actual predictor through predictionReach. The coarse plotted term needs R+4. Therefore common fine/coarse agreement on D ≥3R + 4 + predictionReach at the tested anchor makes the whole lGap equal, with the same actual predictor tags on both sides. For dGap the three banded row-square operators are unchanged on radius 2R; hence the same D suffices for every fixed input array w. These prove the local gap identities used in the chart transfer, without changing any clipping indicators in an unrelated data array. 6.26 A reliable cutofflogarithm Choose a smooth profile χ with values in [0,1], equal to one on the inner unit interval and zero beyond its prescribed outer interval, and set cutLogd(U) = χ.innerShare(d, chord(U)) · log(mat(U)). Here innerShare evaluates χ at (chord(U)/d)2, so it uses a smooth squared matrix norm. Its nonzero values imply chord(U) < 2d. The ordinary logarithm is only required to have its chart properties on this latter shell. The local logarithm estimates give C, s > 0 with exp(log U) = U and ∥log U∥≤C chord(U) when chord(U) ≤s. Choose also a matrix radius t where log is analytic and a color radius R where log(exp a) = a. Set r = 1 2 min(s, t/ colorScale, R/C), γ = C. For every 0 < d ≤r, the entire chord shell ≤2d satisfies all three chart conditions. In particular the logarithms there have norms ≤R. Applying log(exp a) = a to a = Ad(v)log U and a = −log U, whose norms are unchanged, and using exponential conjugation and inversion, proves log(vUv−1) = Ad(v)log U and log(U−1) = −log U on the shell. On chord(U) ≤d the profile equals one, so the cutofflogarithm is the actual inverse chart. At every U its norm is ≤γ · 2d: if the profile vanishes this is immediate, and otherwise use its value ≤1 and the shell logarithm bound. Chord conjugation and inversion invariance extend the two equivariance identities to all U, splitting into the zero and nonzero profile cases. For any smoothly varying matrix-valued group input, the cutofflogarithm is smooth. Inside the closed outer shell this is composition with the analytic log and the smooth squared-norm profile. Outside it, continuity of chord gives a neighborhood where the profile vanishes identically. The same argument proves continuity, hence Borel measurability, as a function on G. Thus no regularity of the chosen logarithm outside its chart is used. 6.27 The sector input and its grafted signal Fix a forest F, coarse background V, tags k and frozen slice values Y. A fine link is hot when it is not a trunk and neither of its coarse endpoints touches the 10B seed dome. Define input(u) by putting exp(ue)(replica(V ))e on hot links, the fixed Y values on other nontrunks, and the prescribed tree value on trunks. This is on the same tree slice for every u. 68 ## Source page 71 A Wilson Yang–Mills construction J. Rodgers Let small(e, U) be the cut logarithm of Ue (replica(V ))−1 e at radius width·t. Define saved to be this logarithm on frozen nontrunks and zero elsewhere; lens(u) is u on hot links and zero elsewhere. The signal is saved+lens(u). Thus signal equals u on hot links and zero on trunks. Shell access gives two facts, uniformly over every forest and actual input: a reliable logarithm on the chord shell of radius width·t, and the shell displacement bound chord(Ue replica(V )−1 e ) ≤ width · t whenever the local Interior condition holds. It is available by choosing width equal to the previously proved shellPrice and then taking x large enough that width·fiery lies in the reliable logarithm radius. Its constants are chosen once after L. On a link satisfying that Interior condition, input(u) = exp(signal(u)) replica(V ). For a hot link this is its definition. On a trunk both the signal and the replica trunk increment are zero or identity as appropriate. On a frozen nontrunk the shell bound makes the cutofflogarithm uncut and its exponential recovers the actual discrepancy, giving the same equality. The reliable cut logarithm also supplies the all-argument bound ∥savede∥≤γ · 2(width · t). Hence local Interior conditions for every link of a radius-B hull, tangent bounds ∥ue∥≤M at its hot links, and γ ·2(width·t) ≤M imply the grafted condition on that entire hull with signal(u). Frozen links use the saved bound; hot links use the given tangent bound. There is a simpler remote version requiring no Interior premise. If the hull center is outside the (11B+1) seed neighborhood, every nontrunk in its radius-B hull is hot: otherwise one endpoint would touch the 10B dome, and its parent lies within one further site, contradicting the separation. Only hot links and trunks occur in that hull. The defining identities above and hot tangent bound therefore prove grafting directly for any M ≥0. These are the required and distant grafted constructions used in the packet annulus and the optional remote letters, respectively. 6.28 A common chart on the grafted hull Suppose the coarse background V is clean on radius d+1 and U is grafted to V by a tangent field bounded by H(x) on radius d, with H of rate −1/2. Flatten V there to exp b in gauge g, with ∥b∥≤v(x) of rate −1/2. If Dch is the exponential chord constant, the gauged coarse links have chords ≤Dch v. Gauge covariance of grafting writes the gauged fine links as exp(Ad(g)u) times the replica of the gauged coarse field. Their chords are therefore at most Dch H + Dch v by the product triangle inequality; replica links are either 1 or a coarse link. The single envelope W = H + v + DchH + Dchv bounds b, both kinds of link chord, and H itself, and has rate −1/2. All chart radii are met eventually. This is the common coarse/fine local chart; it imposes no condition outside the hull. Apply affine accuracy in this gauge after restricting the hull from B/2 to B/4. The enlarged envelope W supplies both its background-log and tangent bounds. Entry covariance, replica co- variance and covariance of the straightened row rotate the entire affine difference by the same root adjoint. Norm preservation then transports the estimate back to the original fields. This proves the strokes bounds M of rate −1/2 and E of rate −1 used in the soft remainder argument. The same chart proves that the actual local branch tags and holonomies lie in any prescribed small neighborhood eventually. Given δ > 0, the finite-stencil branch estimate supplies a radius r and reach k such that fine links of chord ≤r imply site mass 1, pin mass 1, and predictor chord ≤δ/2. Choose x so that k ≤B/2 and W ≤min(r, δ/4). Apply this estimate to the gauged fine field. Site and pin masses are gauge invariant. The actual wire discrepancy is also invariant and 69 ## Source page 72 A Wilson Yang–Mills construction J. Rodgers is at most the sum of coarse-link chord W and predictor chord δ/2, hence at most δ. For a fine plaquette based at the chosen parent, replace links outside the radius-B/2 hull by 1. Its four-step product is unchanged, and its chord is ≤4W ≤δ. Thus the original fine plaquette toll and the actual wire both meet the small-curvature threshold. The uniform branch-cutoffand predictor estimates in this chapter give this finite-stencil estimate. 6.29 Uniform local branch activation At the identity cover field the minimizing frame is 1 for every frame tag, the connector array is 1, and the minimizing predictor is 1. Both minimum tests are strict there. Continuity of the selected minimizers on their open minimum branches therefore gives a neighborhood of the identity cover field on which the site-frame minimum test holds and every connector predictor lies within any prescribed positive chord tolerance δ. This neighborhood can be chosen simultaneously for all axes and both endpoint frame tags because these alphabets are finite. On it every predictor fallback tag yields the same best value. A product-topology neighborhood restricts only finitely many cover links. Choose a finite set I of such links and a common positive chord radius r small enough to meet all their coordinate neighborhoods; the conversion between chord and matrix norm is the fixed factor colorScale. Set d to the largest length of the coordinate-order route from 0 to a root in I. For any torus input, expand it at the chosen embedded coarse root. Every required cover link projects to a fine link whose coarse-parent distance from that root is at most its route length, hence at most d. Thus chord bounds ≤r on this parent-distance hull place the expanded field in the chosen cover neighborhood, independently of the period. The actual connector equals the cover test connector with its actual two endpoint tags. Con- sequently its predictor chord is ≤δ. On the minimum branches the normalized site mass and pin mass equal one: their unnormalized masses are respectively the reciprocals of the numbers of frame tags and connector tags. This proves the finite-stencil branch activation used in the local chart-mark argument without requiring any tags or defaults away from the finite hull. 6.30 Global mismatch estimates from the flat identities Let F be a cut transported row and F0 its full flat comparison, with both row and column sums ≤C and difference sums ≤Cq0. Suppose an input f has linear comparison t with ∥tj∥≤Ch and ∥fj −tj∥≤Chq0. The exact decomposition F Tf −F T 0 t = F T(f −t) + (F −F0)Tt and the column-sum estimates give norm ≤2C2h q0 at every output coordinate. For arbitrary bounded input w, applying this argument to f = Fw and t = F0w gives ∥F TFw −F T 0 F0w∥≤2C2∥w∥∞q0. Choose C to dominate the fine plotted, coarse plotted, predictor-noise and three kernel constants previously proved. Choose the chart radius as their common minimum, and choose τ = min(τkernel, widthtop, widthbelow/L) > 0, q0 = widen(R, h, τ). The plotted-field and predictor Taylor comparisons then all have the common input bounds above; the noise quadratic remainder uses h ≤q0. The F,I,N kernel comparisons likewise have the 70 ## Source page 73 A Wilson Yang–Mills construction J. Rodgers common row and column estimates. The full flat cross-roll identity cancels the linear terms in lGap, leaving two transpose/input errors and one coarse plotted error. Hence ∥lGap∥≤(4C2 + C)hq0. The full flat diagonal-roll identity cancels w against the sum of its three flat row-square operators. Applying the second estimate to F,I,N gives ∥dGap(w)∥≤6C2∥w∥∞q0. The common constant 6C2 + C bounds both. These estimates use the uncut flat identities, so both the transport error and the cutofferror are included in q0. They hold for every cutoffR, including zero, and allow winding paths and arbitrarily short periods. The local transfer described next uses exactly these same gaps. 6.31 Localizing the mismatch bounds Assume, in one gauge, both fine and coarse links have chords ≤h on the radius-d hull of a tested row, with 3R +4+predictionReach ≤d. The bounded logarithm extension constructs globally bounded fields a,b agreeing there with those actual links, and ∥a∥, ∥b∥≤Alog h. Put X = max(1, Alog) and choose h ≤min(slog, rglobal/X). The global logarithmic mismatch estimate then applies at envelope Xh. Locality identifies its tested gaps with the actual gauged gaps because all their reads are in the hull. Gauge covariance rotates lGap by the root adjoint. For dGap it rotates both the row and its arbitrary input data w; this preserves the bound ∥wj∥≤q. Finally widen(R, Xh, τ) ≤X widen(R, h, τ) because X ≥1 and both summands are nonnegative. Thus increasing the global constant C to CX2 gives simultaneously ∥lGap∥≤CX2h widen(R, h, τ), ∥dGap(w)∥≤CX2q widen(R, h, τ). Only the independent supremum bound q on w is used. This matters when w is the actual thinned data: changing the links outside the hull need not preserve that entire data array. The global mismatch estimate and exact local gap identities proved here supply these inputs. 6.32 The quartic curvature deficit The energy estimates separate useful positive income from the remainders that must be paid. “In- come”, “price”, “loan” and “tariff” are bookkeeping names for nonnegative numerical bounds in this construction. They do not denote additional physical degrees of freedom. In particular a packet tariffis a summable upper bound on a packet’s contribution after its annular and normalization expenses have been assigned. The inequalities below show exactly which positive curvature terms absorb which remainders. Let Plie denote orthogonal projection onto the real matrix Lie algebra. For every group element U, not only those near 1, Plie(mat(U) −1) = colorMat(curvature(U)). 71 ## Source page 74 A Wilson Yang–Mills construction J. Rodgers Indeed subtract the skew part (mat(U) −mat(U)∗)/2 from mat(U) −I. The remainder is self- adjoint. Its real Frobenius inner product with every skew-adjoint Lie algebra matrix is zero, by cyclicity of trace. Projecting therefore retains just the Lie projection of the skew part, which is the defining curvature. Pythagoras for tangent and normal projections gives the exact identity colorScale2[chord(U)2 −∥curvature(U)∥2] = ∥mat(U) −1 −Plie(mat(U) −1)∥2. The analytic focus fixes each group matrix near 1, and its derivative there at 1 is Plie. Its Taylor remainder consequently gives constants C, r > 0 such that ∥mat(U) −1 −Plie(mat(U) −1)∥≤C∥mat(U) −1∥2 when ∥mat(U) −1∥≤r. This is an ambient analytic estimate restricted to group points; it requires no logarithm branch on an entire configuration. Substituting the chord normalization in the squared bound and setting K = C2 colorScale2 proves chord(U)2 −K chord(U)4 ≤∥curvature(U)∥2. Choose δ = min(r/ colorScale, p 1/(4K)). If chord(U) ≤δ, then K chord(U)2 ≤1/4, so the last bound is at least 3 4 chord(U)2. Both constants depend only on G and are chosen before the block length. This proves the small-curvature input to the soft and coarse-C estimates. 6.33 Soft curvature and noise remainders Define flaw(g) = [chord(g)2 −∥curvature(g)∥2]/2. It is nonnegative by the curvature norm bound. On a retained fine plaquette whose toll is at most its clipping threshold, expanding the two di- rect/path moment coordinates gives the exact identity crown = direct · flaw(faceU) + X i∈Bool ∥entry(face,i)U∥2/2. Adding the guide cross term and its square completes each square. Thus workF equals direct·flaw plus the two half-squares of bill = entry −guide. The noise identity holds without a retention hypothesis: workI equals penalty·flaw(V Q−1) plus the single bill half-square. These are algebraic identities, using the actual Q = prediction(U, k) and the same guide. The affine comparison already proved bounds entry by M(x) of rate −1/2 and bill−reply by E(x) of rate −1; subtracting the common guide does not change that affine error. The coarse maximum condition bounds guide by A·fiery, so bill has norm at most m = M + A · fiery. For vectors v,w with ∥v∥≤m and ∥v −w∥≤E, factor the difference of squared norms to obtain |∥v∥2/2 −∥w∥2/2| ≤E(2m + E)/2 =: Z. This has rate −3/2. The small-curvature estimate supplies J > 0 and a chord neighborhood with chord(g)2 −J chord(g)4 ≤∥curvature(g)∥2, 3 4 chord(g)2 ≤∥curvature(g)∥2. The local chart-mark estimate places the plaquette holonomy and V Q−1 in that neighborhood. A scaled curvature bound ∥√k curvature(g)∥≤M therefore gives chord(g) ≤2M/√k and 72 ## Source page 75 A Wilson Yang–Mills construction J. Rodgers |flaw(g)| ≤(J/2)(2M/ √ k)4. For the fine and noise terms put respectively X = direct · (J/2)(2M/ √ direct)4, Y = penalty · (J/2)(2M/ p penalty)4. Both have rate −2. The common envelope F = X +Y +2Z thus has rate −3/2 and bounds the difference between each soft energy and its corresponding affine reply half-square sum. The two in the last term pays the two fine coordinates; the noise row needs only one. The small-curvature and local branch-activation estimates above supply this reduction. 6.34 Outer N, C and mismatch corrections On a clean radius-B/4+1 hull, flatten the coarse field and use its bounded logarithmic extension. The guide comparison for the N row has zero linear term and envelope e of rate −1. Gauge covariance preserves its norm. Since workN is its half-square, |workN| ≤e2/2 has rate −2. For the C row, the hypothesis toll(V ) ≤temp makes its clipping multiplier exactly one, also at zero toll. The exact C-row formula and curvature norm bound then show workC ≤0. The quartic curvature inequality above gives |workC| ≤directbelowJ/2(2 fiery)4, because temp ≤2 fiery. This envelope has rate −2. For the L and D mismatch rows, use the common local gauge and logarithmic extensions of the actual fine and coarse inputs on their clean grafted hull. The layered extension gives a common envelope W of rate −1/2. The interior kernel comparison supplies constants C, τ > 0 and, writing e = widen(R, W, τ), ∥lGap∥≤CWe, ∥dGap(vision)∥≤Cde, d = A · fiery . The maximum condition gives ∥vision∥≤d. Hence Cauchy–Schwarz bounds each L inner product by M = d C W e and each D half-inner-product by V/2 = d C d e/2. There are two Bool coordinates, giving |workL| + |workD| ≤2M + V. Broadening preserves the rate −1/2 of W; both products have rate −3/2. This proves the asserted outer-envelope assembly from the layered-extension and interior-gap estimates, with actual U,V,Q shared by the two mismatch terms. 6.35 Light estimates on the annulus The light estimate is required in two forms: a lower bound at every row and an absolute bound when that row is selected. The source decomposes it into four already specified energy types: the soft curvature/noise remainder, the N correction, the coarse C correction, and the L/D corrections. Let their envelopes be E1, E2, E3, E4 respectively. The first and fourth have rate −3/2; the other two are weakened to that rate. Their sum Elight is nonnegative and has rate −3/2. 73 ## Source page 76 A Wilson Yang–Mills construction J. Rodgers For a selected curvature row, its extracted energy is exactly the sum of its two real-path half- squares. The soft curvature estimate consequently gives the required absolute remainder bound. For a selected noise row its extracted energy is its single real-path half-square, and the exact noise split reduces to the soft noise estimate. At an unselected curvature or noise row the extracted term is zero; the remaining energy is nonnegative, being the sum of a nonnegative flaw cost and half-squares. It therefore satisfies the lower bound without an absolute-value claim. The C row has no extracted parent contribution and is bounded by its C, L and D envelopes; the N row similarly uses its N envelope. These statements apply to a clean radius-B coarse hull and a grafted tangent bound w(x) of rate −1/2. Under a signed coordinate change first restrict the grafted hull to radius B/2+1, then turn it to obtain radius B/2; the extra site pays the reversed-link shift. The coarse maximum hypothesis is read in the same turned background. For a packet annulus, take a row in the 200B seed dome but outside the 2B dome, together with the Wide bounds through 300B and tangent control through 304B. Local Interior bounds and shell access give the required grafted condition, provided γ · 2(width · t) ≤w(x). The same Wide bounds give the clean hull and make the original debit equal light: retained curvature selectors are one by the actual local toll bounds. Thus debit ≥−Elight on the entire annulus. This uses no derivative of a hard selector and no globally small gauge. The soft-remainder, outer-correction and grafted-signal estimates of this chapter provide these four envelopes and the required shell access. 6.36 Physical operator and path-series bounds We record why the allowances and maximum responses in the energy argument are uniform in the torus and are chosen before L. Suppose a cover kernel between types α, β satisfies P z eη|z|∥Kαβ(z)∥≤C, with η ≥0. Each physical summand is a scalar kernel coefficient times adjoint transport along its prescribed route, subject to the endpoint congruence and route cutoff. Adjoint transport is an isometry, including on winding routes; its operator norm is at most one. Thus the summand is bounded by the absolute cover coefficient with its endpoint indicator. For a fixed input site and cover displacement, the endpoint congruence selects exactly one output site. For a fixed output site and displacement it selects exactly one input site. Interchanging the finite site sum and absolutely convergent cover sum therefore bounds a physical row by |β|C and a column by |α|C. This argument takes absolute values before periodization, retaining every displacement when endpoints coincide. For the fine-to-coarse kernel, the input type includes its fine residue and its two-layer plaquette type. The per-type cover price is C/L2; there are L4 residues. The row bound is consequently a/L2 and the column bound aL2. Noise and tight kernels have row and column bounds at most a common a. One may take a to be the maximum of one and the cover price multiplied by each of the three finite type cardinalities. This choice is independent of L, W. The support assertion concerns kernel incidence: a nonzero cut term has endpoint distance at most its route length, at most B/100000, hence at most B/50. For fine routes use the parent map, which takes adjacent fine sites to equal or adjacent coarse sites. The route from the lifted fine residue to Lz ends at the embedded coarse site parent(x)+wrap(z); this remains true for wrapping routes by additivity of wrapping and the cover route endpoint identity. No larger support claim about all data read by an operation is needed. For path shortening, retain the entire label index i, writing its coefficient and slot as (ci, si). Define the nonnegative matrix Mpu = X i |ci| 1{d length(si)≤R, landing(si,p)=u}. 74 ## Source page 77 A Wilson Yang–Mills construction J. Rodgers The weighted path hypothesis is P i |ci|eσlength(si) ≤P for each source plaquette type. It gives row sums at most P. For a fixed label and source type, the landing equation fixes its source site uniquely; summing over source types gives column sums at most qP. Absolute convergence permits all these finite-sum exchanges. Curvature norm is bounded by plaquette chord and adjoint transport preserves norm, so ∥shorten(U, p)∥≤ X u Mpu toll(U, u). If the selector keeps p, it bounds the toll at distance k by tek/ρ. Landing distance is at most slot length, and the stage hypothesis σρ ≥1 makes the weighted path sum applicable. Thus the shortening norm is at most Pt. The common stage capacity Astage = (q + 1)P + √ direct dominates row sums, column sums, and the direct clamp scale. Both branches of the thinned field therefore have norm at most Astaget: its shortening branch uses the kept estimate, and its direct branch uses the multiplier min(1, t/r) and curvature norm at most r. At r = 0 the curvature vanishes. The required reach margins hold eventually. If t = temp(x+a) and tek/ρ < 2, then t ≥1/√x + a implies k/ρ < log x once 2√x + a ≤x. For large log x, this gives k + 1 ≤B/50, since B ≥(log x)2. The cut path length till(x)+1 also lies below B/50. Parent distances cost at most the fine distance plus one, by the parent foreshortening inequality. These observations supply both the shortening stencil reach and the discarded-witness reach used above. Finally a = G.slopeConstant log L ≥0: the minus Killing form is a sum of squares, so the Casimir and slope constant are nonnegative. At fixed shift a, eventually temp(x) ≤2temp(x + a); use √x + a ≤2√x and monotonicity of the nonnegative logarithmic factor in the temperature. Put A = Atop + Abelow, and let C be the common operator allowance. The coarse vision has norm at most 2At. Applying the fine and noise operator row bounds to it gives ∥vision∥≤dt, ∥Mv∥≤dL−2t, ∥Mw∥≤dt, d = 2A(1 + C) > 0. Both stage lane counts may use this same d. It is fixed before L, W, while the common late cutoff is taken afterwards. These prove the operator and maximum hypotheses used throughout the preceding energy estimates. 6.37 Fine and noise income, including discarded rows Here is the summation argument producing the fine and noise inequalities used below. Write a > 0 for the direct quadratic coefficient, b > 0 for its linear coefficient, p > 0 for the noise penalty, and t > 0 for the fine threshold. Let ri be the plaquette chord, K the listed plaquettes with ri > t, and Z = P i∈K ri. The actual plot conditions imply that every listed plaquette outside K has t/4 ≤ri ≤t. Any plaquette with ri > t whose parent is in the required five-radius dome is listed. These are local statements about the one configuration under consideration. The two components of a fine row, with their corresponding coarse responses v0 i , v1 i , are Di = clipt(ri) + 1 2∥v0 i ∥2 −⟨v0 i , shooti⟩, Ei = 1kept(i) 1 2∥ui∥2 + 1 2∥v1 i ∥2 −⟨v1 i , ui⟩. Here ∥shooti ∥≤√a ri, and clipping is ar2 i /2 for ri ≤t, and btri otherwise. Suppose the previously constructed response bounds give ∥vν i ∥≤εt. Completing the scalar norm square gives Di ≥1 2(√a ri −∥v0 i ∥)2 (ri ≤t). 75 ## Source page 78 A Wilson Yang–Mills construction J. Rodgers Thus, if ε ≤√a/8, each listed small plaquette contributes at least at2/128. If ε√a ≤b/4, each large listed plaquette contributes at least 3btri/4. Unlisted rows in the summation region have ri ≤t and hence nonnegative direct part. Consequently X i Di ≥3b 4 tZ + a 128t2|costs \ K|. It remains essential to bound the discarded second components after summation. Let ρ > 0 be the fixed relaxation length and Hρ = (8ρ + 1)4 |PlaqAxes|. A row fails the kept condition only if some plaquette j satisfies rj > t exp(dist(i, j)/ρ). Such a row belongs to the ball about j of radius ⌊ρ log(rj/t)⌋, including all plaquette types. For y ≥1, this ball has at most Hρy elements: use 2⌊ρ log y⌋+ 1 ≤(8ρ + 1)e(log y)/4 and raise to the fourth power. The local witness-reach estimate places each witness j in the five- radius dome, so it belongs to K. The discarded set X therefore satisfies t|X| ≤HρZ. This count takes place on the original fine lattice and introduces no factor L4. Let Mij ≥0 be the previously bounded shortening majorant, with both row and column sums at most C. Its support-reach estimate shows that, for i ∈X, every contributing j /∈K has rj ≤t. Hence ∥ui∥≤Ct + X j∈K Mijrj, X i∈X ∥ui∥≤C(Hρ + 1)Z. Kept rows have Ei = ∥ui−v1 i ∥2/2 ≥0. Discarded rows satisfy Ei ≥−εt∥ui∥. Choosing εC(Hρ+1) ≤ b/4 yields X i (Di + Ei) ≥b 2tZ + a 128t2|costs \ K|. In particular this argument never presupposes that a discarded row has a good halo. For noise rows put ni = wirei. Their exact energy is bounded below by 1 2(√p ni −∥vi∥)2. On listed aisles the plot gives ni ≥wt. The uniform noise-response bound ∥vi∥≤Ct, together with C ≤√p w/2, therefore gives pn2 i /8 per listed aisle; all other noise rows remain nonnegative. The order of these choices matters. The response theorem supplies the common constant C before L is chosen and gives ε = C/L2. First take L large enough for the three displayed restrictions on ε; next take w ≥2C/√p; finally take the late scale parameter at those fixed values. The fine coefficient delivered by this construction is exactly a/128 > 0. 6.38 Macro multiplicity and the spending census The macro census used here has a period qualification, already included in the step hypotheses. Put R = j +8. The macro box radius is (j +7)L. If twice this radius is smaller than the fine period, two displacements in that box with the same periodic image coincide, coordinate by coordinate. Thus M(i, z) ≤1. A nonzero multiplicity also implies dist(i, parent z) ≤R: multiply the coarse distance by L, use the distance from the embedded base to z, at most (j + 7)L, and the fine site’s distance to its embedded parent, at most L/2. The triangle inequality gives the stated integer bound. Consequently X i M(i, z) ≤(2R + 1)4. 76 ## Source page 79 A Wilson Yang–Mills construction J. Rodgers The port identity follows by interchanging its finite cover-displacement and plaquette-type sums and grouping identical periodic plaquettes, with their multiplicities. The period hypothesis is used for the column bound, not for that identity. The store’s base set has at most four times its number of shares: a fine plaquette has at most four corner parents, a noise bond at most two pins, and each flag contributes one site. Hence any collection of coarse faces based in its radius-r dome has cardinal at most V (r) shares, V (r) = 4(2r + 1)4q. For B ≥1, V (2B) ≤V (2)B4. This proves the bounds on the outer and affected face sets, using respectively the original store and the spending store. The exact flag condition says each flag is touched by a coarse face with toll above the coarse temperature. Since a plaquette has at most four corners, the number of flags is at most four times the number of those faces. For completeness, every coarse face whose toll exceeds a fixed δ > 0 has an actual spending witness within distance R, once the scale parameter is late enough. Write T for the large spending threshold. If no spending base lies in that neighborhood, then every fine input in its macro port has toll at most T: listed inputs exceeding T would themselves be spending witnesses, while unlisted inputs have toll at most t/2 ≤T. Likewise every one of its four boundary wires is at most T, using the unlisted bound 2Wt ≤T. Its toll is then at most  4 + DL−2q| box((j + 7)L)|  T. At fixed L, W, this is eventually at most δ, a contradiction. All relevant inputs lie in the plot dome by the distance bounds above, so the local predictor comparison applies. The limits used here are explicit. With fine inverse coupling βF →∞, T = β−49/100 F →0, whereas t = β−1/2 F P(βF ) with P smaller eventually than every positive power. For each fixed m, use P(βF ) ≤β1/200 F and m ≤β1/200 F to obtain mt ≤T. Also B = ⌈(log x)2⌉→∞, allowing B ≥5R. One common pre-L census constant is N = V (R) + V (2) + R + (2R + 1)4. It simultaneously bounds the macro columns and reach, the number of δ-large coarse faces by N times spending shares, and the outer and affected face counts by NB4 times their respective shares. This supplies the census used in the departure and point-charge bounds. 6.39 Transfer of affected departure costs Fix the same plot, and let s now denote its affected coarse faces, with m = |s|. Use constants a, d, D ≥0 for the operator allowance, maximum response, and predictor-curvature comparison respectively. These letters are local to this argument. Put B = L−2 X swell(t) toll U + X aisles wire(Q, V ), u = p high.direct + d, v = √ low.direct + d, g = √p. Retain the two nonnegative excess arrays exactly on the fine swell at threshold t and on all listed noise aisles. The macro transfer estimate constructs a nonnegative coarse excess H such that, in the three-radius dome, toll V ≤rt + H, X H ≤b, r = macroSmall(j, D, L, W), b = macroLarge(N, D)B. 77 ## Source page 80 A Wilson Yang–Mills construction J. Rodgers This uses the actual usable-plaquette inequality toll V ≤fuel(Q, V ) + DL−2 port(U). The macro transfer estimate has the following explicit form. Let q = |PlaqAxes|, and let M(i, z) be the integer macro-path multiplicity, counting cover displacements separately even when their periodic images coincide. Its row sum is at most Sj,L = q | box((j + 7)L)|, while the macro-store census gives P i M(i, z) ≤N on the physical torus. The port identity is port(i, U) = P z M(i, z) toll(U, z). For the retained arrays E, F, set H(p) = X e∈∂p F(e) + DL−2 X z M(p.1, z)E(z). The boundary sum has four terms, retaining repetitions. For each fixed plaquette type and each one of its four boundary positions, the map from its base site to that bond is injective, including on the torus. Hence its total is at most 4q P F. Summing the macro term and using its column census gives X p H(p) ≤4q X F + DqNL−2 X E ≤(4q + DqN)B. Thus macroLarge(N, D) = 4q + DqN. In the five-radius dome, the plot conditions give toll U ≤t + E and wire(Q, V ) ≤2Wt + F. For unlisted inputs these follow from the plot’s upper thresholds; for listed inputs they follow by retaining the full input size in the excess. Every face in the three-radius dome is usable: its required coordinate box has radius one and is inside the plot’s listing dome. Each of its four boundary bonds is in the five-radius dome. The four wire terms therefore contribute at most 8Wt + P ∂p F. The macro stencil also stays inside the five-radius dome, by its reach bound and B ≥1 + N; its row sum bounds the small part of the port by Sj,Lt. The predictor-curvature comparison now proves the asserted split with macroSmall(j, D, L, W) = 8W + DL−2Sj,L. The row count follows simply by summing the defining indicator over all cover displacements and plaquette types. It allows coincident periodic images, so no injectivity of the cover box is assumed. The operator row and column bounds transfer the local input split to the 2m tested Boolean slots as follows: X ∥Φ∗Xf∥≤2m aL2ut + 2auB, X ∥(Φ′)∗Z∥≤2m ag(2Wt) + agB, X ∥Xn∥≤2mvrt + 2vb. For the first inequality, separate the uniform small input ut from the plotted nonnegative excess. The wide column bound costs aL2 on the small input, while the row bound a/L2 multiplies the total excess, at most 2u P E. Thus the large-input term retains its L−2 saving. The noise argument uses narrow row and column bounds a, and the coarse argument uses the plotted excess bound directly. In each case the stencil radius takes the tested two-radius dome into the input three- or five-radius dome, so only the given local split is used. The departure L is the slotwise inner product of the vision, of norm at most dt, with Φ∗Xf + (Φ′)∗Z −Xn. Cauchy–Schwarz gives its summed lower bound by minus dt times the sum of the three right sides above. The remaining departure D requires no large-input saving. Indeed its residual vector has norm at most (d + ad + ad + a2d)t, 78 ## Source page 81 A Wilson Yang–Mills construction J. Rodgers since the wide factor aL2 cancels the response bound dt/L2, while the narrow and tight terms cost adt and a2dt. Pairing with the vision and summing the two half-weighted slots gives workD(p) ≥−d0t2, d0 = d(d + ad + ad + a2d). All third-range coarse faces charged by the payments relation are contained in s. Their toll sum is at most mrt + b. As the coarse temperature is at most 2t, their linear charge can consequently be included in the same bound. Writing l= low.linear, the result is ltemp X tides toll V −talll(s) ≤Cpre tB + Cseat t2m, where the exact available choices are Cpre = d(2au + ag + 2v macroLarge(N, D)) + 2lmacroLarge(N, D), Cseat = 2d(aL2u + 2agW + vr) + d0 + 2lr. The first constant is fixed before L, W; the second may depend on both. The store census gives m ≤NB4 shares(spend). Thus the seat term is bounded by (NCseat + 1)t2B4 shares(spend), with a positive constant. This is precisely the affected-departure charge needed in the point-charge assembly. In particular the fine excess here is cut at t, while only the census uses the larger spending threshold. 6.40 The three coarse clipping ranges The curvature estimate above yields the coarse row bounds without differentiating its selector branches. Let r = toll(V, i), s = temp(x) ≥0, a = low.direct > 0, and b = low.linear. Put α = 1 if r = 0, and otherwise α = min(1, s/r). The shortening components cancel exactly: on a kept row their contribution is its shortening half-square, and on a discarded row both tested shortening terms vanish. Expanding the remaining direct component gives workC(i) = a(α −α2/2)∥curvature(V, i)∥2 −clips(r). Choose K, δ > 0, depending on the group alone, so that for r ≤δ, r2 −Kr4 ≤∥curvature ∥2, 3 4r2 ≤∥curvature ∥2. If r ≤s ≤δ, then α = 1 and workC(i) ≥−aKs4/2. If s < r, then 0 ≤α = s/r ≤1, so α −α2/2 ≥α/2. Therefore workC(i) ≥as 2r∥curvature ∥2 −bsr. In the middle range s < r ≤δ, the curvature lower bound and the clip hypothesis b ≤a/8 give workC(i) ≥asr/4. In the outer range r > δ, discard the nonnegative curvature term to obtain workC(i) ≥−bsr. Finally s ≤δ eventually. The supplied temperature estimate x temp(x)3 ≤1 implies this whenever x > 1/δ3. Thus the same late-scale choice gives all three bounds, with small-range debit coefficient aK/2 and a group-dependent fixed middle-range cutoffδ. 79 ## Source page 82 A Wilson Yang–Mills construction J. Rodgers 6.41 Threshold ratios and absorption of stitch errors All limits in this paragraph are taken after fixing L. Set y = βF = x + k, where k is the fixed coupling shift, B = ⌈(log x)2⌉, R = ⌊B/100000⌋, t = temp(y), T = y−49/100. The temperature is v−1/2 exp((log log v)2) for positive v in these late-scale calculations. For every fixed real shift k, temp(x + k)/temp(x) →1. Indeed its logarithm is (log log(x + k))2 −(log log x)2 −1 2(log(x + k) −log x). On [1, ∞) the derivative of (log v)2 has absolute value at most two. The mean value estimate bounds the first difference by twice | log(x+k)−log x|, which tends to zero. Exponentiation proves the ratio limit. For every r > 0 and integer n ≥0, Bn = o(xr) = o(yr), where the latter notation means Bn/yr →0. To see this directly, eventually B ≤2(log x)2, and every fixed power of log x is smaller than xr by a factor tending to zero. The fixed shift does not affect that comparison. The monitor bound gives, for every p > 0, v−1/2 ≤temp(v) ≤v−1/2+p eventually. It follows that Bntemp(x) →0 and Bnt/T →0, BnT →0, BnT 2/t →0. For the first limit choose p = 1/200, leaving the decaying power y−1/200; for the last use the lower bound on t, leaving y−48/100. In particular the square of T, rather than T itself, is the quantity small relative to t. For each τ > 0, e−τR ≤y−1 eventually. In fact B ≤100000R + 100000, while log y ≤2 log x eventually. Since (log x)2 ≤B, the quadratic growth of this lower bound for R dominates 2 log x/τ. Thus Bne−τR →0, Bne−τR/t →0. Define the stitch size h = CB10T for fixed C > 0, and its widening error e = (1 + R)h + e−τR. The preceding limits, with n + 10, n + 20, or n + 1 in place of n, give Bnh →0, Bnh2/t →0, Bne →0, Bnhe/t →0. For the last two statements use R ≤B, expand e, and bound each nonnegative summand by one of the already established limits; in the mixed tail term also use h →0. These are the precise limits that pay the unaffected-row stitch losses after multiplying by the dome census. They involve no torus period or number of store shares. 6.42 Finite-store budget inequalities Write a = lineartop, p = penalty, t = fiery and u = largeT ≥t. Let Ft be the listed fine plaquettes with toll > t, Fu those with toll > u, and R = P Ft toll. Then t|Ft| ≤R and u|Fu| ≤R. The supplied fine tally bound is (a/2)tR+d t2|costs minus Ft|. If m ≤a/8, m ≤d and cF ≤a/4, use 2m + cF ≤a/2 and split costs into Ft and its complement to obtain tallyF ≥mt2|costs| + mtu|Fu| + cF tR. 80 ## Source page 83 A Wilson Yang–Mills construction J. Rodgers This separately pays all original fine shares, the larger spend subset, and the fine mismatch fee. Every listed noise bond satisfies wire ≥Wt ≥t. With RI = P l isted wire2, this implies t2|aisles| ≤RI. For the spend subset wire > u, also tu| spenda isles| ≤RI. If C ≤pW/16, then C t · wire ≤(p/16) wire2 bondwise, since Wt ≤wire. Using m ≤p/32, the supplied noise bound tallyI ≥(p/8)RI therefore gives tallyI ≥mt2|aisles| + mtu| spendaisles | +Ct X listed wire . The listed-wire lower bound comes directly from Plot read at that bond’s endpoint, which belongs to the store dome. For the coarse row split outs into three ranges: toll ≤s = temp, s < toll ≤δ, and toll > δ, where s ≤δ. The coarse row estimate gives respectively a lower bound −Bquartic s4, a positive bound at least 4m t2 when m ≤directb elow/64 and t ≤2s, and a lower bound −linearb elow s·toll. Sum these disjoint ranges, enlarging the small-range count to all outs because its error is nonnegative. This proves tallC + linearbelows X toll>δ toll +Bquartics4|outs| ≥4mt2 · number(s < toll ≤δ). Finally Payments bounds flags by four times the number of s-large tides and bounds the δ-large tide count by K shares(spend). Since the latter tide set is a subset of the former, subtract it and obtain |flags| ≤4[number(s < toll ≤δ) + K shares(spend)]. These are exactly the four finite-store inequalities used in the assembly. They are algebraic consequences of the fine/noise/coarse producers and Payments, with no compatibility assumption on the periods. In particular the fine and noise spend gains are paid at tu, while the original seed shares are still paid at t2. 6.43 Assembly of the point-charge gain Use the same store, actual input, predictor and background in all estimates below. Put t = fiery(x), s = temp(x), u = largeT(x), and B = radiusB(x). The fine energy estimate bounds its tally below by lineartop/2t X largefinetolls toll +dt2 · number(otherlistedfinetolls), with d > 0 after choosing L,W. The noise estimate gives tallyI ≥(penalty/8) P l isted wire2. The coarse estimate, store census and hit-row mismatch estimate supply the remaining terms described below. Choose a positive m no larger than lineartop /8, d, penalty/32 and directb elow/64. The hit mismatch estimate has fine-loss coefficient cF = C/L2 and noise-loss coefficient C: 81 ## Source page 84 A Wilson Yang–Mills construction J. Rodgers tallLD(hit) + cF t X large finetolls + Ct X listed wire +Zt2B4 · shares(spend) ≥linearbelow · s X δ−largecoarsetolls toll . Choose L first so that cF ≤lineartop /4. Choose W afterwards with W ≥1 and C ≤penalty · W/16. Since every listed noise wire is at least Wt, its quadratic positive energy absorbs the linear C t wire loss. The fine positive energy similarly absorbs the cF loss. The coarse inequality pays the large coarse-toll term, leaving only the small quartic coarse error. The flag census is number(flags) ≤4[number(tidessminustidesδ) +K · shares(spend)]. The output-row census is at most K B4 shares. Choose x large so that s ≤δ, t ≤u, t ≤2s, and (4mK + ZB4)t2 ≤mtu, Cquartics4KB4 ≤(m/4)t2. The first inequality lets the large toll/noise spend pay both its own flags and the mismatch remainder; the second pays the small coarse error by one quarter of the desired gain. The unaffected mismatch estimate costs at most another (m/4)t2 shares. The N tally is nonnegative. To obtain these last two scale inequalities, use t/s →1, t/u →0 with B4t/u →0, and B4s2 →0. Dividing the first desired inequality by tu reduces it to a quantity tending to zero. For the second use s ≤2t and the vanishing B4s2. These choices are uniform in the live period. Split the L/D tally into hit and unaffected rows. Add the fine, noise and coarse budget inequal- ities, the flag census, and the two quarter-error bounds. Their positive remainders pay m t2 per listed fine/noise/flag share before the two quarter losses. The result is assessed ≥(m/2)t2 shares . The exact fine/noise/coarse budget rearrangements supplying these positive remainders are developed next. This assembly fixes the quantifiers: the fine coefficient is paid by L, the noise coefficient by W, then m > 0 is fixed, and finally the common delay pays Z and the quartic and unaffected errors. No later change of L or W is used. 6.44 Unaffected mismatch costs in the point-charge proof The point-charge producer combines the E01–E09 energy estimates with one additional unaffected- row estimate. The latter is derived from the mismatch bounds proved above. Fix an unaffected coarse plaquette row, and let t = fiery. The coarse maximum gives ∥vision∥≤A t. The stitching estimate supplies a gauge in which the actual fine and coarse links are bounded on its radius- 2B/5 hull by h(x) = stitchCost(x), with no change of forest or tags. Choose the delay so that this hull contains 3 till(x)+4+predictionReach and h lies in the interior chart radius. Put e = widen(till(x), h, τ). The local gap bounds and Cauchy–Schwarz give 82 ## Source page 85 A Wilson Yang–Mills construction J. Rodgers workL + workD ≥−loss, loss = 2(At)(Che) + (At)(CAte). The factor two pays the two L coordinates; the two D half-inner-products together give the last term. This estimate applies to each row outside the hit subset of the store’s output rows. The store census bounds their number by N B4 shares. The stitching-scale limits give B4he/t →0, B4e →0. Indeed after dividing N B4 loss by t2 its two terms are exactly 2NAC B4he/t and NA2C B4e. Hence, for every prescribed ε > 0, the total unaffected mismatch cost is at least −ε t2 shares eventually. Sum the row bounds with the unchanged actual U,V and predictor. This proves the additional unaffected-row estimate rather than assuming it in the point-charge chain. The stitching bound, its two displayed scale limits, and the census and maximum estimates of this chapter supply its inputs. The final point-charge statement has order of choices: choose a block lower bound n, then L ≥n; choose W sufficiently large, then a positive μ; finally choose x sufficiently large, uniformly over all periods with coarse side at least 100(1+B), all funded step data, forests and actual Plot inputs. It asserts assessed ≥μ t2 shares pointwise. Passing from the supplied grant to this statement just intersects its delay with positivity of x and t and specializes its universal query. The fine, noise, coarse-clipping and flag budgets in this chapter prove this combination of energy estimates. 6.45 Exact assignment of the energy to packet cores A row location has four types: a fine plaquette, a coarse noise bond, a coarse plaquette carrying C together with L and D, or an N row. Their principal values are respectively workF, workI, workC+workL+workD and workN, all at the same actual input, coarse background and predictor. Their coarse anchor is the parent of the fine plaquette root or the coarse root of the other types. A fine plaquette has two parent moment rows; a noise bond has one; the remaining row locations have none. Each moment row occurs as a parent exactly once. Select a signed row when its signed coarse anchor lies outside the 6B ambient seed dome. Its surcharge is the corresponding real affine path half-square if selected and zero otherwise. Extracted is the sum of its parent surcharges. Put debit = principal −extracted and liftSq equal to the orientation average of all extracted rows. The actual kernel localization identity, including the invariant noise sum, now gives wholeGap + X noise monetary = averages X all rowlocationsdebits + liftSq . This is exact recombination: finite subtraction cancels the extracted rows, while C,L,D remain bundled at their common coarse plaquette. Noise covariance under a signed change is accompanied by the actual transformed tags, and its finite bond sum is invariant under that bond permutation. For a store Z contained in the ambient seed place, every posted 2B core row lies inside the ambient 6B dome, so it is unselected. Its extracted value is zero, and its debit equals its principal energy. Summing the four principal types over this dome is exactly assessed(Z): expand the finite sums and group C,L,D at their anchor. This proves the core identity used in the packet gain without discarding any extracted energy. 83 ## Source page 86 A Wilson Yang–Mills construction J. Rodgers Finally, a Wide condition through d ≥5B implies Plot on its 5B restriction and is preserved by a signed rotation of fields, tags, forest and store together. Apply the point-charge inequality in each rotated tree slice to get assessed ≥μt2 shares. Rotation preserves shares. Averaging gives the mean assessed bound used above. This completes the assignment and averaging step, using the preceding point-charge estimate. 6.46 Paying a packet’s annulus and obtaining its tariff For a store Z contained in the ambient seed place, the core identity identifies its assessed energy with the sum of debits on the posted 2B rows. Its assigned energy sums the same debits through 200B. Subtract the two finite sums. Every annulus row has debit at least −Y (x), by the annulus estimate already proved. The number of posted rows is at most shares(Z) ballCost(200B + 1)spots(L) in each orientation, where spots(L) counts the row types per coarse site. Consequently assessed−E(x) shares(Z) ≤assigned, with E = Y ballCost(200B + 1)spots(L). The radius factor has rate zero, so E retains rate −3/2. The point-charge estimate on the actual wide sector gives mean assessed energy at least μ t2 shares(Z). Average the preceding inequality over the same signed orientations. This proves quoteNear(Z, u) ≥[μt2 −E(x)] shares(Z). Its hypotheses include Wide through 300B, tangent control through 304B, and the same shell- access bound. No connectedness of Z was required. The exact core assignment and point-charge inequality proved in this chapter supply this step. The ancillary packet expenses can be bounded uniformly before using this gain. The all-field joint diagnostic bounds the condensation by a constant. The Haar logarithm is eventually bounded by one. Each swept neighborhood and the ticket fee is at most a fixed constant times bill(x)4, and the negative logarithm of the frozen measure is at most a fixed constant times bill(x). Thus for one fixed Kup > 0, steamLog +tollFee · shares ≤−βquoteNear + Kup bill(x)5 shares . Now consider an admitted nonzero packet. Its seed-consistency indicator supplies containment in the ambient seed place and the correct actual flags. Nonzero ticket and sector profiles imply the Wide hypotheses on its 300B dome. Its nonzero enforcement profile supplies hot tangent control 2a t through 304B. The width condition γ · width ≤a supplies the matching frozen tangent bound. Hence the lower bound for quoteNear applies with that same field, forest and tags. Since E has rate −3/2 and β ≤2 betaF, eventually βE ≤1 ≤bill5. Combining the last two bounds and exponentiating yields |Pack(Z, u)| ≤exp [1 + Kupbill(x)5 −βμt2] shares(Z). This is the packet income with cap 1, per-share expense 1+Kupbill5 and per-share gain βμt2. A zero packet satisfies it immediately. The estimate is pointwise before the frozen-Haar and Gaussian integrations and applies to the entire packet factor once, including its own tag and guard costs. 84 ## Source page 87 A Wilson Yang–Mills construction J. Rodgers 6.47 Compulsory packet factors and normalization expenses For a seed store A, let visitd(A) be the fine bonds with at least one coarse endpoint in its radius-d dome, and pens(A) the nontrunk bonds in visit(10B) A. Let mg(β) = chartScale( √ 2π β−1/2)dim(Color). The packet exponent is defined exactly by steamLog = −β[quoteNear(A, u) + farBills(A)] + X e∈visit200B(A) condensation(e) + X e∈plaza(dome304B(A)) gentry(e, u) −|pens(A)| log mg(β). Here farBills is the signed-orientation average of the extracted mean half-squares at the posted 200B rows; it is nonnegative. The entire packet factor is exp(steamLog) times the contraction-order profile, the actual tag tickets on the 300B dome, the sector profiles there, and tangent enforcement on the 304B plaza. It is nonnegative. A nonzero factor forces every individual enforcement cone in that plaza to be nonzero. The order, sector and enforcement profiles lie in [0,1]. The tickets have upper bound exp(tollFee(patch, 300B) · shares(A)). Hence |steam| ≤exp[steamLog +tollFee · shares(A)]. This bound keeps the ticket cost of the whole packet; it is not repeated separately in neighboring log components. Suppose |condensation| ≤Q0, |gentry| ≤C0, and −log mg ≤K0, with these bounds nonnega- tive. Endpoint census gives | visitd(A)| ≤shares(A) ballCost(d + 2)L4|Axis|. The gentry plaza is contained in visit304B(A), and pens in visit10B(A). Dropping the nonpositive term −β farBills yields steamLog ≤−βquoteNear + shares(A)[Q0swept(200B + 2) +C0swept(304B + 2) + K0swept(10B + 2)], where swept(d) = ballCost(d)L4|Axis|. This is the exact ancillary expense to be compared with the packet’s core gain. The measure expense has a uniform logarithmic bound over 0 < β ≤2 betaF. Expand −log mg = −log chartScale −dim(Color) log √ 2π + (dim(Color)/2) log β. Its upper bound depends only on log(2 betaF), so eventually it is at most Cmeasure · Delay.bill(x) for one positive constant. No lower beta-band bound is needed for this upper estimate. The point-charge core bound and the packet annulus estimate show that the gain in quoteNear dominates these counted expenses. 85 ## Source page 88 A Wilson Yang–Mills construction J. Rodgers 6.48 The short correction family The short corrections and Gaussian chains are organized so that locality can be priced before the full connected expansion is summed. A chain letter remembers its genuine path support and itinerary; changing to a smaller nominal support would invalidate the later hit counts. The inverse and logarithm series are absolutely convergent under the displayed smallness conditions, so their pointwise exponent identities may subsequently be integrated. Budget language as inequalities The correction families are organized by the upper bounds they spend and the positive estimates that absorb those bounds. A price is an explicit coefficient in such an inequality; income is a proved lower estimate; and a payment is the displayed domination after the parameters have been chosen in the permitted order. These terms do not describe a new probability law. Retaining their weights and exponents is essential because the same finite store must absorb the fine, noise and coarse contributions simultaneously. The local short letters have three kinds. At a signed row root o outside the 200B seed neigh- borhood, the energy letter is −βorientation · [enforce(2a)(u, booth(o)) · light(u) +extracted(μ)]. It is zero inside that neighborhood. The second letter is the local condensation term, set to zero when either endpoint of its fine bond touches the 200B seed dome. The third is the cut-off Haar logarithm gentry, retained only at hot links outside the plaza of the 304B seed dome. Their geometric sleeves are fixed neighborhoods of their row or bond captions. On the original seeded sector the light term equals the original energy debit at a remote row. The noise row equality is definitional. For the curvature row, turn the actual fields, tags and seed store together; the seeded sector and positive likelihood give the local Interior bounds. If the soft retention threshold at a possible curvature read is below 2, the room inequality places that read in this interior and its plaquette toll is at most t/2 ≤t. If the threshold is at least 2, the universal chord bound supplies retention. Thus the retained high-stage path field is unchanged there, and the exact split-curvature identity identifies light with debit. This reasoning uses actual sector membership only for this equality, not for the all-field bounds below. Let A(x) be the protected-mean path envelope already proved, of rate −1. The extracted term is a sum of at most two nonnegative half-squares of parent path values. At a remote row its absolute value is therefore at most 2(A2/2) = A2. Take J = E1 + E2 + E3 + E4 from “Light estimates on the annulus,” with grafted tangent bound 4a fiery(x). The soft-square and mismatch estimates there have rate −3/2, and the two quartic or squared-guide corrections have rate −2; weakening the latter gives J rate −3/2. Take Econd = dim g pulseHead(repair) from the preceding local plate comparison. Its repair envelope has rate −1/2, its window census has rate zero, and its fixed inverse-window constants do not depend on x, so Econd has rate −1/2. These are the actual two bounds needed here. The Haar estimate gives | gentry | ≤C fiery. To use the light estimate for arbitrary input u, split according to whether its enforcement profile is zero. If it is nonzero, its protection property bounds every tangent value read by the booth by 4a fiery. The distant-field identity then supplies precisely the grafted condition on the common clean radius-B hull. The profile is at most one, so its product with light is bounded by J. Outside the 200B exclusion all these hull and parent-path conditions follow from the same seed separation. For 0 < β ≤2 betaF and 0 ≤orientation ≤1, the energy letter is bounded by 86 ## Source page 89 A Wilson Yang–Mills construction J. Rodgers Wshort(x) = 2betaF(x)[J(x) + A(x)2]. Its rate is −1/2: betaF has rate 1, J has rate −3/2, and A2 has rate −2. The condensation and Haar letters have bounds Econd and C·fiery of rate −1/2 as well. Thus their sum Eshort = Econd + C · fiery +Wshort is a common eventually nonnegative envelope of rate −1/2 for all three kinds, all fields and all admissible β. A bond whose endpoints touch a dome has its parent in the one-site enlargement; conversely a bond not touching it has its parent outside it. These elementary endpoint facts justify the condensation exclusion used here. The light-remainder, remote-condensation and grafted-signal estimates of this chapter establish the three inputs to this assembly, together with the mean and Haar bounds. 6.49 A fixed multiple of the small-field scale for the base stroke Interpreted typography. The bound below is read as 4 pcap(4)( pdirecttop + pathstop)t. Both the direct row and the summed path row carry the common small-field factor t. This repairs the extraction’s missing parentheses and mistyped terminal subscript; it is the convention used in the subsequent base-stroke constant C1. For an integer n define pcap(n) = 32(2(n + 1) + 1)5|PlaqAxes|. Axial flattening on radius n+1 and temp ≤2t make the background links on radius n have chords ≤pcap(n)t. The crucial point is that n is fixed separately for each finite query below, rather than increasing to B. First consider any fine plaquette of replica(V ) with a clean radius-five hull about its coarse parent. Use n = 4. Replica links in that hull are either identity or a coarse link in the chosen gauge, and hence have chords ≤pcap(4)t. Replace links outside the hull by 1. Its plaquette product is unchanged, so its chord is at most 4 pcap(4)t. The same bound applies to its curvature norm. For the fine base moment row, every retained path slot ends at a plaquette whose parent has a clean radius-five hull inside the original radius-B clean region. Its actual path length is bounded by till(x), and the room inequalities ensure this inclusion. Its adjoint transport preserves curvature norm. Summing against the absolute path coefficient budget, and including the direct curvature coordinate, therefore gives ∥XfR(replica(V ))∥ ≤4 pcap(4)( q directtop + pathstop)t =: C1t. No single gauge is needed for every slot: only its curvature norm is estimated. For the noise base stroke, instead fix n = predictionReach(D). Flatten on that fixed hull and extend its logarithm to a global bounded field b with ∥b∥≤A t, A = Alog pcap(n). Its linear copy a has norm at most ∥b∥and exponentiates exactly to replica(exp b). Predictor locality identifies the actual base noise stroke in this gauge with zTest(a, b), with its original tags. Its uniform Taylor estimate gives ∥zTest(a, b)∥≤J∥(a, b)∥2 + J∥(a, b)∥≤2JAt eventually, since At ≤1 and the chart radius. Gauge covariance transfers the bound back. Thus C1 + 2JA is one positive constant bounding every clean-hull base stroke by that constant times t. This proves the crowns input to the global mean bound without spending an extra growing-radius factor. 87 ## Source page 90 A Wilson Yang–Mills construction J. Rodgers 6.50 Global mean and stationary residual The global mean bound uses selected rows, the frozen cutofffield, and the inverse-window bounds; it is separate from the sharper remote path estimate. Let the clean-background base stroke be bounded by A0t, every guide by Mt, the oriented cutoffoperator by N, and the saved field by γ·2Kt, where K is the fixed shell width. On a selected row, seed separation supplies the clean hull needed for the base-stroke estimate. At zero hot input, ∥priced(0)∥≤scale[A0 + M + 2NγK]t =: b1t. An unselected row is zero and satisfies the same bound. This uses the saved-field bound globally, without requiring a common global gauge. The linear bill is the transpose form applied to priced(0). Its row-sum bound Y gives |bi| ≤ Y b1t =: b0t. Let P be the inverse-column pad and μcount = −P Tb. Its column bound Cinv = payE gives |μcount, i| ≤Cinv b0t. Summing its color-coordinate bounds yields ∥μe∥≤dim(Color)Cinv b0t. The stationary residual has a useful exact form. Symmetry of A−1 gives Aμcount + b = −A(P −A−1)Tb. The pad discrepancy and plate row bound Q therefore imply |Aμcount + bi| ≤Q distPay(B)b0t. Choose Cmean = 1 + max(b0, b1, dim(Color)Cinvb0, 0). It simultaneously bounds priced(0), the linear bill and the mean by Cmean t. The residual envelope above has any prescribed rate k: scraps(B), and hence distPay(B), has rate k+1/2, while t has rate −1/2. This supplies the mean/residual hypotheses of the correction budgets. The clean base-stroke and uniform maximum estimates proved here give A0 and M. 6.51 Exact Gaussian exponent rearrangement Let D be the form matrix, Ssleep the dormant diagonal, A = DTD+Ssleep, μ the count-coordinate mean, b the linear bill, and Cref the positive reference covariance. The lifted tangent input is μ+gz, with g = β−1/2. The affine-square sum has the exact polarization identity liftSq(μ + gz) = liftSq(μ) +g⟨z, b + DTDμ⟩+ g2⟨z, DTDz⟩/2. This follows by expanding each squared affine row and summing the finite color coordinates. The dormant mean vanishes, so Ssleep μ = 0. Define ghost(z) = ⟨z, Ssleep z⟩/2. Using βg2 = 1 and βg = g−1 gives −β liftSq(μ + gz) −ghost(z) = −β liftSq(μ) + ⟨z, hattedβ⟩ +⟨z, C−1 ref −Az⟩/2 −⟨z, C−1 ref z⟩/2. The last term is the reference Gaussian exponent. The preceding quadratic term is the correction expanded by the itinerary chain; the linear term is the first correction. The inserted ghost supplies exactly the dormant diagonal needed to complete A, and its normalized dormant Gaussian integral is one as shown earlier. This equality is pointwise algebra and remains valid where a later profile vanishes; it assumes no sector event. 88 ## Source page 91 A Wilson Yang–Mills construction J. Rodgers 6.52 Pointwise Gaussian chain corrections Write A for the completed plate, Cref for the symmetric reference covariance, and Serr = A Cref −I for its small discrepancy. The linear correction vector is hattedβ = −gStep(β)−1(Adrift + b). For each fine link e its scalar correction is P c olor z(e,j) hatted(e,j). Multiply this by aperture(e), the cone profile at twice the chosen enforcement radius on hot links and the value 1 on other links, to obtain qfirst. For an itinerary of n+2 link steps, the quadratic correction is half the bilinear pairing between its starting and ending z vectors with the ordered matrix-block product −Serr · · · −SerrA, containing n+1 discrepancy factors and one terminal A. Multiply it by the two endpoint aper- tures to obtain chain. Summing itineraries at fixed starting link and n restores exactly the matrix product (−Serr)n+1A in that quadratic form: the intermediate link and color sums are precisely matrix multiplication. The terminal A is retained in both the formula and its price. Dormant plate rows are identity rows and the reference covariance has the same identity comple- tion. Therefore Serr has zero entries whenever either endpoint is dormant. The linear bill and mean vanish there too, so hatted is zero at dormant coordinates. A nonzero quadratic chain consequently has hot starting and ending links: every nonzero discrepancy factor forces hot endpoints, and the final A cannot join a hot link to a distinct dormant identity row. Induction over its discrepancy factors proves this for every n. Let ∥μe∥≤K t and g = gStep(β) > 0. At a hot endpoint where its aperture is nonzero, the cone bounds ∥μe + g (atmos(z))e∥≤4a t. The triangle inequality gives ∥atmos(z)e∥≤g−1t(4a + K) = heat(a, K, t, β). Thus an envelope U ≥heat bounds every endpoint color coordinate. The aperture is at most one. If | hatted(e,j) | ≤M, the finite color sum yields |qfirste| ≤dim(Color)UM. This holds at dormant links as well because their linear correction vanishes. For the chain, put Rc = dim(Color), dpay = distPay(η, r, Q, m, B), and Merr = Q dpay. The discrepancy has entry bound Merr, whereas the terminal plate has entry bound Q. Summing its n+1 intermediate color indices bounds each entry of the prescribed block product by Rn+2 c QMn+1 err . The convenient extra color factor is included in the source’s product estimate. The two endpoint color sums therefore bound the half-bilinear form by 1 2U2Rn+4 c QMn+1 err = centreQ(U) · etaSn, centreQ(U) = 1 2U2R4 cQ(Qdpay), etaS = RcQdpay. 89 ## Source page 92 A Wilson Yang–Mills construction J. Rodgers A nonzero term must also be armed: every itinerary hop has parent distance at most 3B. Oth- erwise its corresponding discrepancy or plate block vanishes by the finite-range estimates. Define capQ to equal the displayed center-times-ratio power on armed itineraries and zero on the others. Splitting into zero aperture, zero bilinear form, and nonzero cases proves |chain| ≤capQ pointwise for every z. This explains why these optional terms use endpoint cutoffbounds rather than a separate Gaussian tail event. 6.53 Convergent inverse and logarithm expansions Weighted absolute row and column prices multiply under matrix multiplication. If a matrix E has price q < 1 at a nonnegative width, En has price qn, so its entries form absolutely convergent geometric series. The finite identity P n 0, so this is log det A. Let Ps be its inverse-column pad and Es = Gs Ps −I. The uniform discrepancy price ρ = scraps(B) < 1 gives (A −I)G−1 s = X n≥0 (A −I)Ps(−Es)n. The pad Ps occurs once in each term; only the residual −Es is raised to the power n. This follows from GsPs = I + Es, hence G−1 s = Ps(I + Es)−1. Each diagonal summand is continuous in s and bounded in absolute value by (Q + 1)payE · ρn. The geometric bound is summable independently of s. Hence its interval integrals may be summed: 90 ## Source page 93 A Wilson Yang–Mills construction J. Rodgers the sum of their integral norms is bounded by the same geometric series on an interval of length one. Write lacti(n) for these diagonal integrals and localBasei = lacti(0). We obtain log det A = X i localBasei + X i X n≥0 lacti(n + 1). Thus the local principal-window diagnostic is only the zeroth term; the remaining determinant arms restore the full inverse exactly. For any discrepancy E of price q < 1, apply the same argument to I+sE and expand its inverse as P n(−sE)n. The integrands E(−sE)n have uniform bound qn+1, giving log | det(I + E)| = X n≥0 X i Z 1 0 [E(−sE)n](i,i)ds. In particular E = A Cref −I yields I +E = A Cref. Both A and Cref have positive determinants, so the right side equals log det A + log det Cref. These source-arm terms therefore represent the actual covariance normalization defect. All exchanges here use uniform summable entry bounds, followed by finite coordinate sums; no volume factor is treated as a local price. 6.54 Itinerary receipts and scalar arm identities An itinerary of k hops is a list of k successor fine links, with the initial link retained separately. A hop is armed at radius d when some coarse endpoint of the first link is within d of some coarse endpoint of the next. Each parent is within one site of either endpoint, so an armed hop moves parents by at most d+2. The town of a link is the mB halo of its two coarse endpoints. It contains its parent and lies in the radius-mB+1 ball about that parent, hence in the radius-(m+1)B ball when B > 0. It can therefore be covered at scale B with at most steady(m) = ballCost(m+1) tiles. The visited receipt is the union of the towns of every link in the itinerary, including both its initial and final links. If d + 2 ≤mB, each successor parent belongs to the previous town and to the remaining visited receipt. These two sets meet. The covering inequality for meeting supports and induction give coverNumberB(receipt) ≤(k + 1) steady(m). For its tenure bound, any point in the first town lies within mB+1; a point in the rest is within d+2 of its next parent plus the induction bound there. Thus, when d + 2 ≤mB + 1, every receipt point lies within (k+1)(mB+1) of the initial parent. These establish the two geometric inputs used in the address count. For a sequence of matrices Ps(0), Ps(1), . . . and a fixed itinerary, multiply their color blocks in its link order. Keep only itineraries returning to their initial link, take the color trace of that block product, and integrate s from 0 to 1. Call this scalar its stamp. Expanding matrix multiplication shows that summing stamps over all k-hop itineraries restores the integrated diagonal trace of Ps(0) · · · Ps(k −1). There are finitely many itineraries at fixed k; continuity of the matrix factors gives integrability and permits interchanging these finite sums and the integral. For the determinant arm choose Ps(0) = A −I, Ps(1) = pad(I + s(A −I)), and Ps(j) = −error(I + s(A −I)) for j ≥2. Its order-n label has n+3 hops, so its sum restores the local integrated intensity at grade n+1, retaining n+1 discrepancy factors. For the source arm choose Ps(0) = flawed(A) and Ps(j) = −s flawed(A) for j ≥1. Its order-n label has n+1 hops and restores the corresponding integrated grade-n note. These are exact scalar identities; the finite color trace belongs inside each arm label. If each factor has entry bound lj, the absolute stamp is at most Rk+1 c Q j 0. Its B-covering number is at most (k+1)steady(m), and if a fixed site x belongs to its receipt, its starting link has parent within distance (k+1)(mB+1) of x. The underlying definitions of visited receipts give these covering and tenure bounds; they remain the immediate geometric inputs here. Their effect on the count is as follows. For coefficients 0 ≤w(n,e,p) ≤C Qn on armed addresses and zero elsewhere, the tariffat order n includes exp(2εn) and exp(Asupport coverNumber). Bound the latter by exp(Asupport(n + u + 1) steady(m)). The number of eligible starting links is at most ballCost((n + u + 1)(mB + 1)) density(L). The elementary inequality n + u + 1 ≤(u + 1)2n and the four-dimensional ball formula give ballCost((n + u + 1)g) ≤ballCost((u + 1)g)16n. Multiplying this starting-link count, the armed-hop count, and the coefficient/tariffbounds yields, at each x, an order-n total at most HC(QRprice)n, H = ballCost((u + 1)(mB + 1)) density(L) · abundance(L, d)u exp(Asupport(u + 1) steady(m)), Rprice = 16 abundance(L, d) exp(Asupport steady(m) + 2ε). When Q Rprice ≤1/2, its sum over n is at most 2HC. All terms are nonnegative. To pass from the finite order-n sums to the countable ledger sum, first bound every finite subset within its finitely many order fibers, then apply the convergent geometric bound. This proves the asserted sales inequality and its summability, retaining every address even when different itineraries revisit the same link. 6.56 Pointwise determinant and source arm prices For the determinant arm the ordered factors along its n+3 hops have respective entry prices Q+1, payE, and n+1 copies of scraps(B). Each factor has range at most 3B: the plate term uses its B/2 stencil, the inverse-column pad its window, and its discrepancy the enlarged window stencil. The 92 ## Source page 95 A Wilson Yang–Mills construction J. Rodgers common interpolating parameter s ∈[0, 1] preserves these bounds and their positive floors. The product entry estimate, followed by the color trace and s integral, gives |ArmA(n,e,p)| ≤Rn+4 c (Q + 1)payE · scraps(B)n+1 = centreA · etaAn on armed itineraries, and zero otherwise. Here centreA = R4 c(Q + 1)payE · scraps(B), etaA = Rc · scraps(B), and Rc is the color dimension. For the source arm all n+1 factors have entry price Q·distPay(B); the factors beyond the first also contain the scalar interpolation multiplier, whose absolute value is at most one. Their range remains 3B. Thus |ArmS(n,e,p)| ≤Rn+2 c (Q distPay(B))n+1 = centreS · etaSn with centreS = R2 cQ distPay(B) and etaS = Rc Q distPay(B). These bounds are the actual scalar arm prices before exponential expansion. They use the ordered-product and trace-integral representation of the arms given in the itinerary section; the pointwise estimate sums their finite internal color indices and integrates over an interval of length one. 6.57 From itinerary envelopes to support budgets The general itinerary bound takes an armed-path coefficient of the form Y (x)κ(x)n, supported on its predetermined receipt of halo i ≥4 and fixed number u of extra steps. Let abundance(L, 3B) bound the number of possible next links in an armed hop. Both this abundance and the fixed-step prefactor expoHead(L, i, B, 3B, u, Asupport) have rate zero. The geometric receipt estimate gives the sales bound 2expoHead · Y as soon as 16 abundance(L, 3B) exp(Asupport · steady(i) + 2ε)κ ≤1/2. If κ has any negative rate, that condition holds eventually; multiplying it by the rate-zero abundance does not change its decay. Thus an envelope Y of rate b yields a sales bound of rate b. This numerical deduction is uniform in geometry; the itinerary address count and halo-cover estimate supply the quantities expoHead and steady used here. Apply it to the determinant and source arms separately. The determinant arm uses u = 3, center R4 c(Q + 1)payE · scraps(B) and ratio Rc · scraps(B). The source arm uses u = 1, center R2 cQ· distPay(B) and ratio Rc Q·distPay(B). All centers and ratios have every prescribed negative rate, so their sales budgets can have any prescribed rate b. Their pointwise bounds retain one scalar per actual arm, rather than splitting a trace into independently labeled colors. Applying the same bounds to the blank geometry supplies its determinant arm independently. The pointwise arm identities and the geometric itinerary estimate are still the inputs to this budget calculation. 6.58 Numerical budgets for the Gaussian chain letters The mean/residual hypotheses are retained separately. They assert ∥μe∥≤C·t and |(A drift +b)i| ≤ E(x), together with the shell access bound. For the physical beta band 0 < β ≤2 betaF put 93 ## Source page 96 A Wilson Yang–Mills construction J. Rodgers burst(x) = q 2 betaF(x). Then gStep(β)−1 = √β ≤burst, and burst has rate 1/2. The bounded cutofffield used by the chain estimates therefore has common envelope U(x) = burst(x) · fiery(x) · muRate(C, H), of rate zero. This envelope is independent of β and the Gaussian variable. Its nonnegativity and the mean bound supply the hypotheses of the pointwise chain estimate. If E has rate k−1/2, then b(x) = burst(x)|E(x)| has rate k. The first mean-correction envelope is J(x) = dim(Color)U(x)b(x), of rate k. Its caption is the parent of its fine bond. With support town contained in radius- (m+1)B and fine-link caption census L4|Axis|, its sales price is at most ballCost((m + 1)B)L4|Axis| · J(x) · exp(Asupport ballCost(m + 1)), also of rate k. The higher quadratic-chain envelopes are capQ evaluated at U. Their series sales bound follows from the itinerary estimate with center coefficient proportional to 1 2U2flagP 4Q2payE · scraps(B) and ratio coefficient flagP·Q·payE·scraps(B). The latter has every prescribed negative rate; the former has rate k for any prescribed k because U has rate zero. The itinerary summation consequently gives the chain sales bound of rate k. The support charge is its series index, and halo m ≥4 is retained. The pointwise Gaussian chain and short-correction estimates supply capQ and J; this calculation retains their coupling and residual factors in the common budget. For a finite caption family of density n with supports contained in radius-dB balls, a constant coefficient envelope C0 has sales price at most ballCost(dB)n C0 exp(Asupport ballCost(d)). Indeed a fixed site belongs to at most ballCost(dB)n supports, and each support has B-covering number at most ballCost(d). This is the fixed-envelope counting used for the first correction and local Short letters. 6.59 True path supports and bundle geometry The native support of a walk is the finite set of its visited sites, including its start and endpoint. Every visited site lies within its actual walk length of the start, by induction and the one-step separation bound. A walk of length at most M is therefore covered by one radius-M tile. For a longer walk, split offits first M steps. The two native supports meet at the split endpoint. The meeting-support covering inequality and induction give M coverNumberM(native(w)) ≤M + length(w), M > 0. This uses the walk length with all repetitions and winding, even when the finite native set is much smaller. 94 ## Source page 97 A Wilson Yang–Mills construction J. Rodgers A curvature slot’s actual support is the native support of its transport walk followed by the four-step plaquette. This equality follows by concatenating visit lists and wrapping their cover coordinates. Its slot length is exactly that concatenated length. A bundle of d slots rooted at x is the union of their supports with x inserted. All its supports meet at x, so the same covering inequality gives M coverNumberM(bundle) ≤(1 + d)M + totalLength. Every bundle site also lies within totalLength of x, since each individual slot length is at most their nonnegative sum. For a new support Y contained in halol B(parent(bundle)), the parent-distance inequality gives its extent from parent(x) at most (lB+1)(1+totalLength). Apply the coarse footprint bound with M = B and the last bundle covering inequality. Define baseFee = preCost(l) + conversion(L, l)(1 + d), legFee = conversion(L, l)/B, originFee = lB + 1. Then coverNumberB(Y ) ≤baseFee + legFee · totalLength. The anchored-to-hit principle applies with degree equal to true totalLength, extent originFee, and those two covering coefficients. At each fixed fine root, its input budget is exactly the original slot-family length budget; the root delta and this budget form a product budget, so no volume factor is inserted beyond the parent population L4. This proves bundleh it, including the tail exp(−εm) for totalLength ≥m and its explicit payment. Atom footprints are exactly bundles of their slot legs; an ordered path pair is the two-slot bundle and has the sum of its two actual lengths. 6.60 Connected center traversal and halo tiling The covering currency uses the periodic maximum-coordinate separation. A center set is connected at range b if any two centers can be joined through centers with successive separation at most b. A cover at radius M covers the support by the M halos of its centers and requires those centers to be connected at range 2M+1. Thus disconnected pieces of a support still incur the cost of connecting their cover; it is not an unrestricted ball-counting number. There is a unit-speed line from x to y of duration separation(x,y): in each centered-lift coordinate of y−x, truncate the displacement to the interval [−j,j] at time j, then wrap and add x. Each coordinate changes by at most one per time step, and it stops at y once j reaches their maximum absolute displacement. For a finite center set D connected at range b, build a traversal by adjoining one unvisited center at a time. Connectivity supplies an edge from the visited subset to its complement: take the first exit along any center path to an unvisited point. At the occurrence of its visited endpoint in the current traversal, insert the line to the new center and its reverse. The detour has length at most 2b and preserves all old visits. Starting with one center and repeating gives a unit-speed traversal visiting every center with length at most 2b|D|. The empty case has the zero-length trivial traversal. This proves the ring-path bound used in extent and collapse. For tiling a radius-kB halo of one site, take the image of the integer grid [−k, k]4 scaled by B and translated to that site. Divide each centered coordinate of a halo point by B. Its quotient lies in that grid and its remainder has absolute value at most B, so one grid center covers it. The grid centers are connected through grid steps; wrapping may identify centers and can only reduce their number. Hence at most (2k + 1)4 centers suffice. 95 ## Source page 98 A Wilson Yang–Mills construction J. Rodgers If X has a radius-mB cover with center set D, enlarge its balls to radius (m+1)B. Their original connecting steps are at most 2mB + 1 ≤2(m + 1)B, so these enlarged halos meet along the center connections. Tile each enlarged ball by the preceding grid and join their connected covers at the meetings. This gives coverNumberB(X) ≤|D|[2(m + 1) + 1]4. The same line construction proves that halos of centers connected at range 2M+1 are connected at unit range: the line between two centers lies in their two M halos, with the possible middle adjacent step included. These provide the connected-halo and tiling facts used by the collapse proof without requiring any injectivity of a periodic halo. 6.61 Coarse collapse of a connected covering Write L for the block side. A fine site is within L/2 of its embedded parent. Embedded coarse-site separation is at least L times coarse separation, coordinatewise from centered lifts. The triangle inequality therefore gives Lseparation(parent(x), parent(y)) ≤separation(x, y) + L. For a unit-step fine-site path R0, . . . , RN, sample its parents every LB steps, B > 0. Consecutive sampled parents are within 2B by this inequality, and every parent along the path is within 2B of one sample. The set D of samples is connected at that range and satisfies LB|D| ≤N + LB. This includes the final partial segment and allows repeated samples. Realize the scale-M cover of a support X by a center set A of size coverNumberM(X) whose union of M-balls is connected. Join two centers when their distance is at most 2M+1. This center graph is connected: a lattice path through the union passes from one ball to another only through overlapping or adjacent balls. A finite connected graph has a spanning tree, obtained by adding an edge from the already reached vertices to a new vertex until all vertices are reached. Traverse this tree out and back. Each tree edge is replaced by a lattice path of length at most 2M+1, and each is traversed twice, so the resulting unit-step path visits every center and has length at most 2(|A| −1)(2M + 1) ≤|A|(4M + 2). The empty cover has the constant path and length zero. Join any two sites of X to covering centers by paths of length at most M. Their distance is at most |A|(4M+2)+2M. This proves the extent estimate, on the periodic lattice as well as on the integer cover. Sample the center traversal at scale LB as just described. If M ≤2B, the parent of any support point is within 3B of its center’s parent, and that center’s parent is within 2B of D. Hence halo(sB)(parent(X)) is covered by the radius-(s+5)B halo of D, with its inherited connectedness. Tiling that halo at scale B costs at most |D|ballCost(s+6). Combining the sample count, N ≤ |A|(4M + 2), M ≤2B and B ≥1 gives L coverNumberB(halo(sB)(parent(X))) ≤[L + 10 coverNumberM(X)] ballCost(s + 6). For X empty both sides are immediate; otherwise its realized cover has at least one center, which pays the fixed endpoint term in this inequality. Dividing by L yields exactly the preCost and conversion constants used in the coarse hit budget. Thus all spatial block factors in that budget follow from parent separation, a connected cover traversal, and halo tiling. 96 ## Source page 99 A Wilson Yang–Mills construction J. Rodgers 6.62 Anchored budgets converted to hits The geometric counting principle is the following. Suppose each fine anchor has budget P oi=y bieAdi ≤γ, the parent map has at most P fine sites per coarse site, and x ∈Xi ⇒separation(x, parent(oi)) ≤W(1 + di), coverNumberB(Xi) ≤v + v′di. Let 0 ≤gi ≤Cbi(1 + di)k, and assume a ≥0, ε > 0 and av′ + 2ε ≤A. At fixed degree di = j, the coefficient with its new support weight is bounded by bi exp(Aj) · C(1 + j)k exp(av −2εj). All relevant anchors lie above the coarse radius-W(1+j) ball about x. Their number is at most P ballCost(W)(1 + j)4. Group a finite set of labels by anchor and apply the original anchored budget at each anchor. The resulting degree-j total is at most exp(av)γCP ballCost(W)(1 + j)4+k exp(−2εj). The exponential series gives (1 + j)4+k ≤ε−(4+k)(4 + k)!eεeεj. Summing j ≥m proves a hit budget with tail factor exp(−εm) and payment exp(av)γCP ballCost(W) ·ε−(4+k)(4 + k)! exp(ε)/(1 −exp(−ε)). The argument bounds every finite label subset, so it proves the nonnegative countable summa- bility as well. For a block parent of side L, the site decomposition into parent and residue gives P = L4. For an old support Y at scale M ≤2B, the coarse collapse estimate gives coverNumberB(halolB(parent(Y ))) ≤preCost(l) + 10 preCost(l) coverNumberM(Y )/L, preCost(l) = ballCost(l + 6). Its extent estimate gives separation from the parent of any chosen anchor at most (l+20)B(1 + coverNumberM(Y )). These two bounds supply v,v′ and W in the general principle. The connected traversal and coarse-collapse arguments above prove these covering and extent bounds; the displayed summation then supplies the hit budget. Inspecting a support adds at most four fine-site steps, hence at most five parent steps, so its enlarged support is included after increasing l by five. The reserve inequalities have the required quantifier order. For old exponent A > 2ε, choose L strictly above 10 preCost(l)a/(A−2ε). Then 10a preCost(l)/L + 2ε ≤A, before choosing x. For path supports the coefficient is instead 10preCost(l)/(LB); at fixed L it tends to zero as B →∞, so any σ > 2ε suffices eventually. The same delay ensures the old fine radius radiusB(betaF) ≤ 2 radiusB(x). Fixed powers of these radii and the monitor are subpower in betaF, whereas exp(−c till(x)) and exp(−cmonitorz), c, z > 0, beat every inverse power. These numerical facts justify the strict-exponent absorptions used above, without changing L after the delay. 97 ## Source page 100 A Wilson Yang–Mills construction J. Rodgers 6.63 Support tariffs and path-tail conversion For nonnegative coefficient envelopes wi, supports Xi and integer extra charges ei, the sales condi- tion means that at every coarse site x, X i:x∈Xi wi exp(2εei) exp(a coverNumberB(Xi)) ≤γ, with the corresponding nonnegative series summable. Disjoint alphabets add their γ bounds. A finite additional orientation index costs the sum of the orientation bounds; rotating or translating supports preserves a bound since the site map is bijective and does not increase their covering number. If Xi is nonempty and a, ε ≥0, each wi ≤γ follows by reading the inequality at one site in Xi. To transfer an old anchored bound at scale M, reserve its extra tarifffirst: set vi = wi exp(2ε coverNumberM(Yi)). Then vi exp((δ −2ε) coverNumberM(Yi)) = wi exp(δ coverNumberM(Yi)). The old anchored bound is therefore unchanged after reducing its support exponent by 2ε. If the new Xi lies in the l-halo enlargement of the parent image of Yi, M ≤2B, and the geometric reserve inequality holds, the coarse support estimate applies to v. It allows a new coefficient zi ≤Cwi(1 + covM(Yi))k and pays its new 2ε tariff. Its resulting budget is the explicit footprint payment with fine-child factor L4, extentCost(l, B), degree k, original budget γ, support exponent a and geometric preCost(l). The anchor-to-hit summation and the geometric coarsening proof are given in (NF9)–(NF10) below, with these parameters. For rooted path bundles the same transfer starts from a true length budget P i wi exp(δ lengthi) ≤γ. Set vi = wi exp(2ε lengthi), leaving width δ−2ε for the geometric con- version. The bundle support estimate then supplies sales for any z(root,i) ≤Cwi(1 + lengthi)k on an allowed enlarged bundle footprint. Unlike an anchored activity family, its root varies over every fine site; the payment’s originFee and L4 factors account for that census. An inspected order-r atom footprint lies inside the (l+5)-halo enlargement of its actual path bundle. Applying the path transfer to its magnitude and exponential length budget gives the atom payment. Its degree-k polynomial mark is retained in the payment. For paired tails the mask halo adds only two further coarse halos: enlargementl(pairFoot) is contained in the (l + 2)- halo enlargement of the two-path bundle, by the parent-distance inequality. This supplies the tail payment with r = 2. The tail’s small factor can be computed before support conversion. Write x,y for the two actual slot lengths. A nonzero omitted pair has x d > R or y d > R, so (x + y)d ≥R. The pair-price bound and half of the exponential length reserve give pairPrice · exp((σ/2)(x + y)) ≤2 exp(−σR/(2d)) |ux| exp(σx)|uy| exp(σy). Sum using the product of the two original path budgets, each at most Cpaths. Thus the half- width budget is at most 2 exp(−σR/(2d))Cpaths2. At R = till(x) this has every prescribed negative rate. The remaining width σ/2−2ε is positive under 8ε < σ, leaving the geometric reserve needed for support conversion and the polynomial mark. 98 ## Source page 101 A Wilson Yang–Mills construction J. Rodgers All footprint payments multiply their scalar envelope and original budget by fixed constants and a four-dimensional extent factor. If those envelopes have rates u,v and the extent has rate zero, payment has rate u+v. Consequently any strictly larger exponent absorbs the remaining fixed and subpower factors. This proves the atom and tail rate deductions used by the inherited six-family assembly, conditional only on the geometric footprint conversion. 6.64 Inherited coefficients and their enlarged supports Fix halo l and positive ε with 4ε below the incoming support exponent, and 8ε below both the fine and coarse path widths. A sufficiently large block length L converts the incoming anchored support budget to the enlarged coarse footprints, leaving the prescribed support price Asupport and extra-charge tariff. The same L works for the regular and forced activity classes. This conversion uses the original incoming envelopes, before any new marking reserve is constructed. For a regular occurrence, take its supplied nonnegative envelope bi. On its healthy support its absolute value is bounded by bi using the zeroth-order incoming bound; outside that condition the masked regular occurrence is zero. Its anchor remains the old regular anchor. For a forced coefficient use its incoming envelope di without a health restriction. Choose an inventory point in each forced support as anchor; the incoming hit bound implies the weaker anchored bound at that point. Apply the same coarse support conversion separately to each family. Its geometric payment factor has rate zero, because its extent is a fixed multiple of B and L is fixed. The regular incoming envelope has rate −11/10, so any price betaF u with u > −11/10 is available eventually. The forced incoming bound can be chosen with rate v−1, yielding a sales price betaF v for any fixed v. For an order-k path atom multiplied by a scalar η with |η| ≤D(x), let D have rate u and its total atom budget have rate v. The healthy atom bound gives the pointwise envelope z(site,j) = D(x) fiery(x)kmagnitude(atomj). Its rate before support conversion is u−k/2+v. If this is strictly below a desired exponent r, choose an intermediate exponent s between them, convert the inspected supports, sum over the finite signed orientations, and absorb that fixed number into betaF r. This retains the atom’s actual path length in its extra charge. The omitted path-pair tails have a stronger property: their exponential length budget pays any prescribed power betaF r after a fixed scalar multiplier D of any fixed rate. The support conversion uses 8ε < width and the actual cutofftill(x), not endpoint separation. Sum the finitely many plaquette types and signed orientations at exponent r−1 and absorb their fixed count into exponent r. The detailed atom and tail support-conversion estimates are still the inputs to these two deductions. The inherited logarithmic exponent has six disjoint families: regular occurrences, cubic atoms, quartic atoms, loop atoms, fine path tails, and coarse path tails. Its scalar phase is θ = −orientation · β. The beta band and |orientation| ≤1 imply |θ| ≤2 betaF; the coarse tail uses −θ and obeys the same bound. The cubic and quartic multipliers therefore have rate 1, giving exponents 1 −3/2 = −1/2 and 1 −4/2 = −1. The loop multiplier has rate zero and uses its own incoming loop budget. The two tail families allow arbitrary negative powers. Given r > −1/2, choose −1/2 < s < r. All six families have sales bounds betaF s eventually, with the incoming regular bound weakened to s. Their disjoint-union price is at most 6 betaF s ≤betaF r eventually. The forced family retains its independent betaF r′ price for any r′. This proves the inherited dictionary bound with the actual fine and coarse beta factors already included. Every footprint remains nonempty: regular and forced supports retain their anchor or inventory, atom 99 ## Source page 102 A Wilson Yang–Mills construction J. Rodgers footprints retain the rooted path origin, and tail-pair footprints retain their designated root. These facts support the individual-letter smallness argument used in the optional dictionary assembly. 6.65 The non-warning primitive dictionary Before taking connected components, keep three disjoint kinds of logarithmic letter: inherited fine- scale log letters, local short-correction letters, and Gaussian chain letters. The last kind has five families: the negative actual log-determinant series, the positive blank-reference log-determinant series, the mean-source series, the first linear mean correction, and the remaining quadratic chain correction. In the integrand their signs and factors are respectively −ArmA/2, + ArmAblank /2, +ArmS/2, qfirst and chain. These signs belong to the exponent itself. For an inherited log letter take its actual old exponent evaluated on the lifted fine field and multiply it by the enforcement profile of its own full footprint at parameter 2a. For an inherited forced activity use its actual K coefficient with the same footprint enforcement. These profiles lie in [0,1], so they cannot increase the supplied absolute bound. The full exponent farLog is the disjoint sum of these inherited exponents, the Short correction, and the five chain families. Each optional activity is exp(farLog)−1. Supports are assigned before evaluating any coefficient: inherited letters use their old support enlarged by the prescribed halo; short letters use their sleeves; the chain series use their receipt supports, and the first correction uses its town support. Their extra charges are the old extra charges, zero for short letters and first corrections, and the series index for the other chain letters. Every support is nonempty, including coefficients which happen to vanish. This keeps the support price independent of a field-dependent selection. The numerical assembly has the following exact form. Let the Gaussian-chain estimates give envelope families with a common Sales bound N(x) of rate −1/2. Applying the same estimate to the blank geometry controls the positive reference determinant term with its own envelope, not by identifying it with the actual determinant. Let the local Short estimate have envelope Z(x), whose support sales bound X(x) has rate −1/2. For the latter count, the number of captions per coarse site is at most |Signed|spots(L) + 2itineraryDensity(L), and a sleeve of halo l is contained in the radius-(l+1)B neighborhood of its caption. Its common sales price is bounded by ballCost((l + 1)B) · captionDensity · Z(x) · exp(v ballCost(l + 1)). The radius census has rate zero, so this retains rate −1/2. The inherited log and forced activities have their own preassigned envelopes from the incoming representation. Combine the log envelope, the Short envelope, and the chain envelopes by disjoint union to obtain b0. Their sales prices add. Choose exponents −1/2 < s < b < 0. The inherited log budget can be taken as betaF s, whereas X+N has rate −1/2, so T(x) = betaF(x)s + X(x) + N(x) has rate s. Eventually T ≤1 and 2T ≤betaF b. These thresholds are chosen uniformly before quantifying over the geometry and fields. The combined sales bound and nonempty support imply b0i ≤T ≤1 for every letter. The absolute logarithm is bounded by b0i, and 100 ## Source page 103 A Wilson Yang–Mills construction J. Rodgers | exp(t) −1| ≤2b0iwhen|t| ≤b0i ≤1. Thus yi = 2b0i bounds both the raw logarithm and its Mayer activity, with combined sales price at most betaF b. The inherited forced envelopes retain their separately prescribed betaF u budget for any fixed u. A dummy geometry at the same x, cutoffand width is used only to read the common numerical envelopes; the bounds themselves apply to every admitted geometry, every forest, and every field. This completes the dictionary assembly conditional on its inherited, Short and chain estimates, which are unfolded next. 6.66 Local driving estimate for the nonlinear Gaussian floor The local driving estimate supplies the quantitative bridge between the geometric predictor and the activity expansion. It must be applied to the specific retained and noise coordinates constructed above, on their declared healthy chart. The proof keeps the quadratic positive floor while assign- ing the cubic, quartic and path errors to the finite envelopes already selected. The remaining subsections convert those anchored envelopes into the support-hit bounds required for the marked integration. We now supply the local estimate used in the earlier quadratic-plate absorption argument. Fix one real accepted sector and its forest. For a variation u, let uhot be its restriction to the active bonds. At a coarse site o, define Io(u) =   X dist(o,addr(e))≤3B ∥uhot(e)∥2   1/2 , Ro(u) =   X dist(o,root(p))≤Rmac+2 ∥motion(u)p∥2   1/2 . Each individual term in either sum is bounded by its square root, because all summands are nonnegative. Suppose a gauge transformation z makes all background links within distance 2B of o have chord at most h ≥0. The accepted local stencil estimate then says that the flat row and the background-oriented row, applied to Adz uhot at a rod rooted within Rmac + 2 of o, differ by at most C 2cG(B + 1)h Io(u). This uses only the background links on that local box. Adjoint action preserves the color norm. The rod is selected because an active bond is separated from the forbidden seeds by the prescribed buffer; the triangle inequality places its entire stencil inside the clean region. Gauge covariance identifies the oriented row with the corresponding measured motion component. Its norm is at most Ro(u). Consequently the flat row is bounded by Eo(u) = Ro(u) + 2CcG(B + 1)h Io(u). (NF1) Here is how the local Poincaré estimate turns (NF1) into control of a bond. Write w for the logarithmic color variation of Adz uhot. On every forest trunk uhot is zero, hence its collected logarithmic variation is zero. The forest link estimate therefore bounds the variation at an active bond e, with o = addr(e), by ∥uhot(e)∥≤K X x∈box(Rmac) X α∈PlaqAxes ∥curl w(x, α)∥+ ∥bill w(o, eaxis)∥. The low face row in (NF1) is √qdir curl w, where qdir > 0 is the direct quadratic weight. Its norm is at most Eo(u), so each face norm in this sum is at most q−1/2 dir Eo(u). The pin row is − √ k bill w, 101 ## Source page 104 A Wilson Yang–Mills construction J. Rodgers with k > 0 the fixed penalty, and likewise its bill norm is at most k−1/2Eo(u). All the face roots lie within Rmac + 2 of o, by the parent-distance bound. Thus ∥uhot(e)∥≤UEo(u), U = K| box(Rmac)|| PlaqAxes |q−1/2 dir + k−1/2. (NF2) No global pure-gauge representation of the background has been assumed. The clean-background flattening estimate supplies the gauge in (NF1) with h ≤Y s(x), uni- formly over the actual clean boxes, where s(x) has rate x−1/2 in the source’s rate calculus. The local chord chart supplies this bound by writing each transformed link as the exponential of a Lie-algebra element of norm at most s(x), then using its uniform chord bound. The chart radius is fixed before taking x large. Put M(x) = 2UCcG(B(x) + 1)Y s(x). Equation (NF2), and the trivial zero bound on inactive bonds, give ∥uhot(e)∥≤URaddr(e)(u) + M(x)Iaddr(e)(u). The neighborhood census bounds the sums of the squared local quantities by their global sums times swept(L, d) = ballCost(d)L4| Axis |. Squaring the preceding inequality with (a + b)2 ≤ 2a2 + 2b2, summing, and requiring 2M(x)2 swept(L, 3B(x)) ≤1/2, yields X e ∥uhot(e)∥2 ≤4U2 swept(L, Rmac + 2) X p ∥motion(u)p∥2. (NF3) This smallness holds on a common tail: the radius census has rate zero, while M has rate −1/2, so the absorbed quantity has negative rate. All constants are chosen after the block side L, before the sector and the variation. Finally the real plate is the motion Gram matrix plus the identity on inactive coordinates. The exact normalization is ⟨v, formT form v⟩= scale2 X p ∥motion(u)p∥2. The Euclidean square norm splits into its active and inactive squares. If a > 0 is a common constant in (NF3), the plate therefore obeys ⟨v, plate v⟩≥min(scale2 /a, 1)∥v∥2. This derives the uniform Gaussian floor from one accepted branch’s local forest inequality and clean-background flattening, rather than assuming a floor for the stacked action. The forest inequality used in (NF2) is obtained directly from the nonlinear slice-shell inequality. For a logarithmic direction w that vanishes on the forest trunks, consider Ut = represent(tw). Then Ut remains on the same forest slice for every real t, and U0 = 1. All predictor frame weights are usable near t = 0: the minimum condition holds at the identity and on a neighborhood, and there are finitely many coarse roots. The shell inequality, with comparison background the identity, gives chord(Ut(e)) ≤K porto(Ut) + chord(prediction(Ut)(o, eaxis)). Each local port is bounded by the full sum of plaquette chords over the macro box. This may overcount periodic plaquettes, which only enlarges a nonnegative sum. 102 ## Source page 105 A Wilson Yang–Mills construction J. Rodgers If a group-valued curve Vt starts at the identity and has matrix derivative colorMat(v), the normalization of chord and the color norm imply |t|−1 chord(Vt) −→∥v∥. Indeed the derivative quotient of its matrix is colorMat(v); take its norm and divide by the fixed color scale. Apply this fact to the edge, to each plaquette holonomy, and to the actual predictor. Their derivatives are respectively the collected edge variation, its linear curl, and the predictor bill. Divide the shell inequality by |t|, let t →0 through nonzero real values, and pass through the finite port sums. The resulting inequality is exactly the forest inequality used in (NF2). In particular its pin term is the derivative of the nonlinear predictor chosen by the construction. This argument also gives the fixed-block vertical Hessian gap. If X = box(Rmac), squaring the forest inequality and applying Cauchy–Schwarz to the two finite sums yields X e ∥w(e)∥2 ≤β X b ∥curl w(b)∥2 + γ X b ∥bill w(b)∥2, with β = 2K2L4| Axis | |X|2| PlaqAxes |, γ = 2L4. For fixed offset the map from coarse root and plaquette type to the fine plaquette is injective, so summation over coarse roots costs |X|, even when different offsets represent the same periodic site. Parent fibers have exactly L4 elements. These two observations account for all factors and avoid any period-dependent norm equivalence. On a vertical forest coordinate v, the stretched field has collected value zero on trunks and ve on the complementary bonds. The Hessian of the tested action is bounded below by a X b ∥curl w(b)∥2 + k X b ∥bill w(b)∥2, where a > 0 is the curvature-metric floor. The pin term has this sign because the vertical coarse mismatch is minus the bill. Therefore Hess(v, v) ≥min(a, k) 1 + β + γ X e/∈trunks ∥ve∥2. (NF4) The Hessian restriction follows by composing the joint analytic action with the linear vertical inclusion; its second derivative is the joint second derivative evaluated on two vertical vectors. The local stencil comparison in (NF1) has a separate elementary transport proof. A carrier from x to y has path length at most 4 dist(x, y). When all links on a surrounding radius-d box have chord at most h, and that path lies in the box, the carrier’s chord is at most dh, by the product chord inequality. To apply it using only local hypotheses, replace all links outside the box by the identity; locality leaves the carrier unchanged. The matrix adjoint estimate ∥Adg b −b∥≤2cG chord(g)∥b∥ then bounds straightening of a local variation of size M by 2cGdhM. A bounded stencil of norm at most C, depending only on its declared lookup neighborhood, transfers this to a row error at most 2CcGdhM. One can see the locality step explicitly by cutting the difference of the two variations to that neighborhood; its sup norm is at most the local bound, and the stencil sees the same difference. 103 ## Source page 106 A Wilson Yang–Mills construction J. Rodgers Under a signed coordinate change, a reversed link contributes a further adjoint correction of at most 2cGhM; forward links contribute none. The transformed lookup is contained in the original box enlarged by one. The total comparison is thus ∥orientedRow(V, u) −orientedRow(1, u)∥≤2CcG(d + 1)hM. Taking the box and variation bounds specified before (NF1) proves the asserted estimate. Adjoint action and the signed displacement preserve each individual variation norm. The same transport calculation preserves the entrywise decay needed for the matrix upper price. A single-bond impulse remains a single impulse after the signed displacement and carrier conjugation, and its color norm remains unchanged. The untransported stencil’s impulse decay therefore implies ew dist(e,s focus(p))∥orientedRows,p(V, 1eb)∥≤2Cew(predReach +7)∥b∥, where w > 0 is the common stencil width. The extra one in this exponent pays the possible signed- link endpoint shift. Thus the local comparison does not introduce a volume factor into the matrix estimate. Finally the small logarithmic background in (NF1) is constructed, not assumed. For a clean radius-(d + 1) box, d ≤32B(x), axial gauge flattening gives a gauge z with chord at most u(x) = 16(2(32B(x) + 1) + 1)5| PlaqAxes | 2 fiery(x) on the radius-d box. The accepted temperature comparison bounds the clean plaquette threshold by 2 fiery(x). A fixed local logarithm chart extends these transformed links to a whole logarithmic array b, agreeing on this box, with ∥b∥≤Clogu(x), once u(x) lies in its fixed chart radius. The radius factors have rate zero and the fiery threshold has rate −1/2; hence H(x) = Clogu(x) has rate −1/2 and eventually lies in that chart. This supplies the function s(x) used in (NF3). The logarithmic product estimates used here follow from the analytic local map B(a, b) = log(exp a exp b). It satisfies B(0, 0) = 0 and DB(0, 0)(a, b) = a + b, by differentiation of matrix multiplication and the local inverse relation between logarithm and exponential. A bounded second derivative on a sufficiently small closed ball therefore gives ∥B(a, b) −(a + b)∥≤C∥(a, b)∥2, ∥B(a, b)∥≤C∥(a, b)∥. The local exponential recovers the product on that ball. Applying this at each link gives the same constants for every finite lattice in the sup norm, since the operation is coordinatewise. This justifies use of a genuine left group chart in the comparison, including its quadratic error. The extension of local logarithms used above has a particularly simple form. The analytic matrix logarithm chart satisfies log 1 = 0 and, on a fixed neighborhood, exp(log U) = U, ∥log U∥≤C chord(U). The second inequality follows from the local Lipschitz bound of the analytic chart and ∥mat(U) − 1∥= cG chord(U). Choose the chord radius small enough to lie both in the Lipschitz neighborhood and in the neighborhood of the right-inverse identity. On the requested local set of edges define be = log Ue, and outside it define be = 0. This is a whole finite array with ∥b∥∞≤Ch; its exponential agrees with U on the requested set. No continuous cutoffor global logarithm of U is involved. 104 ## Source page 107 A Wilson Yang–Mills construction J. Rodgers The adjoint estimate in (NF1) is likewise a direct matrix inequality. For a unitary U and a color matrix X, UXU∗−X = (U −1)XU∗+ X(U∗−1). Unitary invariance of the Frobenius norm and submultiplicativity bound its norm by 2∥U −1∥∥X∥. Dividing by the color scale proves the stated physical-color inequality. The path chord inequality follows by induction on path length, using the product triangle inequality and invariance of chord under inversion; it also applies to paths with repeated edges. A related estimate needed in the local vertex remainders is ∥curvature(face(exp b)) −curl b∥≤C∥b∥2 ∞ on a fixed sup-norm ball, with the same constant for every lattice. To prove it consider the analytic finite-dimensional map (a0, a1, a2, a3) 7−→curvature(exp a0 exp a1 exp(−a2) exp(−a3)). It vanishes at zero and its derivative there is a0 + a1 −a2 −a3. Taylor’s theorem with bounded second derivative gives the quadratic error and a uniform bound on this linear part. Substituting the four actual face coordinates gives the result; their maximum norm is at most ∥b∥∞, even when periodicity identifies two or more of these coordinates. 6.67 Common envelopes for the non-packet activity letters The regular-log and forced-letter estimates must use one envelope independent of the background sector, its forest, and the Gaussian variable. The assembly is as follows. Fix ε > 0 below one quarter of the support exponent and below one eighth of each of the two stencil widths, a nonneg- ative footprint weight v, and a retained order l ≥4. Choose the block side large enough for the inherited footprint estimates. After this choice fix the positive Gaussian floor and matrix upper- price parameters, the local chart constants, and the remaining real sector parameters. Suppose the covariance-window error has rate −1. For every exponent −1/2 < b < 0 and any prescribed forced-letter exponent u, there are common nonnegative weights wi for forced letters and yp for the other logarithmic letters such that | bagi(U)| ≤wi, | farLogp(z)| ≤yp, |efarLogp(z) −1| ≤yp. (NF5) Their weighted footprint sums have respectively budgets βu F and βb F . These are the sales inequalities used in the connected expansion: the weights include the specified footprint exponential, overlap price, radius charge and retained derivatives. All the ancillary costs enter before the budget is reduced to the displayed powers. To see the exponent bookkeeping, the local estimates give a base envelope b(0) p , obtained by combining the inherited physical letters, the finite local logarithmic letters and the covariance- window letters. Their combined sales budget has the form T(x) = βF (x)s + X(x) + N(x), −1/2 < s < b, where X, N have rate −1/2. Thus T has rate s; eventually T ≤1 and 2T ≤βb F . The envelope is chosen by evaluating the uniform estimates on an empty-seed dummy sector at the same radius and cutoff, then retaining their background-independent weights. This is legitimate because the sector hypotheses hold there and the estimates themselves quantify over all real backgrounds. It makes no choice based on the subsequently integrated Gaussian value. 105 ## Source page 108 A Wilson Yang–Mills construction J. Rodgers The base envelope bounds each logarithm and is individually at most one, since a nonnegative weighted sales sum bounds each of its summands. For |q| ≤a ≤1, the elementary bound |eq −1| ≤ 2a supplies the Mayer envelope. Consequently set yp = 2b(0) p ; multiplying its sales budget by two costs at most βb F , as just arranged. The forced-letter envelope keeps its separately prescribed budget. The real localization factors all lie in [0, 1], so multiplying by them can only decrease these pointwise bounds. This establishes (NF5) with one set of weights usable in every sector and for both the raw logarithms and their Mayer factors. The letters combined in (NF5) are the following actual terms. Besides the inherited activity logarithms, there are three finite local types: row-energy remainders indexed by a signed orientation and a row, a diagonal logarithmic correction indexed by a fine bond, and a Haar-chart correction indexed by an active fine bond. There are also five chain types: the principal log chains for the real plate with coefficient −1/2, the corresponding blank-plate chains with coefficient +1/2, the covariance correction chains with coefficient +1/2, the affine first-order correction, and the residual quadratic chain. Their chain lengths are their additional charges; the affine bond term has charge zero. Their supports are the corresponding full receipts enlarged by the halo parameter lB, including every lookup used by the term. For a row with coarse center o, the actual short exponent is zero within distance 200B of the seeds, and otherwise is −βθ [go(u) lighto(u) + extractedo(μ)] , 0 ≤go(u) ≤1, 0 ≤θ ≤1. (NF6) The diagonal term is zero when either endpoint of its bond meets the radius-200B seed dome, and otherwise equals 1 2 joint(plateblank) −joint(plate)  e. The Haar term is the localized chart logarithm on active bonds outside the radius-304B seed dome and is zero otherwise. These signs are exponent signs: the activity is eShort −1. On an intact real sector the row quantity light equals the actual debit. Indeed outside the dome the local slice is interior and the predictor is alive. The kept-path test is satisfied on the local region by the clean plaquette bound; outside it the exponential path threshold is at least two and the universal chord bound supplies the test. The exact row-energy splitting can therefore be applied to the actual configuration, with its existing choices. Here is the size estimate for (NF6). A nonzero localization factor supplies the local variation bound required by the grafted-input energy estimate. That estimate bounds |go(u) lighto(u)| by a common J(x) of rate −3/2. The protected-mean estimate bounds each of the at most two parent path values of μ by A(x), of rate −1. The extracted term is a sum of at most two nonnegative half-squares, so it is at most A(x)2. Since β ≤2βF (x), its row contribution is bounded by W(x) = 2βF (x) J(x) + A(x)2, of rate −1/2. The diagonal-log difference has a common bound Ediag(x) of rate −1/2, obtained by local gauge flattening and the principal-inverse comparison for the real and blank plates with the same window geometry. The localized Haar logarithm has bound CHaar fiery(x), also of rate −1/2. Adding these three nonnegative bounds gives a common bound for every short letter. To turn this pointwise bound into its sales bound, the center census is | Signed | spots(L) + 2 density(L). Every support is contained in the radius-(l + 1)B neighborhood of its center. Thus a common short-letter envelope Eshort(x) has weighted budget at most ballCost((l + 1)B) | Signed | spots(L) + 2 density(L) Eshort(x)ev ballCost(l+1). 106 ## Source page 109 A Wilson Yang–Mills construction J. Rodgers Its rate remains −1/2. This is a local census bound, independent of the total number of sites. The inherited letters satisfy their own support estimates before they are joined to these local types. For the incoming regular activities, use their common healthy-sector sup envelopes; the masked activity is zero offthe healthy set. For forced activities choose an anchor in each nonempty inventory, which lies in its support. The incoming support budget then becomes an anchored budget at that anchor. The footprint conversion expands each old support to its inspected coarse footprint, paying the retained overlap and charge weights. Its exponential reserve is available once 4ε is less than the incoming support exponent and L exceeds the fixed conversion threshold. These choices give the regular budget βq F for every q > −11/10 and the forced budget βu F for every prescribed u. A degree-k translated local atom, with coefficient bound of rate a and aggregate magnitude budget of rate b, has pointwise envelope equal to that coefficient bound times fiery(x)k times its fixed atom magnitude. Its sales budget therefore has any exponent strictly greater than a −k/2 + b. Finite signed orientations only multiply the budget by their fixed cardinality. Applying this rule to cubic and quartic terms with coefficient of rate one, and to the loop quadratic term with coefficient of rate zero, places all three below every exponent strictly greater than −1/2. The strict inequalities leave room for footprint conversions and their polynomial charges. True path tails, including the lower-scale tail, have an arbitrary power saving after their positive retained cutofforder is imposed; their additional orientation and plaquette-type sums are finite. The six inherited classes can consequently be combined with an intermediate exponent s < b, and the fixed factor six is absorbed by βb F /βs F →∞. This supplies precisely the inherited part of the common envelope construction (NF5). The ordered-chain support estimate can be stated and proved independently of the matrix formulas. A route of length n is an ordered list of n fine bonds after a starting bond. A hop is allowed if an endpoint of one bond is within coarse distance d of an endpoint of the next. Hence their coarse centers are within d + 2. The number of allowed next bonds is at most Pd = ballCost(d + 2) density(L), density(L) = L4| Axis |. Induction on length bounds the number of allowed routes from any starting bond by P n d , including routes with repeated bonds. Attach to every route node the radius-mB halo of both bond endpoints. The full receipt is the union of these halos. If d + 2 ≤mB, successive halos meet. Each halo has B-cover number at most steady(m) = ballCost(m + 1). The union therefore has cover number at most (n + 1) steady(m). Every point in it is within (n + 1)(mB + 1) of the starting center, by the triangle inequality along the route. Suppose a chain letter has n+u hops, with u fixed obligatory hops, charge n, and magnitude at most Cqn on allowed routes and zero on the others. Fix a coarse site x and sum over all such letters whose receipts contain x, including their charge factor e2εn and footprint factor ev coverNumber. Its possible starting bonds number at most ballCost((n + u + 1)(mB + 1)) density(L). The elementary inequality n + u + 1 ≤(u + 1)2n gives, in four dimensions, ballCost((n + u + 1)(mB + 1)) ≤16n ballCost((u + 1)(mB + 1)). 107 ## Source page 110 A Wilson Yang–Mills construction J. Rodgers Combining the starting-point count, route count, magnitude, charge and cover costs bounds the length-n contribution by HC(qR)n, where H = ballCost((u + 1)(mB + 1)) density(L)P u d ev(u+1) steady(m), R = 16Pdev steady(m)+2ε. If qR ≤1/2, summing the geometric series gives the sales bound 2HC, uniformly in x and in the lattice period. This counts every endpoint and its complete lookup halo before any geometric shortening. Summing over the finite route coordinates at fixed n, then over n, is justified by nonnegativity; finite partial sums obey the same bound. The localization and chart factors used in these estimates have exact identities worth recording. Let γa(u, e) = χinner(at, ∥ue∥), Γa(u, Y ) = Y e∈Y γa(u, e). Both lie in [0, 1]. If at > 0, then ∥ue∥≤at implies γa = 1, and γa ̸= 0 implies ∥ue∥≤2at, hence γ2a = 1. Thus a nonzero Γa bounds every variation on its selected edge set and gives Γ2a = 1 there. For the Haar chart density J, the localized logarithm is γ2a log J. If γa ̸= 0 and 2at lies within the chart radius, its exponential equals J exactly. Its all-argument size bound follows without assuming the argument is in the chart. If γ2a = 0, the localized logarithm is zero. Otherwise ∥ue∥≤4at. Since J(0) = 1, positivity and analyticity on a fixed chart give | log J(ue)| ≤C∥ue∥≤4Cat, once 4at lies in that fixed neighborhood. Multipli- cation by γ2a ≤1 preserves the bound. This proves the uniform Haar contribution used in (NF6), with the shrinking threshold chosen on one common tail. 6.68 Exact warning factors and their vanishing region There are five detector addresses: a fine plaquette, two distinct types of coarse bond, a coarse site, and a fine site. Their raw factors are respectively the fine cutoff, the predictor-noise cutoff, the pin-frame mass, the site-frame mass, and the contracted flag charge. Each address has its fixed district containing its focus, and every district point is within one coarse step of that focus. The warning support is its radius-lB halo. There is one further warning type, the complement of a fine-bond logarithmic cutoff, with the two-endpoint support used above. A detector is compulsory when its district meets the radius-300B seed dome. Its optional warning letter is then zero. Otherwise it is Γ2a(u, plaza(haloB(district)))(rawStorm(u) −1). (NF7) The fine-bond warning is γa(u, e) −1 on active bonds outside the radius-304B dome, and zero on all other bonds. Set Cstorm = 1 + | FrameTag | + | Points |. Every raw detector lies between zero and Cstorm: the two cutoffs lie in [0, 1], each frame mass is bounded by its finite number of summands, and the contracted charge is zero or one. Thus | rawStorm −1| ≤Cstorm. Since all localization factors lie in [0, 1], every warning has absolute value at most Cstorm. Write the Gaussian-coordinate input as u = μ+β−1/2z. A quiet box of radius D about o means |ze,α| ≤ξt p β whenever dist(o, addr(e)) ≤D. (NF8) 108 ## Source page 111 A Wilson Yang–Mills construction J. Rodgers If ∥μe∥≤Mt, this implies ∥ue −μe∥≤nGξt, ∥ue∥≤(M + nGξ)t, where nG is the number of color coordinates. The estimate follows by bounding the color Euclidean norm by the sum of the absolute coordinate values. It uses no independence between coordinates. Here are the choices that make every ordinary detector equal to one on a quiet box. Fix the background guide bound A > 0. First choose τ = √qdir/32 > 0, then take L large enough that 2 √qdir (A/L2 + τ) < 1 4. After L, take W large enough that 2 √ k (A + τ) < W. Choose ξ > 0 so small that nGξ ≤1 and the row-stencil response to the noise in (NF8) is at most τt/3. A concrete admissible choice is ξ = min (nG + 1)−1, τ 3nG scale−1 |Y | + 1 ! , where Y is the uniform row-price bound. The remaining row bill has three contributions: the nonlinear-to-linear row error, the row change from μ to u, and the protected row at μ. The first and third have rate −1 and hence are eventually at most τt/3 each, since the threshold t has rate −1/2. The middle contribution is at most τt/3 by the choice of ξ. Thus every relevant actual row bill has norm at most τt. For a fine plaquette add its guide norm at most At/L2; its weighted curvature is at most (A/L2 + τ)t. The local curvature-to-chord inequality and the displayed fine quota give its chord strictly below t/4. For a predictor bond the same argument, with guide norm at most At and penalty k, gives mismatch chord strictly below Wt. The local small-field tag identities give site- frame and pin-frame masses exactly one. These estimates apply outside the radius-30B dome on a quiet radius-B box and use the actual predictor choices. To deduce vanishing of (NF7), suppose the radius-2B box about its focus is quiet. If the detector is compulsory, the letter is already zero. Otherwise its focus is outside the radius-300B dome. Every point within distance B is then outside the radius-30B dome, and its radius-B box is quiet, by the triangle inequality. The preceding conclusions apply there. The fine and noise cutoffs are on their unit plateaus, and both frame masses are one. Finally a nontrivial contracted flag would supply a touched plaquette with chord greater than t. Its parent lies within the same radius-B neighborhood, where its chord is less than t/4, a contradiction. Hence the flag charge is also one. Each of the five raw detectors is therefore one, so its warning letter vanishes. The fine-bond complement similarly vanishes whenever its protected mean and noise place ∥ue∥ on the unit plateau of γa. Consequently a nonzero optional warning has a coordinate violating a bound of the form (NF8) within its fixed enlarged support. This is the witness statement used by the joint Gaussian catastrophe estimate. It is valid for an individual processed warning regardless of the values of the other cutoffs, so no product of independent warning probabilities is assumed. 6.69 From anchor bounds to support-hit bounds Here is the summation underlying the inherited support budget. Let a map p from fine to coarse sites have at most P fine preimages per coarse site. Labels i have fine anchors oi, nonnegative 109 ## Source page 112 A Wilson Yang–Mills construction J. Rodgers weights bi, and integer sizes di, with X i:oi=y bieAdi ≤γ for every fine site y. Suppose their coarse supports Xi obey, for x ∈Xi, dist(x, p(oi)) ≤W(1 + di), covB(Xi) ≤v + v′di. Let gi ≤Cbi(1 + di)k, where gi, C ≥0. If a ≥0, ε > 0, and av′ + 2ε ≤A, then X x∈Xi di≥m giea covB(Xi) ≤eavγCP(2W + 1)4ε−(4+k)(4 + k)! eε 1 −e−ε e−εm. (NF9) The sum on the left is over labels i, with x fixed. To prove it, first restrict to di = j. All anchors lie over the coarse ball of radius W(1 + j) about x, so there are at most P(2W + 1)4(1 + j)4 possible fine anchors. Sum the given anchored budget over these anchors and use giea covB(Xi) ≤bieAdiC(1 + di)keav−2εdi. The inequality (1 + j)4+k ≤ε−(4+k)(4 + k)! eεeεj follows by retaining one term in the exponential series at ε(1 + j). The degree-j bound is therefore the numerator in (NF9) times e−εj. Sum its geometric tail. This proof also covers infinitely many address variants, since every finite partial sum has the same bound and all summands are nonnegative. For a block ratio L, P = L4: a fine site is uniquely specified by its coarse parent and its four residue coordinates. For the halo enlargement of the parent image of an old support, the geometric cover estimate used earlier reads covB(Xi) ≤v + 10v L di, v = (2(l + 6) + 1)4, where di = covM(Yi), M ≤2B, and the enlargement has halo radius lB. Each point of it is within (l + 20)B(1 + di) of the parent of any chosen anchor in Yi. Thus (NF9) applies with W = (l + 20)B and v′ = 10v/L. In the inherited sales bound the extra tariffis e2εdi; absorb it first into the old weight, replacing the anchored exponent δ by δ −2ε. The reserve condition is then exactly a(10v/L) + 2ε ≤δ −2ε. This gives the stated payment, with its L4 child factor and polynomial-degree cost, rather than an unspecified enlargement constant. The geometric cover estimate just used has the following direct proof. Write L for the block ratio and use the periodic maximum-coordinate distance throughout. A site and the embedded parent of that site are at distance at most L/2. Embedded coarse distances are multiplied by L. The triangle inequality therefore gives L dist(p(x), p(y)) ≤dist(x, y) + L. (NF10) These statements use centered residue representatives and remain valid on winding paths. 110 ## Source page 113 A Wilson Yang–Mills construction J. Rodgers Choose a realizing M-cover A of Y , with |A| = d = covM(Y ). Its centers are connected at distance 2M + 1. A unit-step walk visits them all in at most N ≤d(4M + 2) steps. One way to construct it is to start at one center and repeatedly attach an unvisited center adjacent to the visited connected set, inserting an out-and-back detour of length at most 2(2M +1). For the empty cover use the empty-support case separately. Sample this walk at times LBj, including time zero. Its parent samples form a coarse set D with LB|D| ≤N + LB. Successive samples, and the parent of any intervening walk point and its preceding sample, are within 2B by (NF10). A fine point of Y is within M of a visited cover center, so its parent is within 3B of the parent of that center, since M ≤2B and B ≥1. It follows that the halo of radius lB around p(Y ) lies within (l + 5)B of D, with the required connected covering centers. To replace this larger-radius cover by a B-cover, tile each such ball with points spaced B apart in the four coordinate directions. For a ball of radius mB, the integer coordinate box with |zi| ≤m+1 suffices: coordinatewise integer division puts every point within B of one of its grid points. The grids are connected, and the connections between their centers join their union. Their combined cardinality is at most |D|(2(m+1)+1)4; taking images on a torus only decreases cardinality. With m = l + 5, this gives covB(halolB p(Y )) ≤|D|(2(l + 6) + 1)4. Finally, 4M + 2 ≤10B. Multiply the last inequality by L and use the sampled-walk bound to obtain L covB(halolB p(Y )) ≤(L + 10d)(2(l + 6) + 1)4, as claimed. The same walk bound gives dist(x, y) ≤d(4M + 2) + 2M for x, y ∈Y . Combining it with (NF10) and the halo radius proves the anchor-distance estimate with W = (l + 20)B. This completes the geometric input to (NF9). 7 Marked integration and the coupling reset A marker is a finite algebraic label used to keep track of an insertion in a product. Distinct markers multiply only when their modes are allowed and disjoint. This nilpotent algebra makes the insertion bookkeeping finite, while the underlying scalar activity gas still requires absolute summability. The construction below first restores the actual insertions, then performs a coupling reset as an exact exponent identity, and finally integrates the primitive density with all its marked and forced terms. 7.1 The insertion algebra and its local prices Modes and repeated occurrences The marker algebra keeps track of which insertions occur in the integrand. Its nilpotence is a finite algebraic convention, not a statement about the physi- cal fields. Repeated occurrences of one insertion are assigned distinct occurrence marks when the corresponding product is formed. Their lattice variables can still coincide and remain the same inte- gration variables. This is why disjointness of marker modes does not justify treating the associated scalar factors as probabilistically independent. Fix a finite marker set V and the finite census of allowed marker modes. A mode is an allowed subset A ⊆V ; multiplication joins disjoint subsets and discards products outside the census. Write 111 ## Source page 114 A Wilson Yang–Mills construction J. Rodgers [P]A for the coefficient of a polynomial in this algebra, and eA for the corresponding monomial. The empty mode is the scalar mode and e{v} is a singleton mode. For positive marker weights wv, put tw(A) = Y v∈A wv, ∥P∥w = X A tw(A)|[P]A|. This norm is subadditive and submultiplicative: expand each product coefficient into disjoint pairs, apply the triangle inequality, and use tw(A ⊔B) = tw(A)tw(B). The truncated exponential Exp P, for P with zero scalar coefficient, is the finite exponential series in the nilpotent ideal. Its norm is bounded by exp ∥P∥w. It obeys the exponential addition law for commuting zero-scalar arguments. An insertion representation consists of a density exponent E(U), primary functions Φv(U), forced profiles ωv(q, U), and higher-degree corrections H(S, U), K(q, U). Here S is a support and q = (Sq, Iq) is a support together with its flagged inventory. Let bq(U) be the scalar activity profile, regarded as a polynomial in the empty mode, and let F(U) be the configuration’s flagged set. Restoration means the following literal finite expression: X(U) = eE(U) Exp X v e{v}Φv(U) + X S H(S, U) ! × X P: Pattern(F(U),P) Y q∈P bq(U) + X v e{v}ωv(q, U) + K(q, U) ! . The pattern condition says the supports Sq are pairwise disjoint and the inventories Iq have union F(U). All sums here are finite at fixed torus and census. The corrections have only modes of cardinality at least two. They are gauge invariant, measurable, supported on their stated regions and scoped to markers whose pins lie in those regions. Exponential corrections vanish offthe healthy region and have zero scalar coefficient. A flagged correction can be nonzero only when Sq ∩F(U) = Iq ̸= ∅, with Iq ⊆Sq. These support and permission conditions are preserved by scalar multiplication. The price of a correction family Yi(U), with supports Si, is controlled by nonnegative envelopes ai satisfying |Yi(U)|w ≤ai and the sitewise bound sup x X i:x∈Si aiee cB(Si) ≤f. Here cB(S) is the minimizing B-cover number and e is the support exponent. The definition also includes summability of the envelopes. If every nonzero Yi(U) has some marker pin ov ∈Si, and e ≥0, then X i ∥Yi(U)∥w ≤|V |f. Indeed charge a nonzero term to all marker pins in its support. At least one is present. Since eecB(Si) ≥1, summing the sitewise bound over the |V | pins dominates the desired sum. The scope and positive degree conditions ensure the required pin is present. For the unmarked background profiles the same argument charges to all sites, giving P q ∥bq(U)∥w ≤|Λ|u, where u is their sitewise activity budget. Suppose also 112 ## Source page 115 A Wilson Yang–Mills construction J. Rodgers wv|Φv(U)| ≤x, X q wv|ωv(q, U)| ≤y. The exponential argument then has norm at most |V |(x + f), and the sum of norms of all field factors is at most |Λ|u + |V |(y + f). Dropping pattern restrictions increases the nonnegative norm bound for the field polynomial to X P Y q∈P ∥Tq∥w = Y q (1 + ∥Tq∥w) ≤exp X q ∥Tq∥w. Multiplication with the exponential bound proves ∥X(U)∥w ≤exp{E(U) + |V |(x + y + 2f) + |Λ|u}. The lawful-density bound on E(U) gives the uniform majorant used above. To bound a normal- ized mode integral, use tw(A)|[X(U)]A| ≤∥X(U)∥w, integrate against the probability prior, and divide by the positive density integral. The denominator lower bound yields the final insertion bill; the history treasury bounds that bill by |Λk|τk + |V |(x + y + 2f). This separates the elementary polynomial estimate from the vacuum estimates required to control its density and denominator. 7.2 Restoring the prescribed singleton insertions The correction representation must preserve the actual singleton insertions after every step. The following algebra explains that requirement. Given a represented density R, a primary function Φ, and a summable forced family gi, its singleton density is coinR(Φ, g)(U) = R(U)Φ(U) + eER(U) X i nickelR,i(U; g). The second term is the first variation of the activity gas in the forced direction. Grouping the summable family by its support-and-inventory barcode gives its profile; this grouping preserves the gas variation. In the restored polynomial, the scalar mode is exactly R(U). The coefficient of e{v} is exactly this singleton density: in a product contributing to a singleton mode, at most one factor can supply a singleton and every other factor supplies its scalar mode. The exponential contributes its linear coefficient and scalar coefficient one. Higher corrections contribute neither. These observations prove the scalar and singleton relay identities by finite polynomial expansion and the absolutely convergent grouping of activity coefficients. There is a useful converse. Suppose a polynomial has the form eER(U) Exp(F(U)) Field(K(U), FR(U)), where F has zero scalar coefficient, the scalar coefficients of K are the original background profiles, and its singleton coefficients are the prescribed forced profiles gv. If the singleton coefficients of the whole expression are coinR(Φv, gv), then the singleton coefficients of F are precisely Φv. Subtract the two singleton formulas: their gas-variation terms cancel, leaving R(U)([F(U)]{v} − Φv(U)) = 0. The nonvanishing of R(U) permits cancellation. Projection onto modes of cardinality at least two now splits F into its prescribed singleton part plus the higher exponential correction, and splits every Kq into background, prescribed singleton profile and higher flagged correction. This proves the desired restored representation, preserving the original density pointwise. To adjust an incorrectly placed forced singleton, use attached local responses ψv(S, U) supported on regions containing ov, vanishing whenever that region is healthy. Set 113 ## Source page 116 A Wilson Yang–Mills construction J. Rodgers J(U) = X v,S e{v}ψv(S, U). The closure operation on field factors joins overlapping background factors and attached re- sponses into their connected components, with the resulting support and flagged inventory recorded. Expansion into connected components gives the exact identity Field(Closure(K, ψ)) = Field(K) Exp J. Consequently replacing K by its closure and F by F −J preserves the full polynomial. In the closure expansion, terms containing no response reproduce K. Terms containing at least one response have zero scalar coefficient. Terms containing at least two responses have zero singleton coefficient. The contribution containing just one response equals that response multiplied by the background portion attached to its support. Thus closure leaves the background scalar profile fixed and adds exactly the hook of ψv to its singleton profile. It introduces no unspecified change of singleton field. After this identity, the preceding converse identifies the required primary functions from the known singleton densities. Keeping only higher modes produces the two higher correction families with their locality and permission conditions. Projection to higher modes decreases their weighted prices. The price of closure is controlled by the connected-component estimate: from a field budget W, response budget c, and pin occupancy M, its higher part has budget at most exp{W + M siteFactor(B)c} −1, provided the support-exponent loss pays (2B + 1)4{W + M siteFactor(B)c}. The connected-component forest-price argument below supplies this estimate. A reweighting bound on marker monomials then transfers both higher families to the requested weights. These are the algebraic and numerical operations used in the one-step retained-insertion estimate. The connected-component price estimate follows from a recursion on distinct labels. Here are its details. Let Xi be supports and fi ≥0, with sup x X i:x∈Xi fie(a+w)cB(Xi) ≤γ, a ≥0, (2B + 1)4γ ≤w. For a finite label set D and root region Y , let FD(Y ) be the sum of Q i∈A pi, pi = fieacB(Xi), over subsets A ⊆D every overlap-connected component of which meets Y . Include the empty subset with weight one. For each component choose one label whose support meets Y , and remove that label from the component. Every component of the remainder meets the removed label’s support. Different original components choose different labels. The chosen labels and the corresponding remainders reconstruct the original subset, so this assignment is injective. Allowing all choices of remainders, including incompatible ones, only increases the sum of nonnegative weights. Therefore FD(Y ) ≤ Y i∈D:Xi∩Y ̸=∅ {1 + piFD\{i}(Xi)}. Strong induction on (|D|) proves FD(Y ) ≤ewcB(Y ). Indeed the induction hypothesis makes each factor at most exp{fie(a+w)cB(Xi)}. The sum of these exponents over supports meeting Y is 114 ## Source page 117 A Wilson Yang–Mills construction J. Rodgers at most |Y |γ, by summing the sitewise hypothesis over Y . Finally the cover-volume bound gives |Y | ≤(2B + 1)4cB(Y ), hence |Y |γ ≤wcB(Y ). The same product estimate, with this inductive bound for the smaller label sets, also gives the sharper root-region bound FD(Y ) ≤e|Y |γ. If a nonempty overlap-connected subset A has union containing a fixed site x, it is one of the nonempty terms counted by FD({x}). For such a subset, cB [ i∈A Xi ! ≤ X i∈A cB(Xi). To prove this cover inequality, start with one support and add supports in an order with each new support meeting the preceding union; connectedness supplies such an order. The cover number of two meeting supports is at most the sum of their cover numbers. Induction gives the claim. Since a ≥0, its consequence for products and the preceding singleton-root bound imply X A̸=∅connected x∈∪i∈AXi eacB(∪i∈AXi) Y i∈A fi ≤eγ −1. For countably many labels, first bound every finite subsum by taking D to contain all its labels. Nonnegativity then proves summability and the same bound. No repeated-label expansion or logarithmic series is involved. Apply this estimate to the disjoint union of background-factor labels and response labels, with their respective envelopes. Their sitewise budgets add, so γ = W + M siteFactor(B)c. The en- velope of a connected component is the product of its label envelopes, by submultiplicativity of the weighted polynomial norm. Selecting only the permitted components can only decrease the nonnegative sum. Grouping them by their resulting support and inventory preserves the sitewise bound. This proves the closure price and exponent loss stated above. The exact closure identity also has a direct inverse to its component decomposition. Expand the original field polynomial and Q j(1+uj), choosing background factors and a subset of responses. Join labels whose supports intersect. A nonzero chosen response lies offits healthy region, so its support meets a flagged site. That site belongs to a chosen background inventory; thus the response’s component contains a background factor. Background supports in the original pattern are pairwise disjoint. Each resulting component consequently has a nonempty inventory, and the components’ inventories partition the original flagged set. Conversely, a permitted disjoint collection of such connected components recovers the unique original set of background and response labels by union. The coefficients are unchanged products of the same factors. This bijection proves the identity term by term. For the singleton responses used here, u2 j = 0; hence Q j(1 + uj) = Exp P j uj. This completes both the algebraic identity and its connected-component price estimate. 7.3 Finite transport inventory without changing the insertion The passage to a finite transport inventory uses two different reductions. First, a countable family of local leaves with one fixed dictionary key can be summed in that key. Suppose its complex coefficients are ri, its nonnegative leaf bounds are bi, and P i |ri|biescB(S) < ∞, where all leaves have the same support S. Sum the real and imaginary coefficient families separately. Each real sum is a real leaf with bound at most P i |ri|bi; locality, gauge invariance, measurability and its eligibility condition pass to the absolutely convergent sum. The transported quote also passes to the sum by the sound and tuned transport identities. Use charges (1) and i for the two resulting real leaves. They reproduce the original complex family both before transport and in every transported profile, at total marker price at most 115 ## Source page 118 A Wilson Yang–Mills construction J. Rodgers 2wv X i |ri|bi. An insertion tale uses only finitely many keys. Group its countable leaves by key, apply this construction within each group, and sum over the finite set of keys. The support weight is constant on each group, so this regrouping preserves the summed support budget. It gives finitely many singleton leaves reproducing the primary function minus its scalar displacement, the original forced profiles, and the coarse forced profiles minus their attached response hooks. A tale with price Av yields total support-weighted singleton price at most 2wvAv. Second, higher polynomial corrections can be placed in blank dictionary slots. For a nonzero coefficient function (h(U)) in mode A, write h = Rh + iIh, using two real leaves with that mode and charges (1,i). If |h(U)| ≤bA, their combined price is 2tw(A)bA. The locality, scope, gauge invariance and permission conditions are inherited coefficientwise. Only modes that are nonzero somewhere need be included. Summing their prices proves that a higher correction family of budget f becomes a finite leaf family of budget at most (2f). The dictionary contains a dormant regular slot of zero original coefficient for every nonempty support, and a forced slot of every admissible nonempty inventory with its exact eligibility condition. Place an exponential correction in the first kind of slot, and a flagged correction in the second. A nonzero higher correction has at least two scoped markers, so its support is nonempty. These placements preserve the original correction functions exactly, while the two families together cost at most (4f). Their modes all have degree at least two. The existence of these dormant dictionary slots is part of the history construction; this reduction uses that property explicitly. Combine the singleton leaves from the tales with these higher leaves. Their primary contribu- tions sum to X v e{v}(Φv −νv) + X S HS, and their forced contributions to X v e{v}ωv(q) + Kq. The singleton transported quotes reproduce the prescribed coarse forced profile minus its re- sponse hook. Higher leaves have no singleton quote, since their modes have degree at least two. Occupancy converts the singleton root budgets into a sitewise budget M siteFactor(B)d. Together with the (4f) correction budget this is the quantity T used in the numerical step. The exponential at a regular dictionary key must still be represented exactly. If ui are its finitely many tagged monomials, each has nonempty marker mode and u2 i = 0. Consequently Exp X i ui ! = Y i (1 + ui) = 1 + X i ui + X A: |A|≥2 disjoint allowed marker modes Y i∈A ui. Products with overlapping marker modes vanish, and products outside the census vanish. Each remaining product is a real product leaf at the same dictionary key, charged by the product of charges and carrying the union mode. Its price is the product of the original prices. These extra composite leaves have degree at least two; they do not change any singleton quote. They are added only at regular keys, so forced-slot contributions are untouched. This finite enlargement replaces the exponential by (1+) the enlarged key sum without changing its value. 116 ## Source page 119 A Wilson Yang–Mills construction J. Rodgers Expanding the represented activity gas now gives the finite transport expression. At each dictionary key choose either its old scalar contribution or one marked leaf; the ticket condition is that no two chosen leaves use the same key. Each term is the old unselected activity product multiplied by the chosen marked ingredients. The polynomial normalization of each marked leaf is exact: its positive fee times its normalized monomial is its original charge times its marker monomial and leaf bound. Thus no division by a zero total coefficient occurs. The original activity series is absolutely summable under the lawful-activity bounds; only finitely many keys receive marked perturbations. Their sum is still absolutely summable, so regrouping by barcodes and interchanging the finite ticket sum with the activity series are legitimate. The result is exactly the restored fine density with the scalar displacement factored out as Exp P v e{v}νv. The additional composite-leaf price is bounded as follows. For a key k, let pk be the sum of its unweighted leaf prices. The common support contains a marker pin, so the sitewise budget implies 0 ≤pk ≤T. The sum of all compatible composite prices at that key is bounded by the sum over all nonempty subsets: X composites at k price ≤ Y i at k (1 + fi) −1 ≤epk −1 ≤epkpk ≤eT pk. The penultimate inequality follows by integrating et ≤epk on 0 ≤t ≤pk. All composites at this key have the same support as their factors. Multiplying by its support weight and summing over distinct regular keys therefore gives a sitewise composite budget at most eT T: the disjoint key classes use disjoint sets of original leaf labels. Adding the original budget yields T + eT T. This proves the inventory bound with its stated constant. The normalized monomial used in the ticket expansion has fee c = tw(A)|q| and value eA(q/c). If c > 0, cancellation gives c eA(q/c) = eAq. If c = 0, positivity of tw(A) implies q = 0, so both sides vanish under the source’s zero-inverse convention. Its weighted norm is at most one in either case. This explicitly handles zero charges. Each such monomial squares to zero because A ̸= ∅, which also justifies its exponential expansion above. The remaining input at this point is the Mark-step reserve acting on this finite, fully priced inventory. 7.4 The repeated-occurrence rooted-tree bound Repeated occurrences matter in the rooted-tree estimate: two vertices may carry the same activity label and still be distinct vertices of the expansion. Summing over distinct supports too early would omit those terms or lose their factorials. The parent-map proof below keeps occurrences numbered until its nonnegative sums and factorial estimates have been completed; only then is the passage to connected sets made. Here is the tree estimate used in the cluster argument. Set VB = (2B + 1)4, t = VBγ/δ > 0, and F(Y ) = max{δcB(Y ), 1}. For nonnegative vertex weights wi obeying the sitewise budget at exponent a + δ, define K(Y, i) = 1Xi∩Y ̸=∅wie(a+δ)cB(Xi) tF(Y ) . Every finite row sum of this kernel is at most one: sum the sitewise bound over sites of Y to bound its numerator by |Y |γ ≤VBcB(Y )γ = tδcB(Y ) ≤tF(Y ). Its entries are nonnegative. Represent a forest on m numbered internal occurrences by a parent map p : {1, . . . , m} → {0, 1, . . . , m}, where (0) is an external root with support Y . Only parent maps admitting ranks strictly decreasing toward each internal parent are allowed. The support of every child must meet that of its parent. No condition prevents repeated activity labels at different occurrences. For a 117 ## Source page 120 A Wilson Yang–Mills construction J. Rodgers fixed allowed parent map, the sum over all label assignments of the product of the corresponding K-entries is at most one. To prove it, sum a vertex of maximal rank first. It has no remaining children, so its label occurs only in its own row factor. Its row sum is at most one. Remove it and repeat. Finite label restrictions suffice; arbitrary label sets follow by nonnegative summation. Let kj be the number of children of vertex j, including the root. The elementary exponential estimate gives F(S)kj ≤kj!eδcB(S). For δcB(S) ≥1, this is xkj/kj! ≤ex; for x ≤1, use F(S) = 1, kj! ≥1 and ex ≥1. Multiply over parents. Each internal vertex’s extra factor eδcB(Xi) cancels the extra support weight introduced into K; only the root factor remains. Thus the original overlap-constrained weight product at exponent a is at most eδcB(Y )tm   m Y j=0 kj!   m Y i=1 K(Sp(i), i). After summing labels and dividing by the occurrence factorial (m!), its contribution is at most eδcB(Y )tm Q j kj!/m!. All child factorials are paid together by an exact parent-map count. For maps from m numbered objects into r bins, let Am(r) be the sum of the products of their bin-size factorials. Appending one object to bin j multiplies the weight by kj + 1; summing over bins gives (m+r). Hence A0(r) = 1 and Am+1(r) = (m + r)Am(r). With r = m + 1, X p m Y j=0 kj! = 2m m ! m! ≤4mm!. This upper bound sums all parent maps, so it also bounds the allowed ones. The total forest weight is therefore at most eδcB(Y )(4t)m ≤eδcB(Y )qm. The root may instead be an internal occurrence with nonnegative weight ui. Suppose P i uie(a+δ)cB(Xi) ≤b. Multiply the forest bound rooted at Xi by uieacB(Xi) and sum i. The result is bqm, with factorial (m!) for the other m occurrences. This proves the designated-root estimate used for size m + 1 clusters. Summing the geometric size tail gives bqm/(1 −q). The argument covers m = 0, empty external root and arbitrary branching without changing the normalization. The graph interval construction used to pass from connected graphs to these trees is explicit. Fix an injective numerical ordering of the occurrence vertices and a root (0). For a connected graph H, let (d(v)) be shortest-path distance from (0). For every v ̸= 0, choose as parent the least ordered neighbor at distance (d(v)-1). The resulting parent edges form a ranked spanning tree. Its depth function is uniquely determined by its parent map: it is zero at the root and increases by one at every child. More generally any graph with this breadth-first parent has that same depth function, since every edge changes depth by at most one and the tree realizes the assigned depths. For a fixed tree p arising in this way, let Up be the subset of original graph edges satisfying both of the following requirements. Their endpoint depths differ by at most one. If an edge joins a vertex v ̸= 0 to a preceding-level vertex u, the order of u is no smaller than the order of (p(v)). Then the graphs whose breadth-first parent is p are exactly stem(p) ⊆H ⊆Up. 118 ## Source page 121 A Wilson Yang–Mills construction J. Rodgers One direction follows from shortest-path distances and the least-parent choice. Conversely, containing the stem gives a path of assigned length from the root to every vertex, while the upper constraint prevents any shorter path; hence these are the graph’s actual distances. The second constraint and injectivity of the order give the same least preceding-level neighbor. Thus the breadth-first parent is again p. These intervals partition all connected subgraphs of the original graph. If Up \ stem(p) contains an edge, toggling it pairs every graph in the interval with one of opposite parity. The interval sum vanishes. If there is no such edge, its sum has absolute value one. Summing interval bounds proves precisely the rooted spanning-tree bound on the absolute connected coefficient. 7.5 Passing from numbered trees to distinct connected sets The same forest estimates also bound distinct-set expansions. For a distinct label set D of size m, a parent map on D, and each bijection {1, . . . , m} →D, relabel the parent map and its activities by that bijection. The resulting numbered forest determines all three objects: its label range is D, its label assignment determines the bijection, and transport back through that bijection recovers the parent map. This assignment is injective. There are exactly m! bijections; all have the same product weight divided by m!. Their sum is therefore the unnumbered product weight, with no factorial left. Restricting the numbered forest sum to this injective image proves the distinct-parent-map estimate and its geometric size tail. A finite overlap-connected set whose union meets x can be rooted at the external support (x). Connect the root to labels whose supports contain x, and use a breadth-first spanning tree of the resulting connected graph. The sum over all parent maps includes this one, so it dominates that set’s nonnegative product weight. For the stronger sitewise prefactor γ, instead choose one label containing x as an internal root, charge its activity to the site’s budget, and use the internal-root forest estimate for the other labels. Summing over all possible roots only increases the bound. Numbering the distinct nonroot labels cancels their factorial exactly as above; allowing the root label once more among nonroots in the majorant is harmless. Consequently the distinct connected- set sum rooted at x, with the actual union support weight, is bounded by γ/(1 −q) under the readiness assumptions. This is the distinct-set input used in the positive-multiplicity restoration argument. 7.6 Cluster logarithms with repeated occurrences Let (R(i,j)) be a symmetric incompatibility relation with (R(i,i)) for every label. Let Xi ̸= ∅be its support, with incompatibility implying intersecting supports. For real activities wi, the hard-core gas is Z(w) = X A compatible Y i∈A wi. The empty subset contributes one. In contrast, its logarithm uses ordered occurrence tuples z = (i1, . . . , in), permitting repeated labels. Form the graph of incompatible occurrence pairs and define φ(z) = X H connected H⊆incompatibility graph (−1)|E(H)|, L(w) = X n≥1 1 n! X z∈In φ(z) n Y j=1 wij. 119 ## Source page 122 A Wilson Yang–Mills construction J. Rodgers For one occurrence the connected graph has no edges and coefficient one. Equal labels at distinct occurrence positions are incompatible. This is essential to the logarithmic identity even though the gas itself is indexed by distinct-label subsets. The absolute estimate uses cancellation inside φ. Breadth-first exploration partitions connected subgraphs into intervals with one fixed spanning tree as lower endpoint and an allowed edge set as upper endpoint. In each interval, if an optional edge exists, toggling that edge is a sign-reversing involution, so the alternating sum is zero. Otherwise the interval contains only its tree and its absolute contribution is one. Hence |φ(z)| is bounded by the number of compatible rooted spanning trees. Opening the absolute value before the connected-graph sum would lose this estimate. Here is the resulting root accounting. Put cB(S) = coverNumberB(S), take a ≥0, δ > 0, γ > 0, and suppose sup x X i:x∈Xi |wi|e(a+δ)cB(Xi) ≤γ, 0 < q < 1, 4(2B + 1)4γ δ ≤q. The rooted-tree estimate bounds the size-n cluster sum with a designated root-activity budget b by bqn−1, using the normalization (1/(n-1)!). In an unrooted cluster that contains a label with a chosen property, sum over all possible marked root positions and divide by n. Its (1/n!) normal- ization is then bounded by the same rooted expression. Relabeling tuple positions preserves the graph coefficient and product weights, so the estimate holds for each possible position, including repeated labels. If x ∈S j Xij, at least one occurrence has support containing x. Use that property for the root, with b = γ. A nonzero connected coefficient gives an overlap-connected support union. As in the cover argument above, eacB(∪jXij ) ≤ Y j eacB(Xij ). Thus, summing sizes n ≥m + 1, the absolute logarithmic terms whose support contains x, including their actual union support weight, have total at most γqm 1 −q. Summing root bounds over all sites gives absolute convergence of L at fixed torus, with total at most |Λ|γ/(1 −q). The rooted-tree estimate and its breadth-first interval construction are the remaining combinatorial ingredients behind this bound; the occurrence/root normalization and transfer to the union support have been made explicit here. The real logarithmic identity follows without assuming Z(w) > 0. First restrict to a finite alphabet of s labels. Let P(t) = Z(tw), a polynomial of degree at most s, and write L(t) = P n≥1 lntn for the connected series. The finite graph decomposition into connected components gives the coefficient recurrence n[P]n = n X k=1 klk[P]n−k, obtained by designating one occurrence, choosing its connected component and then the re- maining graph. The binomial choices of positions and the component factorials give precisely these coefficients. The absolute cluster bound gives |ln| ≤Cqn−1. Choose u > 1 with qu < 1; both (L(t)) and its differentiated series converge uniformly on compact subsets of ((-u,u)), by the summable 120 ## Source page 123 A Wilson Yang–Mills construction J. Rodgers majorant Cn(qu)n−1. The recurrence implies P ′ = L′P. Therefore (Pe−L)′ = 0 on this interval. Its value at zero is one, so P(1) = eL(1). For an arbitrary alphabet, restrict to finite subsets directed by inclusion. Every fixed gas subset and every fixed occurrence tuple eventually occurs unchanged. The gas has an absolute majorant obtained by dropping exclusions, bounded by Q i(1 + |wi|) ≤e P i |wi|; the sitewise budgets make that sum finite at fixed torus. The connected series has the absolute majorant above. Dominated convergence passes both finite-alphabet sides to their full sums. Thus Z(w) = eL(w) > 0, log Z(w) = L(w). This also applies after removing all labels incompatible with a specified forced set: the absolute activity budgets only decrease. 7.7 Primitive marked integrals and averaging over local tree choices The primitive activity attached to a marked team is an actual integral. A team is a finite collection of real local leaves with distinct dictionary keys; its weight is the product of their bounds and its ingredient the product of their functions. For a fixed coarse configuration and local tree choice, first choose a permitted flag sector, a local block layout, and a finite list of warning and regular- exponential selections. Their union is the molecule address p. The marked integrand is the product of the team’s paid ingredients and the selected local vertex factors; it is set to zero unless all of the team’s keys occur in p. Its primitive coefficient is its integral over the tree-gauge variables and the Gaussian fluctuation variables. The Gaussian measure has the covariance and density already attached to that same local tree choice. Sector selectors and all eligibility tests stay inside this integral. More explicitly, if X denotes that local data, k the sector, and IX,k,p(h, z) its selected integrand, set aX,k(p) = 1approved(X,k) Z IX,k,p(h, z) dνX(h) dγX(z). The full primitive coefficient averages these values over the finite boundary choices for the layout of p; it is zero for a disallowed layout. Its scalar prefactor is the original local overhead. The local vertex and integral estimates are developed in the packet, non-packet and warning arguments below. The expansion into molecule addresses is justified before the cluster expansion. In a permitted layout, warning factors are expanded by the finite identity Q j(1 + dj) = P D Q j∈D dj. The regular exponential factors use their absolutely convergent subset expansion. Marked regular keys are dormant in the unmarked density: their original regular coefficient is zero, so their unmarked exponential increment e−0 −1 is zero. Consequently one may insert those marked keys into every selected list and delete them again uniquely; they are supplied by the team ingredient rather than by an unmarked exponential increment. Forced marked keys must likewise be present in the sector. If they are absent, both the selected expansion and its marked integrand vanish. The resulting address assignment records the layout, sector, warning list and regular list. It is in- jective on permitted data and has an explicit inverse obtained by reading these pieces from p. Thus each nonzero molecule occurs with exactly the specified coefficient, and disallowed molecule ad- dresses contribute zero. The summable activity majorant bounds the integral norms of these terms by the team’s weight times the product of the unselected primitive tariffs. This gives a summable series of integrable terms. Fubini and the theorem on integration of a series with summable integral norms therefore give 121 ## Source page 124 A Wilson Yang–Mills construction J. Rodgers X p overheadX aX(p) = P(team density)(U). This is the primitive team expansion. In Gaussian coordinates the change of measure is explicit: dγX(z) = θX(z) dz. The transformed integrand contains that same θX and its tribute prefactor, so the Gaussian integral agrees with the original local fluctuation integral by substitution of the density. No independent Gaussian approximation is made in this identity. To obtain compatible products of primitive coefficients, average over independent local tree choices at coarse sites. A molecule’s integral depends only on the tree choices in its enlarged ground support. To prove this locality, project the fluctuations to the finite coordinates needed by that ground. Two local data sets agreeing there have the same projected covariance. Their Gaussian pushforwards under the coordinate projection are therefore the same measure. The integrands depend only on those projected coordinates and agree on each fiber, by their local footprint formulas. Their integrals coincide. The same argument applies to the gauge variables through their local measure-preserving coordinate maps. Thus changing exterior tree choices leaves the molecule coefficient unchanged. Compatible molecule grounds are disjoint. For functions fp of independent finite choices, each depending only on its ground, finite product averaging gives Av Y p∈A fp = Y p∈A Av fp. This follows by factoring the finite sum over assignments on disjoint coordinate sets; unused coordinates contribute one. The local scalar overhead is invariant under the tree choice. Averaging the team expansion consequently yields the product of the averaged molecule coefficients with the original compatibility and inventory conditions. Finally expand a finite leaf inventory into tickets with distinct keys. On a compatible set of molecules, every chosen leaf key lies in exactly one molecule; selecting it there gives a unique global ticket, and conversely a global ticket restricts uniquely to each molecule. Multiplication of the leaf amplitudes factors over this partition. Hence the product of primitive marked molecule sums is exactly the ticket sum of the team’s compatible product. Finite linearity of the Mark integral then proves X A compatible,permitted overhead Y p∈A X D ticket Y i∈D vi aD(p) ! = P X D Y i∈D vi teamDensityD ! (U). The primitive-integral bounds and support-hit estimates in the nonlinear and marked- integration chapters give these summable integral-norm and footprint bounds. The address regrouping, Gaussian marginal comparison, independent tree averaging and ticket factorization show how they produce the literal Mark expansion used in the gas/log argument. 7.8 Fusing attached logarithms into the forced dictionary The dictionary fusion used twice in the coarse construction has an explicit formula. Let the old forced labels be i ∈N, with supports Si, nonempty inventories Ii ⊆Si, and coefficients Ki(U). Let attached logarithms lj(U) have nonempty supports Tj, and suppose lj(U) ̸= 0 only if Tj meets the configuration’s flagged set. Both families have summable uniform envelopes. Form the disjoint atom alphabet of old forced labels and attachments, with factors 122 ## Source page 125 A Wilson Yang–Mills construction J. Rodgers hf,i = Ki(U), ha,j = elj(U) −1. A new label is a finite overlap-connected atom set containing at least one old forced label, with the old forced supports mutually disjoint. Its new support is the union of all atom supports, its inventory the union of the old forced inventories, and its coefficient the product of its atom factors, restricted to its exact inventory and original eligibility conditions. These labels form a countable set; it is infinite since each singleton old forced label is one. Choose an enumeration by N. This choice only names the labels and does not change their functions or bounds. Expand the old forced gas times exp P j lj using Q j(1 + (elj −1)). Absolute summability follows from the uniform logarithm envelopes and the old forced envelopes. In a nonzero term, every attachment meets a flagged site. The exact inventory coverage of its old forced set puts that site in an old forced inventory, hence also in its support. Therefore each connected component of the selected atom supports contains an old forced label; no free attachment component occurs. Components are precisely the new labels, with disjoint supports and inventories covering the original flagged set. Conversely a permitted set of new labels has disjoint atom sets, since every atom support is nonempty; their union recovers the original old forced set and attachment subset. The exact inventory condition also implies the original local inventory conditions: another disjoint forced support cannot supply a flag within the given support. Thus decomposition and union are inverse on all nonzero terms, with identical products. This proves Gasnew(U) = Gasold(U) exp X j lj(U). The regular dictionary and its original coefficients remain the regular part of the fused repre- sentation. The attached logarithm is consumed only through the displayed exponential and the new forced factors. Gauge invariance and measurability pass to finite products and measurable eligibility indicators. Locality uses the union support. Widening a dictionary beforehand is simply an injective placement of its old labels in a universal dictionary with zero coefficients at all un- used indices. It preserves the gas and the regular-log sum by reindexing the absolutely convergent families. The fusion estimate for switched coefficients follows from the distinct-component forest bound already proved. Suppose the old switched forced coefficients have paid sitewise budget B0 at support exponent (s+1), and the fixed attached logarithms have envelope budget w0 there. Choose κ ≥0. Assign inflated envelope eκbi to an old forced label and envelope 2cj to an attachment with |lj| ≤cj ≤1. The latter bound is justified by |ex −1| ≤2|x| (|x| ≤1), which is used on the source’s smaller admitted logarithm range. The combined sitewise budget is γ = eκB0 + 2w0. Each new connected label contains an old forced atom. Undoing its inflation supplies at least e−κ; every other undone factor is at most one. Thus its paid envelope is bounded by e−κ times the product of the inflated atom envelopes. Its eligibility indicator is independent of the switches and does not increase the price. If VBγ ≤1, the earlier connected-component forest estimate, spending one support-exponent unit, bounds the new sitewise paid budget at exponent s by 123 ## Source page 126 A Wilson Yang–Mills construction J. Rodgers e−κ(eγ −1). The selected payment inequalities require this quantity to fit the requested output allowance. This proves transfer of the paid switching family through each fusion. The family’s unmarked value is the fused unmarked coefficient by the same formula. An exterior-switch silence identity passes through fusion term by term: if an atom’s union support omits the required parent site of a switch, each factor is unchanged on erasing that switch; so is their product and its eligibility indicator. These facts supply the value, price and scope properties needed after the two actual coarse dictionary changes. The scalar normalization needed after these fusions is an equality of exponents, not a comparison of partition functions. Let E0(U) be the intermediate frozen exponent, let R(U) be the coupling- reset receipt, let α be its coefficient, and let L1(U), L2(U) be the attached-log sums consumed by the two fusions. The path reset gives Efinal(U) + L2(U) = E0(U) −αR(U). The unmarked local integral normalization gives exp{E0(U) + L1(U) −αR(U)} = overhead(U)Z0(U). Combining these equations and the exponential addition law proves eEfinal(U)eL1(U)eL2(U) = overhead(U)Z0(U). In particular both sides use the same unmarked local gas Z0 that appears in the switched formula. Subtracting its logarithm in the switched regular increment cancels precisely this normalizer. The path-reset identity and unmarked local normalization proved in this chapter retain the stationary path and vacuum choices used here. 7.9 The coupling-reset exponent identity The reset changes the represented coupling and compensates that change by explicit local countert- erms. The total density is preserved by the displayed exponent identity. Its projected coefficient is read from the affine Hessian with the source’s chosen normalization. A probe value is a different operation and is not silently identified with that projection. We now specify the reset in the scalar identity. Let v be the source on a listing X, let R be the same path family used in the kinetic energy, and let p fix the threshold, support scale and Taylor order. All masks below are the actual healthy-field masks at that threshold. Write ∆= δ(readX(v)), and let α be the already committed loop coefficient. The new coupling and vacuum scalar are β′ = β + α + ∆, τ ′ = τ − 1 |Site| X i logReadi(v; 1). The receipt R(U) is the averaged sum of the masked plain low path terms, with their original paths. It is not obtained by changing paths to match a later choice. The regular-source decomposition separates each log read into its value at the identity, a local projected payout, a fault, and a masked rebate. Absolute summability of the original paid activities bounds these four series respectively by the original total bill and fixed multiples of it. Summing 124 ## Source page 127 A Wilson Yang–Mills construction J. Rodgers the payouts first at their anchor sites gives ∆P o headsp(o, U), by the calibrated projection identity. Consequently additivep(readX(v); U) = X i logReadi(v; 1) + ∆ X o headsp(o, U) − X i dusti(U) + X i untaggedi(U). The projected-payout identity uses the simple-group projection hypothesis and the stipulated period n > 20. There are also low and high path corrections. Denote their averaged sums by F∆and Hα,∆, and the low loan sum by Lα,∆. Their defining termwise identities, summed with the same masks, are (α + ∆)KR + F∆+ Hα,∆= αR + ∆ X o headsp(o, ·) + Lα,∆. Indeed the true kinetic base is the sum of live low terms and high summits. The high correction is minus α + ∆times the summit, so those terms cancel. For each low term its defining loan identity reads (α + ∆) live + fade∆= α maskedPlain + ∆sketch + loanα,∆. The sketch sum is the head sum. The low live, masked plain, proxy and fade series are absolutely summable by the path bounds; this identity then also proves summability of the loans. The high summit series is absolutely summable as well. Averaging over the finitely many seats preserves these properties. Hence the cancellations are valid equalities of convergent sums. The reset dictionary’s additive sum is the untagged sum plus F∆+Hα,∆; its attached logarithm is the dust sum plus Lα,∆. Substitution into the representation exponent τ ′|Site|−inChannels(β′)− additive, followed by the two displayed identities and the channel decomposition, gives Ereset(U) + X i logi(U) = Efrozen(U) −αR(U). This is the path-reset identity used above, including its constant term and signs. Exponentiation gives the same identity for unnormalized evaluators. It holds even when the activity gas vanishes, since the gas multiplies both sides and is never divided out. The subsequent dictionary fusion absorbs the attached-log exponential by the already established fusion identity. The lower-level projection, live-base and loan formulas supply the mathematical content of this reset decomposition and must retain the same threshold and paths. 7.10 Gauge invariance at the flat germ Let F(u) = f(increment(1, u)). Regularity gives a twice continuously differentiable germ at u = 0. Constant gauge transformations act there by the linear map ue 7→Adg ue, because exponentiation commutes with conjugation. Hence DF(0), restricted to a single edge’s color space, is an invariant linear form. Such a form vanishes. Indeed differentiating invariance along exp(tx) shows it kills every bracket [x, y]. Its kernel is an ideal; simplicity makes this kernel zero or the whole algebra. The zero case would make the form injective and force all brackets to vanish, contradicting non- abelianness. Summing single-edge directions proves DF(0) = 0. Twice differentiating the same constant-gauge identity proves adjoint invariance of D2F(0). For a color-valued site field φ, let (∇φ)x,i = φ(x + ei) −φ(x). Gauge the link exponential of u by the site elements exp(−tφ(x)), and take its local logarithm. This defines a smooth chart C(u, t) near zero with C(0, 0) = 0, DC(0, 0)[u, t] = u + t∇φ. 125 ## Source page 128 A Wilson Yang–Mills construction J. Rodgers The derivative follows by differentiating the three factors on each link: the contributions are −tφ(x), ux,i, +tφ(x + ei). The local logarithm recovers the gauged configuration near the iden- tity. Gauge invariance gives F(C(u, t)) = F(u). Differentiate this equality in directions (0, 1) and (u, 0). In the second chain rule the term involving D2C is multiplied by DF(0) = 0, while the right side has no t-dependence. Thus D2F(0)[∇φ, u] = 0. Its symmetric counterpart follows from symmetry of the second derivative. Locality implies that Hessians agree when the tested direction fields agree on the support: first compare their diagonal exponential lines, then polarize. These arguments concern smooth unmasked germs throughout. The color-bilinear assertion used below also admits a direct finite-dimensional proof. If B is adjoint invariant, differentiation gives B([x, y], z) = −B(y, [x, z]). Its antisymmetric part A = B −BT satisfies A([x, y], z) = A([x, z], y). Cycling this relation and using bracket antisymmetry makes this value equal to its negative, so it is zero. For fixed z, the linear form A(·, z) kills brackets, and the kernel argument above makes it zero. Hence B is symmetric. Represent B(x, y) = ⟨Tx, y⟩by a self-adjoint real operator T. Invariance and skew-adjointness of the adjoint action give T[x, y] = [x, Ty]. A nonzero real eigenspace of T exists by the finite- dimensional spectral theorem and is an ideal by this commutation identity. Simplicity makes it the whole color space. Thus T = cI and B = c⟨·, ·⟩. Skew-adjointness here follows directly by writing the color inner product as the real matrix trace and cyclically moving the commutator factors. This proves the invariant-bilinear-form input without an assumption about a complex or infinite-dimensional color representation. 7.11 Reduction of affine tests to constant curl Use centered lifted coordinates zi(x) = lift(x −o)i. A linear jet is Ai(x) = ai + X j zj(x)bij, Fij = bji −bij. On a support satisfying 10 dist(x, o) < n, with n ≥4, the centered lift has its ordinary increment under one positive or negative lattice step. This follows from uniqueness of the centered represen- tative: both the old lift plus that step and the new lift wrap to the same site and stay within the centered coordinate range. Thus the discrete curl of A on this support is F. If F = 0, the slope matrix is symmetric. The color-valued polynomial φ(x) = X i zi(x)(ai −1 2bii) + 1 2 X i,j zi(x)zj(x)bij has forward difference φ(x+ei)−φ(x) = Ai(x) there. The diagonal correction removes the extra half- diagonal term in the quadratic difference; symmetry combines the remaining cross terms. Hence every curl-free affine field agrees on the tested support with a gauge gradient. Locality of the germ transfers its Hessian to agreeing directions, and gauge invariance annihilates its pairing with that gradient. Two affine fields with equal curls differ by this construction, so the Hessian depends only on the two curls. An explicit primitive for a flux is obtained by adding the wedge fields: for a positive plane a = (i, j), its wedge has only component j, equal to ziwa. Its curl is wa in plane a and zero in every other positive plane. Summing therefore gives a primitive with exactly the prescribed flux. Its threshold-scaled unit-color wedge is an admissible paid direction, since |zi(x)| ≤dist(x, o) and the permitted direction growth dominates 1 + dist(x, o). The average of two such scaled wedges is admissible by the triangle inequality. 126 ## Source page 129 A Wilson Yang–Mills construction J. Rodgers For a signed permutation with signs εi and permutation π, pullback of an affine jet changes its slope to b′ ij = εiεjbπi,πj. Its constant term is a′ i = εi aπi −1{axis i reversed}bπi,πi . The subtraction is necessary because a reversed positive link is read from the shifted base. Direct substitution of the lifted-coordinate increment proves this formula on the nonwrapping support. Its curl is the signed plane reframe of the old curl. Pullback of exponential link increments is a linear pullback of their direction fields; the Hessian chain rule at the critical identity therefore gives precisely the signed-plane covariance claimed for the projected form. No global identification of a nonperiodic affine polynomial on the torus is used. 7.12 Affine-Hessian coefficients and their bounds Fix a unit color vector ξ, which exists because the simple gauge algebra is nonzero. For each positive coordinate plane a and color x, let Wa(x) be the affine wedge field with constant curl x in that plane and zero curl in the other planes. It is linear in x and commutes with adjoint color transformations. For the regular germ f at the identity, define cab(f, o) = D2f(1)[Wa(ξ), Wb(ξ)]. For fixed a, b, the bilinear form B(x, y) = D2f(1)[Wa(x), Wb(y)] is invariant under simultaneous adjoint transformations, by gauge invariance of the germ. Differentiation makes it invariant under the infinitesimal adjoint action. The simple-algebra invariant-bilinear-form result therefore makes it a scalar multiple of the color inner product. Evaluating at ξ, ξ fixes the scalar, giving D2f(1)[Wa(x), Wb(y)] = cab(f, o)⟨x, y⟩. Since the primitive of a flux is the sum of its wedge fields, bilinearity gives D2f(1)[primitive(w), primitive(z)] = X a,b cab(f, o)⟨wa, zb⟩. The invariant-bilinear-form argument and the reduction of affine Hessians to curls are the simple- algebra and gauge-calculus inputs of this formula. The constants in the projection estimate follow from actual normalized directions. The direction tWa(ξ) is admissible in the paid second derivative bound; hence t2|caa| ≤cf. So is u = t 2(Wa(ξ) + Wb(ξ)). At the critical identity, its repeated ordered left derivative equals its diagonal Hessian, and 4D2f(1)[u, u] = t2(caa + cbb + 2cab). Bounding the derivative on the left by cf and each diagonal term by cf yields t2|cab| ≤3cf. These estimates use only the paid directions at the identity, rather than a limiting periodic Fourier test. The actual projected matrix is cab if the enlarged anchored support is nonwrapping, and zero otherwise. Its entry bounds hold in both cases. On a nonwrapping support it has exactly the flux-Hessian formula above. This cutoffis preserved by signed rotations and translations, as is the enlarged radius-two support. The pullback formula for flux Hessians then gives covariance of its quadratic form under the signed plane action, including the zero case. The paid calculus used for these statements bounds all ordered left derivatives up to p.order in the specified anchored direction class, on the closed healthy set. The underlying functions remain smooth on full neighborhoods of each such configuration. Thus it differentiates the unmasked germs, 127 ## Source page 130 A Wilson Yang–Mills construction J. Rodgers not their discontinuous indicators. Constant multiplication multiplies the bound by the absolute constant; sums and differences add bounds. For an inner product the ordered Leibniz rule has at most 2p.order splittings, giving pairingCost(E, k) = 2k∥innerSLE∥. Linearity of the derivatives and Hessian under subtraction follows on these full neighborhoods; enlarging the healthy support only restricts the configurations tested. These facts justify the residual and clover bills, including boundary points of the healthy sets. 7.13 Isotropy of the summed coupling projection Let A be the actual countable activity listing, with its regularity, paid bound and label permutations implementing every signed lattice motion. Define the matrix at anchor o by To(a, b) = X {i: anchor(i)=o} Πi(a, b). These sums are absolutely convergent: the coefficient bound t2|Πi(a, b)| ≤3ci is dominated by the summable activity bill. Finite sums over plane types can therefore be interchanged with this series. Reindexing by the label permutation for a signed motion, using its exact anchor, support and value transformations, gives QTaffine(s,b)o(w, z) = QTo(reframesw, reframesz). In particular the nonwrapping cutoffin each projected entry has already been transported before the label sum is taken. Test this equality with fluxes supported in single planes and equal there to the fixed unit color vector. Translation and a permutation taking the reference plane (0, 1) to any chosen plane show that every diagonal entry at every anchor equals ∆= T0((0, 1), (0, 1)). For two distinct positive planes, there is an axis occurring in one and not the other. Reflecting just that axis changes the sign of their mixed pairing and fixes their plane labels. Its coefficient is consequently its own negative and is zero. Therefore To(a, b) = ∆1{a=b}, X anchor(i)=o QΠi(w, w) = ∆ X a ∥wa∥2. Applying this with the actual clover flux, and the same singleton healthy indicator on every term at that anchor, proves the projected-payout identity used in the reset exponent. It would not justify replacing different individual support indicators by that singleton indicator; the faults above are precisely what account for that change. The diagonal coefficient estimate gives the sharper bound t2|∆| ≤c, where c is the listing’s anchored paid budget. Indeed the triangle inequality bounds the anchor-zero diagonal sum by its local mass, and that mass is at most c. The off-diagonal factor three is not spent here. Thus both the scalar and its bound are obtained at the original finite period from the original label orbits. 7.14 The local clover counterterm The petal flux at an anchor o is the based clover: the signed average of the four oriented plaquette curvatures in each coordinate plane, all based at o. Every such oriented square touches o, and all its visited sites lie within distance two of o. Thus a healthy singleton anchor bounds each square chord 128 ## Source page 131 A Wilson Yang–Mills construction J. Rodgers by t, and the petal flux reads only the radius-two clover. Its gauge transform is adjoint transport by the gauge value at o; signed lattice motions act by the corresponding signed action on plane indices. For a word of length l, curvature is the linear color projector applied to its matrix word minus the identity. The projector norm is at most the inverse color scale. The paid word derivative bound therefore gives the curvature bill t colorBill(G, l, P, k), colorBill = G.colorScale−1feeWalk(G, l, P, k). Adding the four clover terms with coefficients of absolute value 1/4 gives this same bill with l = 4 for each petal. The ordered product derivative rule for their inner product gives pairBill(G, p)t2, pairBill = pairingCost(G.Color, p.order) colorBill(G, 4, p.power, p.order)2. Here the pairing cost accounts for all ordered derivative splittings already used in the paid calculus. For a real matrix q on plaquette types, the counterterm is Qq(U) = 1 2 X a,b qab⟨petals(U, o)a, petals(U, o)b⟩. Its bill is at most 1 2pairBill t2 P a,b |qab|. It is smooth, gauge invariant, supported on the radius-two clover, and linear in q. Its normalization follows directly from affine tests. For the exponential link line U(t) = exp(tv), differentiating a based word at zero gives the signed link sum along that word. For a square this is the discrete curl [vj(x + ei) −vj(x)] −[vi(x + ej) −vi(x)]. For the nonperiodic affine field associated to a linear jet, on a nonwrapping neighborhood this equals its constant curl. The signed clover average consequently has derivative equal to the jet’s positive- plane flux w. The period n > 20 ensures the clover neighborhood has the required affine-coordinate increments. Hence Qq(U(t))/t2 −→1 2 X a,b qab⟨wa, wb⟩. Gauge invariance for the simple group makes the identity a critical point of every regular germ. Taylor’s formula then identifies the left limit with one half its diagonal logarithmic Hessian. There- fore the counterterm Hessian on affine directions is exactly the quadratic form q on their curls. Symmetry of the Hessian and polarization recover all mixed affine pairs from these diagonal tests. This establishes the counterterm normalization used in the residual decomposition. 7.15 Masked residual decomposition For a regular term f anchored at o with support D, enlarge the support to D+ = D ∪halo2(o). Let qf be the counterterm built from its projected affine Hessian and the local petal flux. Define rf = f −f(1) −qf, df(U) = ( 0, D+ is healthy, f(1) + 1{o} healthyqf(U) −1D healthyf(U), otherwise. Then the exact identity is 1D healthyf = f(1) + 1{o} healthyqf −df + 1D+ healthyrf. 129 ## Source page 132 A Wilson Yang–Mills construction J. Rodgers On a healthy enlarged support all three smaller health tests hold and the equation reduces to the definition of rf. On its complement the enlarged-support term vanishes and the equation reduces to the definition of df. This is why the constant f(1) is retained even outside the original healthy support. If f has paid supremum and derivative bound cf, its projected matrix entries obey |Πf(a, b)|t2 ≤ 3cf. The quadratic counterterm bill is thus at most Mpcf, where Mp = 1 2q2 3 pairBill(G, p). Indeed its half-weighted double type sum has q2 entries, and each petal pairing has bill pairBill(G, p)t2. It follows by the triangle inequality that both the rebate’s paid bound and the fault’s all-configuration absolute bound are at most (2 + Mp)cf. The value f(1) costs at most cf. The fault vanishes on a healthy enlarged support, and hence is a forced term with an actual failure witness. The rebate is regular on D+, vanishes at the identity and has zero affine Hessian: the projected counterterm has exactly the affine Hessian of f, while subtraction of f(1) changes no derivative. This identification uses the nonwrapping support and period hypotheses of the affine-curl construc- tion. Gauge invariance and measurability pass to the counterterm, rebate and fault. The fault uses the enlarged support’s field data and its final-threshold flags, precisely as permitted for forced terms. Signed rotations and translations send the petal flux and the projected matrix covariantly, and send healthy sets by their actual affine images, so they commute with this decomposition. Pro- jection and the counterterm are linear in f; consequently differences of rebates and faults are the rebate and fault of the difference. These facts justify applying the decomposition to the common parameterized source series in the reset. 7.16 Low-path atoms and the loan cancellation The low indices are a plaquette type together with either one direct symbol or an ordered pair of path-row labels (i, j). For a direct index let its mark be m = c.direct, its length zero, and its plain value the plaquette price. For a pair let m = cicj, l = length(si) + length(sj), P(U) = 1 2⟨si(U), sj(U)⟩. Its analogue on a flux w is ⟨wa, wa⟩/2 in the direct case, and ⟨wplaq(si), wplaq(sj)⟩/2 in the pair case. The proxy Q(U) is this analogue evaluated on the local petal flux at the anchor. The home H is the anchor’s radius-B halo together with both true path stencils in the pair case; in the direct case it is just the halo. Let hH and ho be the healthy indicators on this home and on the anchor singleton. The live value J(U) is the clipped plaquette price for a direct index, and 1kept mP(U) for a pair. The freshness and loan functions are exactly F∆= ∆m(Q −P), Lα,∆= ( 0, hH = 1, (α + ∆)J −∆mhoQ, hH = 0. The masked plain value is hHmP. On a healthy home the selector is kept and the direct plaquette is below threshold, so J = mP. Also ho = 1. Substitution gives (α + ∆)J + hHF∆= αhHmP + ∆mhoQ + Lα,∆. 130 ## Source page 133 A Wilson Yang–Mills construction J. Rodgers On an unhealthy home the same equality is the defining formula for the loan, with both home- masked terms zero. This proves the low loan cancellation in all cases, including a healthy anchor inside an unhealthy home. These functions have the required locality despite the selector’s apparent infinite test. Outside the radius-B aperture its exponential threshold already exceeds the maximum chord two, by the decent-budget hypothesis. Inside it the plaquette corners lie in the aperture unless the test is automatically true. Thus the kept predicate is determined locally. Slot values read their true route and plaquette stencils, and proxy values read only the anchor’s radius-two clover, contained in the home. Gauge conjugation preserves chords, prices, kept predicates and the inner products of the transported slots. All these functions are measurable; a loan is a finite combination of continuous functions and measurable healthy/kept indicators. Its possible extra flag read is exactly the final-threshold read allowed in the definition of the forced dictionary. Write t = p.threshold and ρ = c.relaxation. A kept slot of length li has norm at most teli/ρ. Using the chord bound two and the clip kink inequality for the direct term, both cases give the all-configuration estimate |J(U)| ≤|m| 2t el/ρ. On a healthy anchor the proxy has the derivative bill 1 2pairBill(G, p)t2. Consequently |Lα,∆(U)| ≤|m|el/ρ  2t|α + ∆| + 1 2|∆| pairBill(G, p)t2 . A nonzero loan therefore has an actual unhealthy home, and its bound retains the path length and relaxation factor needed for its support charge. It is linear in (α, ∆), as is the freshness in ∆, so their difference estimates use the same formulas with parameter differences. Finally the freshness has zero affine Hessian. Along an affine exponential link field, a slot divided by its scalar parameter tends to the corresponding constant plaquette curl. This follows by differentiating the finite link product; the two nonconstant affine edge terms cancel to the constant curl. On a nonwrapping home, both the plain value divided by the parameter squared and the proxy divided by that square tend to the same flux analogue. Their difference therefore has zero quadratic limit. The freshness is smooth in a healthy neighborhood, vanishes at the identity, and the affine Hessian limit identifies its quadratic coefficient with this limit; polarization gives zero mixed affine Hessians. The nonwrapping and period hypotheses are retained here, as in the reset projection argument. 7.17 Recovering the actual kinetic base from the path atoms The low pair sum is the square of the original path row, with all ordered labels retained. If fi = cisi(U) and P i ∥fi∥< ∞, then X i,j 1 2⟨fi, fj⟩= 1 2 X i fi 2 . Absolute convergence follows by domination by 1 2(P i ∥fi∥)2; continuity of the inner product then permits the two sum exchanges. Applying this to each row proves that the total live low terms are exactly the clipped direct term plus the kept shortening half-square. The pair selector is common to its whole row, so the assertion holds on both selector branches. For flux analogues the same square identity gives half the squared norm of the actual zero- momentum symbol row. If Sab = R.symbol0(a, b), the path normalization is c.direct δbi + X a SabSai = δbi. 131 ## Source page 134 A Wilson Yang–Mills construction J. Rodgers Expanding the symbol-row norm squares and applying this matrix identity proves X low d md analogued(w) = 1 2 X a ∥wa∥2. All series converge absolutely by the weighted path coefficient bounds. Multiplying by the common healthy singleton indicator and evaluating on the petal flux is therefore legitimate. The layer operation is the sum over translated signed views divided by the number of signed orientations; translations are summed, not probability averaged. For each view-dependent term, translation covariance changes its anchor-zero value to its value at the translated anchor. Reindex- ing the finite sites by the signed permutation then proves that the low live layer plus the cubic and quartic masked layers is exactly the stipulated kinetic energy. It also proves that the low proxy layer is P o headsp(o, U): signed reframe preserves the sum of squared plane norms, and the orientation count cancels its inverse. Thus both identities used in the reset’s path cancellation come from the original symbol normalization and actual path rows. On an everywhere healthy configuration every masked plain low atom is unmasked. Their layer is then the original unclipped second term, namely the orientation average of c.direct price+∥l∥2/2. This specialization does not remove masks in a configuration with any flags. The high freshness terms retain the actual cubic and quartic atoms on their own footprints and have coefficient −(α + ∆). An atom of arity r = 3 or 4 has paid bill its startup coefficient times tr; since 0 < t ≤1, both are bounded using t3. Along any exponential link line every slot is first order, so the atom divided by the line parameter to the power r has a finite limit. Dividing instead by its square leaves a factor tending to zero. Taylor’s formula at the critical identity then gives zero diagonal Hessian for each high freshness term, and polarization gives zero mixed Hessian in arbitrary directions. This proves the required vanishing without changing the cubic/quartic masks or attaching an extra boundary term. 7.18 The unmarked local scalar The normalization in the reset uses the same local pool C(V ) > 0 as the density expansion. Its trained connected-log series has sum log C(V ) and is absolutely convergent by the paid log estimate. The incoming-to-current label equivalence preserves the actual union support. For each of these fixed labels use minus its stream as the regular coefficient on that support. The support-mask identity for the trained stream says that this masked coefficient is minus the corresponding trained log term. Reindexing its absolutely convergent series therefore gives the regular sum −log C(V ). Adjoin the literal path-score terms with their original supports. Their masked sum is −γR(V ) −β billcubic(V ) + billquartic(V )  −billloop(V ). Thus the combined regular dictionary has additive sum equal to this expression minus log C(V ). No anchor is selected from an evaluated field: the anchor for each stream support is chosen once from that nonempty support, and its address is retained along the label equivalence. All these complete sums are invariant under a signed translated view. For the path bills, reindex the finite signed-seat average. For the pool logarithm, reindex the trained connected labels by the exact incoming-label motion; its support and eligibility data move with it. The regular dictionary now uses a separate full-seat average: each pulled coefficient is multiplied by the reciprocal of the number of seats, including translations. The healthy mask is pulled with its actual support. Each view has the same complete sum, so summing these averaged coefficients preserves that sum exactly. This full-seat normalization differs from the orientation-normalized layer operation on kinetic paths. 132 ## Source page 135 A Wilson Yang–Mills construction J. Rodgers The field-independent scalar token is the original sector prefactor times the incoming vacuum exponential times the fixed deducted chart factor. It is positive. Set τ = |Sitec|−1 log(token). The actual scalar overhead is then O(V ) = exp ( τ|Sitec| −β averageorientations X i baseTerm(V, i) ) . This follows by separating the token from the exact clipped-base exponential in the overhead; the threshold in that base is the same one in the dictionary. Insert the combined regular sum into the frozen exponent, whose channel decomposition contains exactly that base, the cubic and quartic bills at price β, and the loop bill at its stated price. The latter three cancel the path-score terms. The receipt correction −γR cancels the receipt in the regular sum. The remaining exponent is log O(V ) + log C(V ). Therefore exp{Efrozen(V ) −γR(V )} = O(V )C(V ). If an earlier dictionary fusion has already been performed, its attached-log sum is added to the exponent on the left, giving exactly the scalar normalization formula stated above. Positivity of C(V ), supplied by the convergent real log identity, justifies exp(log C(V )) = C(V ). The forced gas in this same frozen representation is the canopy-pool sum. Its convergent density identity is O(V )C(V ) gas(V ) = push(R; V ). Multiplying the scalar equality by this gas proves the early evaluator identity for every field, including fields where the gas vanishes. This identifies the unnormalized push without dividing by its gas or its value at a selected field. 7.19 Pool averaging and the literal score series The regular pool uses integrated averages of the single-forest activities. If every forest value has absolute value at most a common tariff, the probability average has the same bound, by the triangle inequality and cancellation of the finite averaging cardinality. Thus the original thin and thick tariffs also bound the averaged entries and deposits. The renewed reserve gives the connected- log convergence hypotheses for these averaged entries, with the same supports and compatibility relation. Applying the already proved real gas/log identity gives log C(V ) = X connected labels q Gas.term(entriesPool(V ), q), C(V ) = exp X q Gas.term(entriesPool(V ), q) ! > 0. It is this averaged gas, not an average of logarithms of separate forest gases, that occurs in the scalar proof. For the full density, split each nonzero molecule family into components with nonempty flags and components with empty flags, retaining the requirement of nonempty ground. The two subtype maps have an injective union map and recover every nonzero family. Thus the integrated full gas reindexes as a sum over thick families and thin families. With a thick family fixed, excluding the thin supports it meets amounts to setting those thin entries to zero. Their absolute bounds can only decrease, so the same log theorem applies. The omitted-cluster restoration identity factors this excluded gas as C(V ) exp − X q missingA(q, V ) ! . 133 ## Source page 136 A Wilson Yang–Mills construction J. Rodgers Multiplying by the banked thick-family factor and summing gives the pool density identity used above. Throughout this step the forest averaging is performed at the level justified by disjoint- ground factorization, rather than assuming arbitrary products commute with a common forest average. The literal path score can also be read directly term by term. A low score label consists of a signed translated seat and a low atom. Its coefficient is −γ/|Signed| times the atom’s marked plain value in that view; its support is the view of that atom’s full home. Pulling its healthy mask through the view gives exactly the corresponding summand of −γR. A cubic, quartic or loop score label consists of a seat and an atom label, with its actual transformed footprint. Its coefficient is respectively −β/|Signed|, −β/|Signed| or −1/|Signed| times the unmasked atom. Applying the footprint mask produces its placed atom. The absolute atom budgets justify summing these labels. Reindexing the seat’s translation by its signed action identifies the seat sum with the orientation average of the sum over all anchors. This proves the exact score total used above, with no change of masks. Finally the channel decomposition is simply the expansion of its definition: inChannels(β) = β average X baseTerm + β(billcubic + billquartic) + billloop. These are convergent equalities before any smallness estimate or cancellation of cubic coefficients is imposed. 7.20 Why a nonzero detector forces a warning The quiet-watch condition at o says every fluctuation coordinate with anchor within B of o has |zi| ≤ξ t/g. Since the lifted field is μ+gz, each color coordinate of its difference from μ is at most ξt. If the color dimension is d and ∥μe∥≤Mt, the elementary Euclidean bound by the sum of coordinate absolute values gives ∥(μ + gz)e −μe∥≤dξt, ∥(μ + gz)e∥≤(M + dξ)t on those links. A watch on radius 2B gives watches of radius B at nearby sites, by the distance triangle inequality. Choose a positive error margin τ and the fine/noise plateau bounds 2/√cdirect(A/L2 + τ) < 1/4, 2/√cpenalty(A + τ) < W. These choices are possible in the required order. First take τ = √cdirect/32; next choose L large enough that A/L2 ≤τ; finally choose W larger than 2(A + τ)/√cpenalty. The stationary maximum A is chosen before this block-length choice. The physical local path estimate has three contributions to its discrepancy bill: the effect of the fluctuation away from μ, the protected mean bill, and the nonlinear path-approximation remainder. Choose ξ > 0 with dξ ≤1 and with the first contribution at most τt/3. The source’s coefficient bound makes this possible by taking ξ no larger than τ divided by one plus three times its displayed linear-response coefficient. The mean and approximation errors have reference-coupling order −1 and are eventually each at most τt/3: multiplying either by sqrt(reference coupling) tends to zero, while t is at least the inverse square-root coupling. Their sum is therefore at most τt. Away from the mandatory bad-seed region, the clean-patch and grafted-field estimates apply to these same links. They place the likelihood site and pin tags on their plateau and identify 134 ## Source page 137 A Wilson Yang–Mills construction J. Rodgers the entry-cost rows with the actual fine curvature and prediction-mismatch curvatures. Add the background guide bound A/L2 · t to a fine curvature row and A·t to a noise row. The curvature comparison inequality then bounds the plaquette chord by 2/√cdirect(A/L2 + τ)t < t/4 and the prediction mismatch by 2/√cpenalty(A + τ)t < Wt. Hence the fine cutoff, noise cutoff, site tag and pin tag all equal one. The forced-charge detector also equals one there: any putative flag would be witnessed by a nearby plaquette with chord greater than t, contradicting the t/4 bound. Each nonmandatory detector is its bounded auxiliary cutoff times (one of these quantities minus one), so it is zero under the radius-2B watch. A mandatory detector is defined to be zero already. The active-link cutoffcomplement likewise vanishes when the lifted link lies within its plateau, using the mean bound and dξ ≤1. This proves the detector-to-warning implication from the local protected-mean, physical path- accuracy, clean-patch tag and curvature-comparison estimates, with their three error contributions kept separate. Those local geometric estimates remain dependencies of this paragraph; the Gaussian probability argument does not supply them. 7.21 The joint Gaussian warning estimate Let z be a centered finite-dimensional Gaussian vector of covariance A ≥0, and suppose vTAv ≤ M∥v∥2 for every real v. No independence of its coordinates is required. For a coordinate set D and a sign choice εi ∈−1, 1, let v have those signs on D and zero elsewhere. Then ∥v∥2 = |D| and E exp(sv · z) = exp(s2vTAv/2) ≤exp(Ms2|D|/2). If εi zi ≥t at every i ∈D, then v · z ≥t|D|. Markov’s inequality, for s ≥0, bounds this orthant event by exp((−st + Ms2/2)|D|). The union over its 2|D| sign choices gives P(|zi| ≥t for all i ∈D) ≤2|D| exp((−st + Ms2/2)|D|). This also covers a degenerate covariance and an empty coordinate set. Warnings are located at sites oj, Gaussian coordinates at sites gi. Suppose at most m warning addresses and at most p coordinate addresses lie over any one site. A warning at oj is witnessed by some coordinate gi within distance R whose absolute value is at least t. For a finite warning set W, choose one such coordinate for every warning. There are at most (VRp)|W| witness assignments, where VR = (2R + 1)4 bounds the number of sites in a native torus ball. For an assignment let D be its distinct coordinate image. Then |D| ≤|W|. Each coordinate can witness at most VR m warnings, so |W| ≤VR m|D|. Choose H ≥0 with M/2 + HVRm ≤t and apply the preceding rectangle estimate with s = 1. Its exponential factor is at most exp(−HVRm|D|) ≤exp(−H|W|), and 2|D| ≤2|W|. A union bound over witness assignments proves P(every warning in W has a witness) ≤(2VRp exp(−H))|W|. 135 ## Source page 138 A Wilson Yang–Mills construction J. Rodgers Thus shared witnesses and overlapping warning neighborhoods have been counted explicitly. They are not counted as independent Gaussian events. For the actual warning family R = 2B, where B is the support width. Its address census is the sum of the fine plaquette types, two coarse bond types, coarse and fine site types, and fine-link itinerary types. With block length L it is bounded by m = L4|PlaquetteAxes| + |Axis| + |Axis| + 1 + L4 +itineraryDensity(L). The Gaussian coordinate census p is the tubular coordinate count of that same block. The covariance upper bound M comes from its actual fluctuation covariance estimate. The warning event uses threshold t = ξ tfield/g, where g = β−1/2. Band comparability gives tfield/g at least one half of the fine monitor P. Put H = ε√P, ε = ξ/4. Eventually V2Bm ≤√P and M/2 ≤εP, because the monitor dominates the fixed covariance cost and the polynomial width cost. Hence M/2 + HV2Bm ≤2εP ≤t. The preceding bound gives warning probability at most E|W|, with E = 2V2Bp exp(−ε √ P). The monitor’s growth makes this E smaller than every prescribed reference-coupling power, after the finite block and stage parameters are fixed. This is the rapidly decaying warning allowance used by the activity prices. Its scheduling uses the original monitor and covariance bound, with no reselection of a Gaussian for a given molecule. The event is connected to the detector factors by a pointwise implication. A nonzero detector means the corresponding local quiet-watch condition fails. Unfolding that condition supplies a coordinate within distance 2B with |zi| at least ξ tfield/g. Thus a product with all its warning detectors nonzero lies in the joint event just bounded. The preceding plateau calculation proves this implication: under the quiet watch the curvature is at most one quarter of its threshold, the prediction mismatch is within its plateau, no nearby flag can occur, and every nonmandatory detector is a cutofftimes a vanishing plateau complement. A mandatory detector is identically zero. Taking the contrapositive therefore gives precisely the witness used in the probability bound. 7.22 The primitive integrand and its local extension The primitive integrand is a function of actual group links, chart variables, masks and insertion factors. On the protected interior its logarithm has the stated relation to the group-valued readout. Ordered group multiplication is retained throughout. In particular, from exp(l) = UB−1 one obtains exp(l)BU−1 = I; there is no step that commutes B with U. The extension away from the protected interior retains its cutoffand discrepancy rather than claiming this identity on its full domain. Here are the finite objects to which the integration estimates apply. Fix a coarse-to-fine step, its fine tree F, a seed inventory, a coarse configuration V , local discrete choices k, and the off-tree group variables Y . The active off-tree links are determined by the tree and seed inventory. For a color array u, the fine configuration has link exp(ue)(replica V )e on an active link, the value supplied by Y on an inactive off-tree link, and the fixed tree value on a tree link. The logarithmic signal equals u on active links, zero on tree links, and the chosen cutofflogarithm of Ue(replica V )−1 e on inactive off-tree links. Thus the group configuration and the logarithmic signal are different objects. Their identification by exponentiation is asserted only on the interior domain where the cutofflogarithm agrees with the genuine chart inverse. 136 ## Source page 139 A Wilson Yang–Mills construction J. Rodgers A letter is of one of four disjoint types: a block packet A, an old forced index q, a regular logarithmic index λ, or a warning index w. A molecule p is a finite set of letters; repeated copies of a letter are not allowed. Its forced inventory is the set of its forced indices. Its seed inventory is the union of the seeds of its packet letters. The ground H(p) is the union of the enlarged read footprints of all its letters. In particular it contains every seed and every forced read. Admission requires all of the following: the overlap graph of letter footprints is connected, the packet letters form the permitted layout of their combined seed inventory, the forced inventory satisfies its sealing and incidence conditions with those seeds, and flags(V ) ∩H(p) = flags(seed inventory of p). (PRI1) We spell out the discrete tests in this assertion. A seed is a member of the disjoint union of fine plaquettes, coarse bonds and coarse sites. These types remain distinct even when their spatial positions coincide. A store A is the triple of finite sets (Aplaq, Abond, Amflag); its number of seeds is the sum of their cardinalities. The read set of a plaquette seed is its coarse projected start set, that of a coarse-bond seed its two pin endpoints, and that of a site seed its singleton. The place of A is the union of these read sets; its dome A[r] is their coarse radius-r halo. Give the seed set the graph in which two seeds are adjacent when their radius-lB halos intersect. A connected packet means a nonempty connected set in this graph. A permitted layout L for A at parameters l, B is a finite set of stores with the following three properties: each A′ ∈L is a connected packet, (A′)[lB] ∩(A′′)[lB] = ∅ (A′ ̸= A′′), seeds(A) = [ A′∈L seeds(A′). (PRI6) It exists by taking the connected components of the seed graph and converting each component back to its store. Conversely any permitted layout is this one: an edge cannot connect distinct members because their halos are disjoint, and each member is connected. Thus the packet layout is determined by the combined seed inventory. It does not introduce a multiplicity of choices or a choice of a graph witness. For a finite set I of old forced indices write Pi for its actual fine support and Ji ⊂Pi for its declared inventory. Sealing means Pi ∩Pj = ∅for distinct i, j ∈I. Incidence means ∀i ∈I, ∀y ∈Ji, ∃p ∈Aplaq with y ∈touched(p). (PRI7) Here touched(p) is the fine set inspected for that plaquette seed in the original sector partition. This condition uses the inventory, whereas sealing uses the full supports. Their roles cannot be interchanged. The selector used in the packet integrand is defined at each fine site y by cI(U; y) = ( 0, y ∈flagsθ(U) \ S i∈I Pi, 1, otherwise. OI,Z(U) = Y y∈Z cI(U; y). (PRI8) For the packet, Z consists of fine sites whose parents are inside its 300B dome in the prescribed sense. For the full gas take all fine sites. If a product of old forced factors is nonzero, their individual permission identities give Ji = Pi ∩flagsθ(U). On a sealed family, the full selector is one exactly when the union of those inventories is all flags. Otherwise it is zero. This proves the exact rewriting of the old gas occurrence as the sealed indicator times (PRI8) times its old forced product. If the product is zero, both expressions are zero without using a permission identity for a vanishing factor. 137 ## Source page 140 A Wilson Yang–Mills construction J. Rodgers The footprint of a packet letter is its enlarged dome; that of every other letter is its prescribed inspected, projected halo. Define two letters to be adjacent when these coarse footprints intersect. The molecule ground is their union, as in (PRI1). Its admission is now completely specified by nonempty connectedness in this graph, the layout (PRI6) of its packet letters, sealing and incidence (PRI7) of its forced indices, and (PRI1). The empty seed inventory is allowed when there are no packet letters; its permitted layout is empty. Compatible molecules have disjoint grounds. For such a finite family, admission of its union is equivalent to admission separately in every member, together with the global flag coverage. To prove the nontrivial localization of incidence, suppose a forced inventory site in one molecule is touched by a seed attributed to another. The parent of that site lies in the forced footprint of the first molecule and in the seed footprint of the second. Disjointness of grounds forces the two molecules to coincide. The same parent argument localizes the sealing test. Layout restrictions localize because the enlarged packet domes are contained in their grounds; joining the local layouts gives a global layout. Equation (PRI1) localizes by intersecting the global flag set with disjoint grounds. Finally every finite set of letters has a unique partition into connected components of its footprint graph. These statements prove that expansion into finite letter sets and regrouping into compatible connected molecules are inverse operations, with multiplicity one. Absolute summability proved below permits this finite combinatorial identity to pass to the countable alphabets. All these admission tests remain discrete masks and are not differentiated. For the common lifted fluctuation u = μ + β−1/2z, the four vertex values are, respectively, vA = 1{packet connected}1{packet eligible}SA(u), vq = Kq(U(u)) E2a,q(u), vλ = eLλ(u,z) −1, vw = Dw(u). (PRI2) Here E2a,q is the product of fluctuation cutoffs on the entire forced read footprint. Packet eligibility requires its place to lie in the chosen seed place and its declared flags to agree with the actual coarse flags throughout its 300B neighborhood. To define the profile factors, let χ be the fixed smooth cutoffand put hs(r) = χ((r/s)2). For a tested fine plaquette use ht/4(toll(U, p)) if p /∈Aplaq, and its complement if p ∈Aplaq. For a tested coarse bond use hWt(wire(prediction(U, k), V ; b)) if b /∈Abond, and its complement otherwise. Multiply these factors once over every plaquette whose projected start set meets A[300B] and every bond whose pin set meets that dome. These are the profiles in [0, 1] in the following formula. Let nF , nP be the cardinalities of the finite frame-tag and predictor-point sets. At a site x, the ticket factor is nF times the actual frame-choice probability at x. At a bond it is nP times the actual predictor-choice probability for its connector. Multiply the site factors over A[300B] and the bond factors over all pin sets meeting it. They are nonnegative and bounded respectively by nF , nP , rather than by one. With the uniform average over the complete choice space, likelihood(U, k) = 1 |K| Y x siteMass(U, k, x) Y b pinMass(U, k, b). (PRI9) Indeed |K| = n|Λc| F n| Bondc | P , which cancels the inserted cardinality factors. This identity also includes zero-likelihood choices; their ticket product is zero. The frame probability is uniform over the admissible minimizing tags when that set is nonempty, with the prescribed uniform fallback; the predictor probability is its prescribed nonnegative normalized weight with its corresponding fallback. Thus the ticket retains the actual choice denominators, including the fallback branch. The packet value is explicitly SA(u) = e −β(Qnear(A,u)+Qfar(A))+P visit200B A ce+P active304B A je(u)−NA log mβ × OA(U) TA(U, k) PA(U, k, V ) Ea,A(u), (PRI3) 138 ## Source page 141 A Wilson Yang–Mills construction J. Rodgers where mβ = chart scale ( √ 2π β−1/2)dim g, and NA counts the off-tree frozen links in the 10B visit set. The terms ce are the link condensation terms; je are the local logarithmic Haar and cutoff corrections. The far term is the signed-orientation average of the extracted nonnegative row costs on the 200B posted set. The remaining four factors are the forced-order selector, the finite layout ticket, the product of sector profile factors, and the active cutoffproduct on the 304B neighborhood. The selector, profiles and cutoffs are in [0, 1]; the ticket has its finite combinatorial toll bound. The normalization deficit in (PRI3) is per frozen link, and is retained before that toll is estimated. The regular logarithms Lλ comprise the old regular activity and cubic, quartic, loop and quadratic-tail energy pieces, the remote row and local chart terms, and the finite ledger terms from Gaussian completion and covariance reweighting. Their signs are the signs in the exponent of the original density: in particular an old regular occurrence contributes its negative value. A warn- ing vertex is zero when mandatory; otherwise it is its fluctuation cutofftimes the corresponding sector-profile complement. The Gaussian warnings are also retained in this warning alphabet. All factors in (PRI2) use this single configuration, signal and Gaussian variable. Put bp(Y, z) = Q α∈p vα(Y, z). The primitive value before the finite choice average is the literal integral ap = Z bp(Y, z) dνF (Y ) dγC(z), (PRI4) with the product tree-gauge probability measure and the one covariance C prepared for this sector. There is no normalization by the mass of an admissible event. Local discrete choices are subse- quently averaged with weight 1/|K|; their probabilities do not replace any of the profile or ticket factors. For extension to coarse configurations outside the healthy domain, use the same finite formulas with the extended local row expressions and retain the admission masks at their actual enlarged output footprints. For a chosen ground H, retain only the finite Gaussian coordinates I(H) read there. The submatrix CI is positive definite at an actual admissible sector and has density (2π)−|I|/2 exp  −1 2 log det CI −1 2zT C−1 I z  dz. (PRI5) Integrate the extended product against this density and the tree-gauge measure, then take the uniform finite choice average. Set it to zero if the structural connectedness, layout or incidence tests fail. The marginal extension has its own fixed support and is not presumed positive or energetic at other coarse configurations. On a healthy ground this agrees with (PRI4). The integrand is constant on fibers of the coordinate projection to I(H); its Gaussian pushforward has covariance CI. The remaining local data agree on the ground. Integrating over these fibers and using (PRI5) proves the identity. The same finite-choice fiber argument proves independence of choices outside H. For disjoint grounds the finite choice space is a product; writing its cardinality as the product of the cardinalities of the factors proves that the average of their product is the product of their averages. The tree variables have the analogous product-measure identity. No independence of overlapping Gaussian warning events is used. The later continuity passage uses these same primitive integrals. Along an ultrafilter all finite seed stores and discrete tags can be fixed on one member of the ultrafilter. Keep the tree-gauge input Y fixed, and compare the two histories on this fixed finite data. Their covariance densities and vertex factors have the asserted pointwise limits. The primitive bounds below give a common integrable majorant, so dominated convergence applies to each molecule. Since this holds along every ultrafilter tending to infinity, the difference of the primitive values tends to zero in the ordinary sequential sense. This explains why the fixed finite data in the continuity argument can be selected 139 ## Source page 142 A Wilson Yang–Mills construction J. Rodgers after fixing Y , without claiming that a continuously varying configuration has a globally constant sector. 7.23 Integrating the primitive vertex bounds The primitive molecule body is a finite product of four kinds of vertices: connected block packets, old forced factors, regular exponential increments, and warning detectors. Fix a molecule p and let B(p) be its warning subset. The prepared estimates give nonnegative pointwise tariffs fi for each vertex and integrated tariffs ti, with Y i∈p ti = h|B(p)| Y i∈p fi. Only warning tariffs receive the factor h. The packet tariffis zero unless the packet is connected in the specified enlarged geometry. Otherwise it is its displayed block-packet income. Forced tariffs are the old forced envelopes, regular tariffs bound both the local log and exp(log)-1, and warning pointwise tariffs are the fixed storm bound. The source proves these four pointwise inequalities on the same local data and sector. Write Fp = Q i∈p fi. The triangle inequality for a finite product gives | bodyp(Y, z)| ≤Fp for every tree-gauge variable Y and Gaussian fluctuation z. If bodyp(Y, z) is nonzero, every warning detector in B(p) is nonzero. The warning estimate then places z in the joint catastrophe event CB(p), whose Gaussian probability is at most hˆ|B(p)|. The event and its covariance are common to all the factors; no factorization of its probability is used. Therefore pointwise |bodyp(Y, z)| ≤Fp1CB(p)z. The body is measurable, and the uniform finite bound makes it integrable against the product of the probability tree-gauge measure and the reference Gaussian measure. Integrating the displayed indicator majorant and then integrating Y proves Z |bodyp(Y, z)|dνY dγz ≤FpγCB(p) ≤ Y i∈p ti. This proves the primitive integrated tarifffrom the four vertex bounds and the joint catastrophe estimate. In particular an empty warning set costs no probability factor. Correlated warnings are paid by their actual joint event. For a marked team, remove the team’s selected keys from p and multiply that remaining body by its paid ingredient product. Each paid leaf ingredient has absolute value bounded by its assigned leaf weight, so their product is bounded by the team weight WM. If the team keys are not all in p, define the term to be zero. Otherwise the preceding argument gives the marked bound Z |markedBodyM,p| ≤WM Y i∈p\keys(M) ti. The corresponding uniform pointwise bound uses the fi instead of ti. Taking these products over all finite molecule subsets is summable whenever the primitive tarifffamily is summable: the sum is bounded by WM times the product of 1+ti, hence by WM exp(P ti), with finitely many compulsory keys held present. This supplies the summable integral-norm bound used in the primitive Mark expansion. Its remaining analytic inputs are the block-packet bound, old forced and regular-factor envelopes, warning storm bound and joint Gaussian catastrophe estimate. 140 ## Source page 143 A Wilson Yang–Mills construction J. Rodgers 7.24 Factoring the regular gas and restoring forced exclusions At one Mark step split primitive activity labels into regular labels with empty flag inventory and forced labels with nonempty flag inventory. Fix a compatible forced set A, and let YA be the union of its enlarged supports. The allowed regular gas in the presence of A removes the regular labels meeting the forced exclusion region. Denote the full regular gas by Z0, and its convergent cluster logarithm by L0. The removed logarithmic terms are precisely regular occurrence clusters whose union meets that region. If their sum is MA, the restricted gas is Z0,A = exp(L0 −MA) = Z0e−MA. This is an equality of absolutely convergent series: cutting the regular label family decreases its absolute budgets, and subtracting its cluster sum from the full cluster sum leaves exactly the indicated missing clusters. It retains their original occurrence indices and factorials. Multiply by the forced product and sum the compatible forced sets with the required total inventory. Expand e−MA by positive multiplicities of missing-cluster addresses. The resulting term consists of the forced set and a finite multiplicity family of negative regular clusters; its weight is the product of the forced activities and the cluster factors (−h)r/r!. Each selected missing cluster intersects the forced exclusion region. Absolute summability follows by the restoration bound, or at fixed volume by the summable exponential of the absolute missing-cluster series. Therefore grouping these terms does not change their sum. Decompose every such term into overlap-connected components of its atom supports. Every component containing a missing cluster also contains a forced atom: that cluster meets the union of the selected forced supports. A component consisting only of forced atoms is already allowed. The resulting components have pairwise disjoint supports; their forced inventories partition the original inventory. Conversely, union of any permitted disjoint collection of these components recovers the original forced set and each missing cluster with its multiplicity. The multiplicity factorials factor over components because no atom address belongs to two distinct components. Thus the map is bijective and preserves the product weights. The original gas, multiplied by its external scalar prefactor, is consequently Z0 times the gas of these connected restored forced components. This proves the regular-log factorization and forced regrouping identity once the primitive Mark expansion has been supplied. It uses exact inventories and actual overlap exclusions; the more permissive repetition count used for the absolute bound does not alter this equality. For switched leaves the same argument holds at each real switch value 0 ≤x ≤1, with the same supports, threshold masks and inventory tests. The paid bounds established above apply uniformly there. Two subsequent dictionary fusions put the restored components into the final coarse dictionary. Each fusion preserves the gas up to the exponential of its recorded unmarked regular log contribution. These extra logarithms do not depend on the switch selection. The unmarked normalization identity says that the final coarse exponent, its fusion logarithms, and L0(0, ∅) reproduce the original scalar prefactor. Substitution into the factored switched gas identity yields eEA(U)eL0(x,R)−L0(0,∅) GasA(K(x, R); U) = P(Buy(x, R))(U). All divisions in the regular-gas ratio are legitimate because the real gas/log identity proves Z0 > 0, including the unmarked branch. The primitive Mark expansion, dictionary fusion and unmarked scalar normalization proved in this chapter establish the scalar equality, with the stated exclusion restoration and component regrouping. 141 ## Source page 144 A Wilson Yang–Mills construction J. Rodgers 7.25 Paid regular and forced cluster terms A switched primitive activity is a finite ticket sum. Each ticket carries switch dependence x|D|1D⊆R and a scalar factor bounded by the product of its marked leaf weights and all unmarked activity weights not selected by the ticket. Its switching model is concentrated at D. Summing tickets gives a paid bound equal to the product of the augmented activity tariffs. The selected mixed-activity estimate enlarges this bound by its displayed support and polynomial-moment weights; it sets it to zero when the activity’s eligibility conditions fail. These are pointwise paid bounds uniform in configurations. A regular cluster term is the product of its n switched primitive activities, multiplied by φ(z)/n!. The product rule for paid models gives the bound |φ(z)| n! n Y j=1 uij, where ui are the selected primitive paid bounds. The rooted cluster estimate just proved gives a support-weighted sitewise sum at most P0/(1 −ρ) for their regular cluster envelopes, when the selected primitive budgets are P0 and the tree ratio is ρ < 1. Subtracting each cluster’s unmarked value doubles its paid bound. These differences are the regular log increments in the switched identity; they are not independently selected regular activities. The forced restoration terms combine forced primitive activities and negatives of regular clus- ter terms, each with its explicit occurrence multiplicity. For an atom occurring r ≥1 times the factor is hr/r!; different atom factors are multiplied. The paid product and power bounds there- fore give exactly the corresponding product of envelope powers divided by factorials. Restrict to connected terms with at least one forced atom and the required flag inventory. These restrictions depend on configuration and on the fixed combinatorial term, not on the switching parameter, so multiplying by their indicators preserves the paid bounds. The omitted-cluster restoration and primitive-integral arguments in this chapter control their support-weighted sum. 7.26 Restoring omitted clusters with positive multiplicities The restoration exponential uses a different indexing from the occurrence tuples in the logarithm. Its index is a finite set D of atom addresses together with a positive multiplicity ri ≥1 at every i ∈D. Its weight and total size are Wh(D, r) = Y i∈D hri i ri! , N(D, r) = X i∈D ri. An absent atom is absent from D, rather than being included with multiplicity zero. For nonnegative hi ≤σ < 1, summing each positive multiplicity gives X ri≥1 hri i ri! ≤ hi 1 −σ. Indeed ri! ≥1 and hri i ≤hiσri−1. Multiplying these inequalities over a fixed distinct set D reduces the repeated-atom bound to a distinct-set bound with inflated vertex weights hi/(1 −σ). The sitewise smallness hypothesis guarantees each required hi ≤σ: choose a site in its nonempty support and retain its single summand in that site’s budget. All extra support factors and (1−σ)−1 are at least one. 142 ## Source page 145 A Wilson Yang–Mills construction J. Rodgers The distinct connected-set tree estimate then bounds the sum of repeated connected terms whose union contains a fixed site by η/(1 −q′), provided the inflated vertex activities satisfy the readiness condition with budget η ≤σ and tree ratio q′ < 1. The actual union support weight is paid by the product of its vertex support weights: connected cover subadditivity applies to D, and each multiplicity ri ≥1 only increases the available exponent sum. Nonnegative summation justifies all these successive sums. For omitted-cluster restoration take two kinds of atoms. Forced atoms have nonnegative weights di, nonempty supports Yi, and sitewise budget c at exponent A + δ′. Regular-log atoms are occurrence-indexed clusters, with their absolute cluster weights and union supports. The preceding logarithmic tail estimate gives their sitewise budget γ/(1 −q) at that exponent. Their combined repeated products bound the absolute value of every signed restoration product; forgetting original exclusions and retaining additional repeated forced atoms only enlarges the upper bound. To obtain both a size tail and a compulsory-forced-atom gain, choose t ≥1 and μ ≥0. Multiply every atom weight by t, and every forced atom weight additionally by eμ. After the geometric multiplicity allowance, the combined sitewise budget is at most teμc 1 −σ + tγ (1 −σ)(1 −q). Require this to be at most η ≤σ, with 0 ≤σ < 1, δ′ > 0, and 4(2B + 1)4η/δ′ ≤q′ < 1. Apply the preceding repeated connected estimate to these augmented activities. A term of total size N ≥m acquires the factor tN ≥tm. If it contains a forced atom, the exponent summed with multiplicities acquires at least μ, so its additional factor is at least eμ. Dividing the augmented bound by these factors proves X connected (D,r), N(D,r)≥m at least one forced atom x∈∪i∈DXi eAcB(∪i∈DXi)Wh(D, r) ≤t−me−μ η 1 −q′ . This is the omitted-cluster restoration bound used for the forced paid terms. Its applications use t = 2, σ = 1/2, and q′ = 1/2; at m = 0 the right side becomes 2e−μη. The compulsory atom is charged at least once, regardless of its multiplicity. The remaining input in this argument is the distinct connected-set tree estimate under the stated readiness condition, which belongs to the same rooted-tree bounds as the regular cluster estimate. 7.27 Switching real source formulas into the marker algebra The nilpotent substitution in the marked expansion is defined through real switching values, not by a formal differentiation of an unbounded series. For a finite switch set I, a real coefficient array gx(A), A ⊆I, gives switch values f(x, R) = X A⊆R x|A|gx(A), 0 ≤x ≤1. At x > 0 the array is uniquely determined by these values. Induct on (|R|): subtract all proper- subset terms from (f(x,R)), then divide by x|R| > 0. Equivalently the inverse is the finite Möbius formula gx(A) = x−|A| X R⊆A (−1)|A|−|R|f(x, R). 143 ## Source page 146 A Wilson Yang–Mills construction J. Rodgers A model for f is such an array with each gx(A) →g0(A) as x ↓0. The switching jet is Jf(A) = g0(A), independently of the model by uniqueness for x > 0. A paid model includes nonnegative bounds (b(A)) with |gx(A)| ≤b(A) for 0 ≤x ≤1 and P A b(A) ≤c, uniformly over any additional configurations. In particular |f(x, R)| ≤c on that interval. For square-zero elements zi of the finite marker algebra, define Sz(f) = X A⊆I Jf(A) Y i∈A zi. This substitution is additive on modeled families. To prove multiplicativity, multiply the two switch-value formulas. Their coefficient convolution is (g ∗x h)(A) = X B∪C=A x|B∩C|g(B)h(C). At x = 0 only disjoint pairs survive. On substitution, every overlapping pair also vanishes because some zi occurs twice. Hence Sz(fh) = Sz(f)Sz(h). Constants substitute to their scalar polynomial. Paid bounds multiply by the corresponding product bound, and addition adds their bounds. Countable sums pass through this operation when their paid bounds cn are summable. For each subset A, the corresponding coefficient arrays are dominated by summable bounds, so dominated convergence for series gives their limit at x = 0 and commutes it with summation. There are only finitely many A; applying the finite linear substitution proves Sz X n fn ! = X n Sz(fn). Apply this to the exponential series. The n-th term has paid bound cn/n!, which is summable. Multiplicativity gives Sz(ef) = exp{Sz(f)}. If the scalar mode of Sz(f) is zero, the right side is its finite nilpotent exponential. This establishes the exponential identity without dividing by a density. The price estimate is coefficientwise. If |zi|w ≤1, define the envelope of mode A to be the sum, over switch subsets D, of (b(D)) times the absolute coefficient of Q i∈D zi in that mode. The triangle inequality bounds the substituted coefficient by this envelope. Summing its weighted price and using submultiplicativity gives at most P D b(D) ≤c. Thus a paid scalar switching model produces the local polynomial price required by restoration. When every zi has zero scalar mode, the scalar mode of Sz(f) is f(0, ∅). Its singleton marker coefficient receives contributions only from switch subsets of size one; products of two or more zero-scalar factors cannot have singleton degree. These two facts prove the scalar and singleton coefficient readings used below. Locality also passes at the coefficient level. If changing switch i leaves a scalar switching family unchanged, its Möbius coefficients indexed by subsets containing i vanish. If the output support omits a site in the parent support of leaf i, the local silence identity gives exactly that switch inde- pendence. A nonzero coefficient can therefore involve only leaves whose required parent supports lie in the output support. Their scoped marker monomials then have all their pins there. The finite substitution preserves gauge invariance, locality and measurability. If every scalar switching value vanishes offa prescribed healthy region or inventory event, the Möbius coefficients and their limits 144 ## Source page 147 A Wilson Yang–Mills construction J. Rodgers vanish there as well. These statements supply the support and permission properties of the marked corrections. 7.28 Reading the marked Mark-step output The reserve used by a retained step is constructed for the very finite leaf inventory just described. Give each leaf its fee and normalized square-zero monomial zi. The parent support contains all pins in the monomial’s scope, and |zi|w ≤1. The marked activity expansion produces a family K(t, A) indexed by subsets of the finite leaf inventory. Its mixed jet in the leaves is evaluated on products of the zi; because z2 i = 0, only distinct-leaf derivatives survive. Thus the marked polynomial is a finite substitution in these jets. The coefficient bounds on the mixed jets yield local polynomial bounds by the triangle inequality and ∥Q i zi∥w ≤1. The output has an exponential correction and a forced activity correction. The latter is grouped by the coarse support-and-inventory barcode. Its scalar mode remains the unmarked background profile. For a singleton marker v, only single-leaf jets can contribute; the single-leaf jet identity is ∂iKj = feei Dj(Mi), where Dj(Mi) is the transported hinge of that same leaf. Multiplication by its normalized monomial cancels the fee, including the zero-charge case established above. Hence the singleton coefficient of the grouped forced correction is exactly the sum of the leaves’ quoted transported profiles. Absolute summability of the hinge families follows from the sound-door estimate: each hinge has a forced envelope whose total weighted mass is bounded by a fixed multiple of its leaf load. Summing over finitely many leaves and over the barcode fiber therefore commutes with coefficient extraction. This proves the singleton-profile identity used in the restoration step. The coarse exact density identity is likewise an evaluation identity for the marked expansion: the Mark push of the priced ticket expression equals the exponential output times the coarse field polynomial. It is multiplied by the factored scalar displacement, which commutes with the Mark integral since it is a fixed finite-dimensional polynomial. The exponential output has zero scalar mode. The coefficient bounds and scoped locality give its served envelope; the forced mixed-jet bounds give the field envelope. These are the reserve’s two output budgets. The same construction is additive under summable grouped leaves because the underlying hinge map respects those sums. The switched marker expansion, primitive integral and connected-cluster arguments in this chapter construct the marked activity jets and prove their exact Mark identity. The exact marked identity follows from the real switching identity once its paid bounds have been established. At a coarse configuration U, let a = eEA(U), let L(x, R) be the sum of the regular log increments relative to the unmarked branch, and let G(x, R) be the gas of the switched forced activities. The real identity is a eL(x,R)G(x, R) = P Buy(x, R) (U), 0 ≤x ≤1, with P the actual Mark push. The paid regular increments have a summable total budget. The forced-activity bounds give a summable paid expansion of the gas. Apply Sz to this identity, commute it with those summable series, and use its product and exponential laws. Since the regular log increment is zero at the unmarked switch, its substituted scalar mode is zero. The left side becomes a Exp X S HS(U) ! Field(K(U), FA(U)). 145 ## Source page 148 A Wilson Yang–Mills construction J. Rodgers For the right side, expand Buy into finitely many tickets. A ticket with leaf set D has switch dependence x|D|1D⊆R times its scalar coefficient; its switching jet is concentrated at D. Its substi- tution is therefore that coefficient times Q i∈D zi. Each scalar ticket coefficient is integrable against the Mark kernel. Finite linearity of integration proves that substitution commutes with the push of this ticket sum. Its value is exactly the push of the priced polynomial inventory. This proves the marked Mark identity from its real counterpart and paid envelopes, with all infinite summations justified by their paid series bounds. The shipping bound ensures that this finite inventory is admissible to the activity estimates. If a fine leaf has cover cost v = cB1(S), its enlarged coarse birth support obeys (q + 1)cB∗(birth(S)) ≤(q + 1)Cfoot + 5 4εv. The tariffalso spends 2εv. With s1 ≥100, the selected relative width and exponent satisfy 13 4 ε ≤s1−4. Thus the original fine support weight pays both costs, leaving only the fixed multiplier e(q+1)Cfoot. Charge each leaf to one of its scoped pins, use the parent-pin occupancy bound M, and convert the rooted budgets to a sitewise bound. The remaining numerical demand is M siteFactor(B∗)e(q+1)Cfoot v′ ≤ribbon, where v′ is the inventory’s input budget. This is the shipping price. It pays the aggregate tariff after grouping equal activity labels, since those groups simply sum nonnegative leaf prices. Adding the unmarked activity reserve gives the admissible marked tariff. The geometric birth-support inequality and the scalar connected-activity estimates providing the ribbon, paid log increments and switched gas identity remain the lower-level inputs to this construction. 7.29 The numerical form of a retained insertion step Write s1, s0 ≥100 for fine and coarse support exponents, B1, B0 for their support widths, and M for the occupancy bound. Choose 0 < d ≤1. First replace fine marker weights Iv by dIv. Since every correction has degree at least two, its new monomial weight is at most d times its old weight, so its unit price budget becomes at most d. Reduce the support exponent to s1 −4. The finite transport inventory then has cost controlled by T + eT T, T = M siteFactor(B1)d + 4d. Purchase a Mark-step reserve whose input budget is at least this quantity, and whose output field budget is u. The response transport bounds must satisfy 2Ivav ≤1 and Ivbv ≤1, where av, bv are its two local envelope prices. After shrinking, these become 2dIvav ≤d and dIvbv ≤d, precisely the bounds required by the reserve. At the coarse level put Q = u + M siteFactor(B0)d. The numerical requirements are (2B0 + 1)4Q ≤1 2, eQ −1 ≤1. The first pays the closure loss in support exponent, and the second bounds its final correction price by one. The matching condition on coarse weights Jv pays another half unit of support exponent when passing from dIv to Jv, with price multiplier at most one. The reserve supplies the coarse exponential exponent s0 + 1 2 and field exponent s0 + 1; these two half-unit expenditures leave at least s0. The singleton feed identity and the restoration argument above identify the coarse primary and forced insertions. This proves the retained unit-price served conclusion conditional on the finite transport inventory and Mark-step reserve. The finite transport inventory and marked reserve constructed in this chapter provide these inputs. 146 ## Source page 149 A Wilson Yang–Mills construction J. Rodgers 7.30 Separation pays for complex inflation The large analytic disk in the omission argument is a geometric consequence of separated markers. Let B ≥8, let A ⊆V have at least two elements, and suppose distinct pins in A have torus max-distance at least B2. If a support S contains these pins, then B|A| ≤64cB(S). To prove this, choose a minimizing cover with D = cB(S) centers. Its halo admits a walk visiting the centers in at most (D(4B+2)) unit steps. For each pin choose a visit time whose center is within B of it. Two chosen times differ by at least g = B2 −2B > 0, by the distance triangle inequality and the unit-step walk bound. Thus their integer quotients by g are distinct and (|A| −1)g ≤N, where N is the walk length. Since |A| ≥2 and B ≥8, |A|B2 ≤4(|A| −1)g ≤4N ≤4D(4B + 2) ≤64DB. Division by B > 0 gives the assertion. Notice that the walk visits covering centers; visiting the pins themselves is unnecessary. Every nonzero higher correction has a nonzero mode with at least two markers, all pinned in its support. Hence cB(S) ≥B/64, and SB := eB/64 ≤ecB(S). For |z| ≤SB, multiply a nonzero correction envelope ai by (|z|), and set the new envelope to zero for identically zero corrections. Then |z|aie(e−2)cB(Si) ≤aie(e−1)cB(Si) ≤aieecB(Si). Thus multiplying every higher correction by z preserves the price budget while reducing the sup- port exponent by two. Locality, gauge invariance, measurability, scope and permission conditions survive scalar multiplication, and no scalar or singleton mode is introduced. This proves precisely the inflation estimate needed for holomorphy and retained transport in the omission comparison. It leaves the primary and forced singleton insertions at their literal values. 7.31 A positive denominator and an explicit vacuum bill Let μG be normalized Haar measure and dM the real dimension of the ambient matrix space of the chosen representation. Normalize the matrix chord by the constant matrix norm γ > 0, so chord(g) = |g −I|/γ. For 0 < r ≤1, μG{g : chord(g) ≤r} ≥(r/4)dM . Here is an ambient-space proof, requiring no coordinates on the group. A collection of group matrices with mutual distances greater than rγ gives disjoint ambient balls of radius rγ/2, all inside the ball of radius 2γ. Comparing Euclidean volumes bounds its cardinality q by q(r/4)dM ≤1. A finite maximal such collection exists: otherwise successive insertion would violate this cardinality bound. Its radius-rγ closed balls cover the group. By unitary invariance of the matrix norm, their intersections with the group are left translates of the chord shell. Haar invariance and the union bound give 1 ≤qμG(shell(r)). Combining the inequalities proves the assertion. On a finite four-dimensional torus Λ, take the shelter of configurations with every positive oriented link in the chord shell of radius s/4, where 0 < s ≤1. Product Haar measure gives this shelter measure at least 147 ## Source page 150 A Wilson Yang–Mills construction J. Rodgers (s/16)dM|E|, |E| = 4|Λ|. The chord of a product is at most the sum of the chords, and inversion preserves the chord. Every plaquette in the shelter consequently has chord at most s. If s ≤θ, the field threshold, there are no flags there; its activity gas is exactly one. Signed translations preserve the shelter as they only permute links and invert some of them. Write C for the path bound, Λ0 for the loop cap, δ for the regular-activity allowance, κ for the forced allowance, β ≥0 for the coupling, and Cdir for the direct quadratic clipping coefficient. Define l= β(Cθ3 + Cθ4) + Λ0θ2, b(s) = |P| Cdirs2 2 + C2s2 2 ! , e0 = δ + l, where P is the finite set of plaquette-axis pairs. Each healthy atom of degree k is bounded by its coefficient magnitude times θk: every transported curvature leg has norm at most the threshold, since its footprint is healthy and parallel transport preserves the norm. Absolute summation with the atom budgets bounds the cubic, quartic and loop remainders per site by Cθ3, Cθ4, Λ0θ2, respectively. The quadratic base is nonnegative. Thus the channel-averaged action is bounded below by −|Λ|l. On the shelter, every curvature leg has norm at most s. Summing the absolute path coefficients gives norm at most (Cs) for the linear path field, so the quadratic base per site is bounded by (b(s)). The same remainder estimate bounds the channel action above by |Λ|(l+ βb(s)). The regular activity exponent has absolute value at most |Λ|δ: charge each occurrence’s weighted budget to its designated root site and sum those root budgets. If v0 is the scalar vacuum exponent, the full exponent ER therefore satisfies ER(U) ≤v0 + |Λ|e0 everywhere, ER(U) ≥v0 −|Λ|(e0 + βb(s)) on the shelter. The represented density is nonnegative. At the finest scale this follows from the positive bare Wilson density; downward induction preserves it because each Mark kernel is positive. Integrate over the shelter, where the gas is one, and use nonnegativity outside it. The resulting denominator bound is ZR := Z R(U) dμ(U) ≥exp{v0 −|Λ|(e0 + βb(s)) −4|Λ|dM log(16/s)} > 0. Combining this with the insertion norm bound proves the explicit normalized estimate tw(A)| R [X(U)]A dμ(U)| ZR ≤exp |Λ| 2e0 + κ + βb(s) + 4dM log(16/s)  + |V |(x + y + 2f) . The scalar v0 cancels. This bound applies to the full denominator, with no change of law or restriction of the normalized numerator to the shelter. For a selected history use s = bgk = bβ−1/2 k , with β ≤2 bβk. The loop price is monotone in nonnegative β, as is βb(s). Hence the coefficient of |Λk| in this bound is at most the explicit treasury τk = 2{δk + lk(2 bβk)} + κk + 2 bβkbk(bgk) + 4dM log(16/bgk). 148 ## Source page 151 A Wilson Yang–Mills construction J. Rodgers This establishes the root estimate used in the omission argument. Its extensivity is explicit; no volume factor is silently absorbed in an error tending to zero. The treasury also has a useful stagewise bound. Set P(s) = exp{(log log s)2} and Θ(s) = s−1/2P(s). For every fixed b and every p > 0, eventually P(s)b ≤sp, because b(log log s)2/ log s →0. Consequently sΘ(s)3 ≤1, sΘ(s)4 ≤1, and 2P(s)vΘ(s)2 ≤1 uniformly for v ≤1, eventually. Substitution into the actual thresholds and loop caps gives lk(2s) ≤4Ck +1, while δk, κk ≤1. Also sbg2 k = 1. Therefore, eventually, τk ≤2{1 + (4Ck + 1)} + 1 + |P|(Cdir,k + C2 k) + 4dM{log 16 + 1 2 log bβk}. Stage parameters are kept visible in this inequality; it asserts logarithmic growth in the reference coupling only after those parameters are fixed. 8 Convergence of the interacting scale histories Convergence requires one common numerical schedule for all fixed finite demands. The order is: choose the finite stage profile, then its block length, then its post-length constants and coupling- entry delay, and only afterwards the terminal cutoffand physical period. Increasing a delay does not reselect a previously fixed block. This prevents circular parameter choices in the stationary estimates, shooting construction and actual coupling running. 8.1 A common analytic stationary graph and its response estimates The stationary graph solves the vertical critical-point equation in a common analytic neighbor- hood. Positivity of its vertical Hessian gives the inverse used in the response estimates. Retained coordinates and vertical coordinates have different norms; the explicit color-coordinate maps and their bounds below justify each passage between them. No extensive scalar action is assigned a volume-independent zeroth-order norm. We specify first the actual score to which the analytic argument is applied, at a fixed derivative order n ≥2. A retained coordinate is a coarse oriented bond together with a real color index. A vertical coordinate is a free fine bond of the fixed forest together with such an index. Their disjoint union is the joint coordinate set. Arrays carry the maximum norm. Splitting the union is an isometry; replacing real color coordinates by their Euclidean color vector defines a linear isomorphism paint. Its norm is at most 1 + dim g, and its inverse has norm at most one. Moreover the sum of squared vertical real coordinates is exactly the sum of squared Euclidean color norms. Denote the resulting retained and vertical vectors by (tint x, tint v). The actual scalar score in these coordinates is F(x, v) = tested(tint x, tint v). (AS1) Thus it is the main fine action plus the prediction penalty, with the fine and outer configurations constructed from these same variables. The connector is not an independent variable. A retained coordinate is addressed at its embedded coarse site, and a vertical coordinate at its fine site. Each address fiber has at most h = |TypeFace1×ColorIndex|+|Axis×ColorIndex| members: orientation and color distinguish the members, and the coarse embedding is injective. Here is the local-to-global estimate, including the counting loss. For addressed coordinate slots q1, . . . , qk, put S(y, q) = P i dist(y, address(qi)). An anchor estimate of price M and width a means |Dkf[eq1, . . . , eqk]| ≤Me−aS(y,q) (k ≤n). (AS2) 149 ## Source page 152 A Wilson Yang–Mills construction J. Rodgers If a term depends only on coordinates within distance R of y, and its operator derivatives through order n are bounded by M0, a coordinate derivative vanishes whenever one of its slots is outside that set. Otherwise S ≤nR. Thus (AS2) holds with M = M0eanR. For path series the common exponentially weighted length budget dominates these derivative sums absolutely; differentiation term by term follows on a smaller common analytic neighborhood. This proves the same estimate for the cubic and quartic series, not merely for finite path sums. Sum such terms over anchors with multiplicity at most h0. Fix a marked slot qj. The triangle inequality gives P i dist(address(qj), address(qi)) ≤(n + 1)S(y, q) and dist(y, address(qj)) ≤S(y, q). Split the exponential in half and use the lattice counting estimate c(b) = 16 6!e2bb−6. The summed derivative has price Mh0c(a/2) and width a/[2(n+1)], measured from any marked slot. This is the Scatter bound used here. At order zero there is no marked slot; no volume-independent bound for the total scalar action is being asserted. For the actual main score take a = μ 4(n + 1)(1 + L). At a signed site its density is the sum over planes of pdirect price(U□) + ∥l(U)□∥2/2, together with the cubic and quartic path series. The plaquette has length four and locality radius 5 + L. The matrix product rule costs 2n per product; inversion costs one for unitary matrices. In particular, writing Cn(K, C) = n!K(1 + C)n, a walk of length at most r has derivative price Tn(r, C) = Cn e∥1∥[1 + 2n(1 + r)]n, C . The Wilson price has bound P(C) = 1+∥traceCharge∥C, where traceCharge(M) = −Re tr(M)/m. If o is the purchased outer-link jet bound, the fine-link bound is Cf = H + 2nH2, H = o + ∥1∥, by the same product rule. These are ordinary finite-dimensional derivative bounds on the unit real coordinate ball. To make the path prices explicit, let rμ be the common path analyticity radius at width μ/2, and set Jn = Cn([1+2n/rμ]n, Cf). If sG is the color-slot coefficient and cG the real color projection, the row and atom prices are Rn = Jn|sG|∥cG∥eμ and An,k = Jn|sG|keμ. These are the convergent weighted path bounds obtained by spending half the exponential length width. Choose Z′ > 0 bounding all derivatives through order n of w 7→∥w∥2/2 on ∥w∥≤CRn; any number exceeding 1 + CRn + (CRn)2/2 suffices. Put Nn = Pn i=0 |OrderedFinpartition(i)|. The per-plane and per-site prices are Qn = |pdirect|P(Tn(4, Cf))ean(5+L) + NnZ′(1 + CRn)n, Mn = 6Qn + CAn,3 + CAn,4. (AS3) The second term is the ordered-partition chain rule for the squared row. Signed site maps are injective, so their anchor multiplicity is one. Averaging over the signed symmetries cancels their number. The main-action Scatter price is therefore Mnc(a/2). For the prediction term choose derivative inputs through order n + 1 first. If C+ f is the fine-link price at that order, put F∗= 2 alignCost pathChange + L variation, ρ = min ( ρframe, √predictorCut F∗ , r colorScale F∗ ) . (AS4) 150 ## Source page 153 A Wilson Yang–Mills construction J. Rodgers Here ρframe is the positive local minimizer radius for the frame score with the purchased (n+1)-jets and half its root gap. The constants pathChange and variation bound the actual connector paths under a unit coordinate displacement; alignCost converts the two endpoint frames. The three in- equalities in (AS4) keep the frame minimizer in its implicit chart, keep the focus below its predictor cutoff, and keep its group logarithm inside radius r, respectively. Consequently the selected con- nector agrees there with the analytic minimizer. One differentiates that analytic expression, rather than the discrete selection tags. Here are the local frame constants in this formula. Let NP be the number of frame points, with the central frame fixed to the identity. The blank score’s quadratic form is the average over ordered point pairs of squared differences of their logarithmic frame coordinates. The pairs with one endpoint at the fixed center contribute twice the sum of squared free coordinates. Consequently its counting gap is gP = 2/N2 P . Put en = Cn(K, 1), and, for input link jets bounded by C, set Sn = P (2n[2nenTn(8R, Cn(C, 1))]en) , RH = min  1, gP 2(1 + Sn)  . (AS4a) There are two frame factors and an intervening route of length at most 8R; their product gives exactly this jet bound for the frame score. The mean-value estimate for its Hessian on the radius- RH ball changes the blank Hessian by at most SnRH ≤gP /2. Its inverse there has counting-norm bound 2/gP . Let J = 1 + 8RC/colorScale, the path chord price, and let A = alignCost J. For the purchased logarithm derivative bound D, valid inside chord radius δ, the frame radius is ρframe = min  RH, frameCut 3J , δ A, RH NP DA  . (AS4b) The last term puts the logarithmic coordinate of the selected minimizing frame inside the Hilbert ball; the preceding two ensure its branch and logarithm agree with the chosen analytic chart. These are the finite local frame and alignment constructions in the nonlinear geometry section. Applying them with order n + 1 is why the penalty estimate purchases that extra order before choosing (AS4). The implicit derivative polynomial used here is also explicit. For branch value bound R0, equation jet bound M, and inverse linearized-equation bound C0, define I0 = R0, Ia+1 = Ia + C0M{1 + |OrderedFinpartition(a + 1)|(1 + Ia)a+1}. (AS4c) Differentiate the implicit equation a + 1 times. Its single-block term is the vertical linearized equation applied to the unknown highest branch derivative. Every other ordered-partition term uses only branch derivatives of orders at most a, and is bounded by M(1+Ia)a+1. Multiplication by the inverse bound proves the recursion; the added Ia retains all lower jets. In the frame application R0 = 1, M is the frame-score price and C0 = 2/gP . This defines the polynomial used in the next paragraph without an unpriced implicit differentiation step. More explicitly, let In be the implicit-derivative polynomial for the local frame score, evaluated with its score-jet price and inverse half-root-gap. The frame transport price is Hn = Cn(K, In). Then the connector price is Kconn = 2n[2nHnTn(L, C+ f )]Hn, by its two endpoint factors and inter- vening path. Here K bounds the local exponential chart jets, and W below bounds the focus-map jets on the radius kept by (AS4). Also on+1 = Cn+1(Z, 1 + dim g), where Z is the purchased group- chart jet price. Each prediction bond is supported within radius 3L + 16R, where R is the patch radius. Its anchor price is explicitly Npred = P (2non+1Cn(W, Kconn)) ean(3L+16R). (AS5) 151 ## Source page 154 A Wilson Yang–Mills construction J. Rodgers There are four coarse bond orientations per embedded anchor, so summation has multiplicity four. Thus the penalty Scatter price is 4|ppenalty|Npredc(a/2). Combining the two terms gives the actual price C∗= 1 + |Mnc(a/2) + 4|ppenalty|Npredc(a/2)| , a∗= a/[2(n + 1)]. (AS6) All constants in (AS3)–(AS6) are fixed from the group, stage, path budgets, local frame jets and patch geometry before the torus period. In particular no number of lattice sites occurs in them. Finally the origin slope and gap are consequences of the actual score expansion. The quadratic line limit is one half of the signed average of P □[pdirect∥dz∥2 + ∥llinz∥2] plus one half of the penalty times the squared linear prediction residuals. Cubic and quartic terms start in their re- spective orders. Analytic Taylor expansion therefore gives zero first derivative at zero and this quadratic form as the Hessian. Restrict to vertical vectors. Keeping the identity signed orienta- tion gives coefficients b = pdirect/|Signed| and ppenalty in front of the curl and prediction-residual energies. The forest counting estimate reconstructs a vertical field from these energies, with loss K = curlFactor(L, box, linkCost)+probeFactor(L). Its tree edges vanish; the harness link estimate bounds the remaining reconstructed links. Hence DvF(0, 0) = 0, D2 vF(0, 0)[v, v] ≥min(b, ppenalty) 1 + K X i v2 i . (AS7) For an explicit count, let X be the finite cover box of radius macroR, let VL = L4, and let c1, c2 > 0 be the two local harness constants. With kernel predictor cutoffq∗> 0, put α = 2 max(1, q−1 ∗)(2c1 + c2), EL = (8L + 1)c2L + 2c1L2 + α/L2. The local link reconstruction bounds a nonnegative link norm by EL times the sum of curvature norms in its translated X-box, plus its coarse prediction-residual norm. Squaring uses (a + b)2 ≤ 2a2 + 2b2, and the curvature sum uses Cauchy–Schwarz with 6|X| terms. Summing fine bonds, reindexing their coarse parent and residue, introduces 4VL positive-axis residue bonds; a fixed translated curvature can occur at most |X| times. A residual occurs VL times. Thus Kmcurl = 2E2 LVL 4 (6|X|) |X|, Kmprobe = 2VL, K = Kmcurl + Kmprobe. (AS8) This derives the finite denominator in (AS7); it may grow with the blocking ratio and fixed patch, but never with the macroscopic torus period. The positive numerator is part of the actual step hypotheses. The equality of color and coordinate squared sums above gives precisely this counting norm, with no dimension conversion. Also tint(ei) is the corresponding physical color frame, so the vertical Hessian block is the physical one. Invertibility of tint shows that the zero vertical derivative in (AS1) is exactly the original stationary equation. Thus (AS4), (AS6), the address multiplicity h, and the gap in (AS7) provide the actual initial radius, width, price, multiplicity and gap for (US1)– (US8). Their stationary graph, inverse-Hessian and determinant estimates consequently apply on the same common domain used in stationary integration and the reference chart. We make explicit the uniform implicit construction used in the stationary integrations. Split the finite coordinate set into retained coordinates ι and vertical coordinates τ, assign every coordinate a lattice site by l, and assume at most K coordinates have any given site. Arrays carry their maximum norm. Write f(x, v) for the real analytic cost, S(x, v) = Dvf(x, v) for its vertical gradient, and H(x, v) = D2 vf(x, v). The analytic scatter bounds established above say that its coordinate derivatives, through a purchased order N ≥3, are bounded by a constant C > 0 times the exponential connecting weights with width a > 0, on the maximum-norm ball of radius r > 0. 152 ## Source page 155 A Wilson Yang–Mills construction J. Rodgers Assume S(0, 0) = 0 and the counting-measure gap X ij uiH(0, 0)ijuj ≥m X i u2 i , m > 0. (US1) All these inputs were obtained for the actual cost; a bound on the extensive value f(0, 0) is unnec- essary. Take the explicit four-dimensional counting bound c(a) = 16·6!e2aa−6. A distance shell of radius k has at most (2k + 1)4 sites. The exponential-series inequality ea(k+2) ≥[a(k + 2)]6/6! bounds its contribution by c(a)/[(k + 1)(k + 2)]. Summing the telescoping series gives P y e−a dist(z,y) ≤c(a), uniformly in the period. Bounded multiplicity changes this to Kc(a). Put D = Kc(a). The scatter third-derivative bound and integration along a segment give ∥H(z) −H(0)∥∞→∞≤∥z∥∞CD2. (US2) The same row and column sums bound the corresponding quadratic form: apply 2|uiuj| ≤u2 i + u2 j to each entry. Thus the gap remains at least m/2 on the ball of radius RH = min{r/2, m/[2(1 + CD2)]}. (US3) We need an inverse in the maximum norm with a price independent of the number of sites. Suppose a matrix B has entries bounded by Ce−a dist(li,lj) and counting gap M > 0. For a fixed site z, conjugate it by the diagonal weights wi = eb dist(z,li). The triangle inequality and |et −1| ≤|t|e|t| show that, if 0 < b ≤a/4, the difference from B has entries bounded by Cb(4/a)e−(a/2) dist(li,lj). Choose b = min  a/4, M 2{1 + C(4/a)Kc(a/2)}  . (US4) The quadratic-form price of the perturbation is at most M/2; hence the conjugated matrix has counting coercivity M/2. It need not be symmetric. To estimate its inverse column, solve Bu = ej, take z = lj, and put vi = wiui. Coercivity gives (M/2) X i v2 i ≤vj. In particular (M/2)v2 j ≤vj, so vj ≤2/M, and then |vi| ≤2/M for every i. Undoing the weight gives |(B−1)ij| ≤(2/M)e−b dist(li,lj). (US5) Summing a row yields a maximum-norm inverse price (2/M)Kc(b). This argument uses the true counting gap and exponential summability throughout, and introduces no volume-dependent equiv- alence between the maximum and Euclidean norms. Apply this with M = m/2 on (US3), and denote the resulting width by b. Set J = 1 + (4/m)Kc(b), RV = min{RH, [4J(1 + CD2)]−1}, RX = min{RV , RV /[4J(1 + CD)]}. (US6) For ∥x∥< RX, consider the map Yx(v) = v −H(0, 0)−1S(x, v) on ∥v∥≤RV . Its derivative is H(0, 0)−1{H(0, 0) −H(x, v)}. Equations (US2) and (US6) give its norm at most 1/4. Integrating 153 ## Source page 156 A Wilson Yang–Mills construction J. Rodgers the mixed Hessian in the retained variable gives ∥S(x, 0)∥≤CD∥x∥, so ∥Yx(0)∥≤RV /4. Therefore ∥Yx(v)∥≤RV /2 on that closed ball. Iteration is Cauchy with successive differences reduced by 1/4; its limit g(x) is its unique fixed point. Since H(0, 0) is invertible, being a fixed point is exactly S(x, g(x)) = 0. We have g(0) = 0, ∥g(x)∥≤RV /2, S(x, v) = 0, ∥v∥≤RV =⇒v = g(x). (US7) At every point of this graph the vertical Hessian is invertible. The local analytic implicit function theorem gives an analytic branch there; its continuity keeps it inside the same vertical ball nearby, and uniqueness in (US7) identifies it with g. These local identifications prove analyticity of the single graph on the whole retained ball. This also identifies any retained critical germ with this graph wherever both satisfy the corridor bounds. For completeness, its higher response estimates follow directly from the critical equation, rather than from a dimension-dependent implicit-function norm. Let Z(x) = (x, g(x)). Differentiate S ◦Z = 0 k + 1 times. In the ordered-partition chain formula the one-block term containing Dk+1g is H(Z)Dk+1g; every other term contains graph derivatives of order at most k. At a fixed purchased order use the common width b∗= min{weakerN(a), weakerN(b)} > 0, where weakerN(t) = t/[2(N +1)]. Combining the joining-tree weights loses only this width once for the whole induction: the span of any one resulting tree is at most the sum of its constituent spans, and splitting each available exponent between this connecting span and the summation variables leaves the remaining half for the counting sums. At most N +1 vertices are joined at the purchased order. If Pk prices graph derivatives through order k, the other terms have price at most Fk = C + | OrderedPart(k + 1)|[C{1 + PkKc(a/2)}]k+1. Multiply by the inverse kernel (US5) and sum its output coordinate. The recursion P0 = 1, Pk+1 = Pk + Fk(4/m)Kc(b/2) (US8) therefore bounds every coordinate response through order k+1, with its connecting-tree exponential of width b∗. Every constant in (US3)–(US8) depends on the purchased finite orders and local input prices, but not the period. Differentiating f◦Z uses the same ordered partitions; its terms containing a vertical first derivative of f vanish by stationarity. This gives the on-shell action bounds used in the nonlinear integral and the reference estimates. The separate stronger strip estimates for generated precision kernels retain their own widths; (US8) does not replace them. 8.2 The Fourier normalization and the kinetic readout The two-field determinant film has an internal momentum and two opposite external momenta. Its internal average is normalized Haar average, namely stock F = R [−π,π]4 F(k)d4k/(2π)4; its finite- period average is shopn F = n−4 P k F(2πk/n). The curvature picture called profitable F is one half of the internal average with the two external link and color indices in their specified order. The factor one half is the quadratic Taylor factorial. Translation invariance makes the actual Hessian entry zero unless its two Fourier momenta sum to zero, and otherwise that entry is the site count times the finite-period average of the film. To compare the two averages, expand the internal variable in its absolutely convergent Laurent series. Finite averaging retains exactly the coefficients whose four indices are divisible by the 154 ## Source page 157 A Wilson Yang–Mills construction J. Rodgers period; Haar averaging retains only the zero coefficient. Every remaining index has length at least the period. Spending half the common strip for summation and a quarter for the tail gives a fixed strip price times e−σn/4. Multiplication by the site count gives the recorded-Hessian error. Thus this internal integral represents the finite Hessian with a quantitative error and a fixed normalization. Compose that picture with the momentum chart for real initial links. The readout leapTip(P)(e0, 1, (a, a)) is the divided coefficient D2P(1,a),(1,a)(0)[e0, e0]/2!. Reality makes this diagonal coefficient real. Color invariance and simplicity make its color bilinear form a scalar multiple of the fixed invariant form, so its value is independent of the normalized color. The factor two in (DP1) is additional to the factor one half in the internal film average; both are retained in that definition. Here is why this one transverse entry determines the leading kinetic tensor. Subtract its multiple of the normalized Wilson tensor, whose readout is one. Gauge invariance gives transversality. Signed covariance includes the phases from reversing initial links; at the first nonzero homogeneous order those phases have constant term one. The constant and linear homogeneous terms vanish by transversality and permutation covariance. Let Quv(p, q) be the symmetric polarized degree-two term of the difference. Reflecting one axis shows that a component can survive only when its momentum occurrences and link occurrences have the same parity on every axis. Permuting axes moves the zero transverse diagonal readout to all transverse diagonal entries on single momentum axes. Longitudinal entries vanish by transversality. Reflection parity then kills mixed momentum entries in a diagonal link component and same momentum entries in an off-diagonal link component. The only remaining possibility has its two distinct momentum axes equal to its two link axes. Transversality at the sum of those two coordinate vectors makes twice that entry zero, since the diagonal entries have already vanished. Bilinearity gives Q = 0. The degree-three term vanishes under total inversion, whose momentum sign is odd and whose two link signs multiply to one. The difference thus starts at degree four. This proves the one-axis kinetic identification used below, including the Taylor normalization and reflected-edge convention. 8.3 Analytic exterior division with the origin jets retained Division retains the harmonic obstruction Exterior division first solves a contraction equa- tion on the regular momentum set. Analytic extension requires the stated vanishing along the relevant harmonic coordinate planes, not merely a value zero at the single full origin. The division formula below keeps the origin jets and the constraints from the other momentum blocks. These data are needed by the two-leg correction that follows; deleting a harmonic component instead would change the supplied stationary identity. We give the analytic division used in issuing the loop coefficients. For one momentum coordinate set qi(p) = eipi −1. If x is a Laurent index, define Li,x(p) = (Pxi−1 a=0 eip·(x[i:=a]), xi ≥0, −P−xi−1 a=0 eip·(x[i:=−a−1]), xi < 0. The empty sum is zero. Multiplication and telescoping give the entire identity qiLi,x = eip·x −eip[i:=0]·x. (ED1) If F is periodic analytic and bounded by A on a strip of width σ, its Fourier coefficient bound on an intermediate strip makes G(p) = P x bF(x)Li,x(p) absolutely and locally uniformly convergent on width σ/4. Indeed the finite sum has at most |x| terms and size at most |x|e(σ/4)|x|; the spare 155 ## Source page 158 A Wilson Yang–Mills construction J. Rodgers width absorbs |x|. Summing the geometric coefficient bounds in each momentum coordinate gives the explicit price ∥G∥σ/4 ≤A 4 σ  21 + σ/4 σ/4 v , (ED2) where v is the number of momentum coordinates. Thus qiG = F −F|pi=0, including at every zero of qi; no singular quotient is defined on a mesh. If the numerator has zero jets below order s + 1, comparison of its analytic Taylor series with qi = ipi + O(p2 i ) gives zero jets of G below order s. Equivalently evaluate on nearby directions with nonzero i-component, divide their one-variable series, and extend the resulting polynomial-jet identity to all directions by continuity. For exterior coefficients let C be contraction by the vector q, and Ii exterior multiplication by the i-th coordinate vector with its orientation sign. The coordinate anticommutation rule gives CIi + IiC = qi id. Apply (ED1) componentwise to a cycle u, and call the divided difference Diu. Substituting in this anticommutation rule shows that CIiDiu = u −combi u, where combi u evaluates at pi = 0 and deletes every exterior component containing i. The identity is first algebraic where qi ̸= 0, and extends to the zeros by analyticity. The residual is again a cycle. Indeed, after separating components containing i, its contraction at momentum p equals its contraction with the i-th symbol set to zero, since every i-containing component has been removed; the original cycle equation at pi = 0 supplies the remaining equations. Evaluation and deletion preserve the original strip, coefficient bound and Taylor vanishing order. Repeat on the residuals in the four axis directions, adding the four lifted differences. Every residual still has the original width σ, price A, and order s. Each new lift is therefore computed directly at width σ/4, price at most A times (ED2), and order at least s −1. There is no iteration of four strip shrinkages or four losses of Taylor order. Summing gives width σ/4, price four times (ED2), and order s −1. The contraction of this sum telescopes to the original cycle minus the final residual. In positive exterior degree the latter is zero because every orientation contains a processed axis. In degree zero it is the original coefficient with all coordinates of this block set to zero; the required hypothesis is its vanishing on that entire coordinate plane. Positive order at the full origin implies this hypothesis only when this is the sole momentum block. This is why the exact harmonic-set check in (TEL2) is needed. Suppose additionally that each coefficient vector lies in a prescribed linear subspace depending only on the other momentum blocks. Evaluation in a divided coordinate preserves that subspace. The divided Fourier sum also does so: where qi ̸= 0 it is the scalar multiple (u −u|pi=0)/qi, and at its zeros continuity and closedness of the finite-dimensional subspace give the same conclusion. Exterior multiplication is a signed selection of these coefficient vectors. Hence every residual and lift retains the other block constraints. For a dependent momentum equal to minus the sum of the independent ones, the linear shear exchanging that block with the dependent block is involutive. If there are t + 1 independent blocks, its missing coordinate is the negative sum of their coordinates. A strip of width σ/(t+2) is mapped inside width σ, since (t+1)σ/(t+2) < σ. Apply the preceding coordinate construction on that narrowed strip and shear back with the same narrowing factor. This gives the dependent-block width and price used in (TEL3). 8.4 The two-leg correction with both momentum blocks retained Here is the precise two-leg use of the exterior division. There is one independent four-momentum p; the other is −p. Write Er = ΛrC4, with its ordered plane basis, and let Cr −: Er+1 →Er be 156 ## Source page 159 A Wilson Yang–Mills construction J. Rodgers contraction by the lattice symbol at −p. Thus Cr −Cr+1 − = 0. A coefficient with one remaining potential index belongs to Er ⊗E1; one with no such index belongs to Er. These are coefficient- valued periodic analytic functions of the same complexified momentum. Let H : Er ⊗E1 →Er contract the second index by the symbol at p. Since these maps act on different tensor slots, HCr −= Cr −H. (TEL1) There is no exterior sign in this convention. A coefficient has order a if all its real Taylor coefficients of total degree below a at zero vanish. Each contraction increases this order by one. The given two-potential tensor is u ∈E1 ⊗E1, of order s+2, satisfying both gauge contractions C0 −u = 0 and Hu = 0. Apply the dependent-block division to obtain X ∈E2 ⊗E1, C1 −X = u, ord X ≥s + 1. The coefficient HX ∈E2 has order at least s+2, and is a dependent cycle because C1 −HX = Hu = 0. A second dependent-block division gives Y ∈E3, C2 −Y = HX, ord Y ≥s + 1. Next lift the independent contraction: choose D ∈E3 ⊗E1 with HD = Y and order at least s. The degree-zero obstruction for this last division must be checked on its actual harmonic set. There is just one independent momentum block, so that set is exactly p = 0, rather than an unspecified collision locus. Since s + 1 ≥1, every coefficient of Y vanishes there. This proves the required obstruction is zero. In the general many-leg construction the other independent contractions de- fine coefficient subspaces that remain constant when this block varies; the coordinate homotopy preserves them. For two legs there are no other independent indices, so this preservation condition is vacuous. Now put V = X −C2 −D ∈E2 ⊗E1. (TEL2) It still has order at least s + 1. Its represented tensor is unchanged, since C1 −V = u −C1 −C2 −D = u. Its remaining contraction is removed, since HV = HX −C2 −HD = HX −C2 −Y = 0. Apply the independent-block division separately to the six E2 coefficients of V . This gives g ∈E2 ⊗E2, of order at least s, whose two curl contractions equal u. All identities hold at zero momentum as analytic identities, as well as at nonzero real momentum. For clarity the width and norm losses can also be followed without any existence convention. Put slimt(a) = a/(t + 2), and, for a coordinate set of size m, Tm(a) = 16 a 2(1 + a/4) a/4 m , P(a) = T4(a/2), E(a) = a/16, Br(a) = 4 r + 1 ! 4(ea + 1). Here Tm is four times the coordinate-division price in (ED2); Br bounds the finite sum defining a curl contraction on the strip. If u has strip width σ > 0 and coefficient bound A ≥0, set τ = E(σ), w = E(τ), μ = w/16, b = P(σ), c = bB0(τ), d = cP(τ), e = dT4(w). The bounds for X, HX, Y, D are respectively Ab, Ac, Ad, Ae; their widths are τ, τ, w, w/4. Restrict- ing to μ, the bound for V is A{b + eB2(w/8)}. (TEL3) For the final independent division there is exactly one remaining leg. Apply the coordinate-wall lift (ED1)–(ED2) with the four independent coordinates to each E2 component of V . The other 157 ## Source page 160 A Wilson Yang–Mills construction J. Rodgers coefficient indices are passive, so taking the maximum of their coefficient bounds introduces no extra factor six. The construction gives explicitly η = μ/4, ∥g∥η ≤A{b + eB2(w/8)}T4(μ), ord g ≥s. (TEL4) To see the contraction identity componentwise, denote the lift of the independent index by Wa(b, p), where a is the dependent two-form index and b the independent two-form index. Then P b symbol1(p)b,iWa(b, p) = Va,i(p). Contracting this equality with symbol1(−p)a,h and summing over a gives uh,i by (TEL2). This is precisely the two-curl representation, with no omitted polarization or identification of the two momentum slots. More generally repeated independent divisions obey the recursion η0(σ) = σ, K0 = 1, ηt+1(σ) = ηt(σ/4), Kt+1(σ) = Tm(σ)Kt(σ/4), where m is the ambient coordinate count. Each step preserves the contractions in the other independent blocks because their symbols do not vary with the divided block. For this application only t = 1, m = 4 is needed. These prices depend on the input strip, not on the torus period or on s. The tensor spaces, the harmonic-set check and the two identities in (TEL2) give the two-leg construction used in both loop-moment applications below. 8.5 Production of the fresh loop moments The moment normalization of the fresh determinant loop is obtained before its path representation is issued. Let f be the common analytic two-field picture of the determinant and Haar term, R the normalized coarse metric row with R(0) = 1, and g the inherited-loop picture. The latter vanishes to order four in real external momentum. Set α equal to the kinetic readout of f and subtract α times the normalized metric picture. The one-axis identification proved above shows that this difference, and therefore h = f −α slots(R) + g, vanishes to order four. It retains its two gauge-contraction identities, reality and color invariance. Its analytic norm is bounded by A+|α| bedCharge(σ, M)+B, where A, B bound f, g and M bounds the metric row. Thus this normalization does not discard the inherited loop price. Apply (TEL1)–(TEL3) with s = 2: the input has zero jets below order four, and its two- curvature coefficient has zero jets below order two. In particular the second-contraction correction uses the harmonic set p = 0 checked there; the coefficient norm and positive strip width have the period-independent prices in (TEL3). Write the resulting coefficients, for fixed plaquette and color types, as a real Laurent family P(z). Taking real parts replaces an analytic coefficient on real momenta by the average of it and its conjugate at opposite momentum. This preserves the represented real tensor and its vanishing order. Project its finite color coefficient vector onto invariant tensors; this is a fixed real linear projection, of finite absolute matrix-sum price, which leaves the already invariant represented tensor unchanged. It commutes with reading the absolutely convergent Laurent series and with its derivatives. Thus bP(p) = X z∈Z4 P(z)eip·z, bP(0) = 0, D bP(0) = 0, X z |P(z)|eη|z| < ∞ (LM0) for a fixed positive output width η. Exponential summability also gives P |ziP(z)| < ∞, by spending part of η. The series may therefore be differentiated on the real momentum axes. Its value and first derivative at zero are P P(z) and i P ziP(z). Equation (LM0) proves both sums zero, separately for each plaquette and color choice. The actual path issue enumerates these types and offsets. Its first leg is based at zero, its second at z, and each leg includes the four plaquette edges. Its length is |z| + 8, so the 158 ## Source page 161 A Wilson Yang–Mills construction J. Rodgers path price differs from the Laurent price only by the fixed factor e8η and the finite type and color counts. The displacement in the moment is exactly z. Reindexing by this enumeration proves the summable zeroth and first unwrapped displacement moments of the issued atoms. These are the fresh-loop identities needed in the next subsection and in (TF17)–(TF19). 8.6 Production and transport of the inherited loop moments The full-family identities (TF17)–(TF19) require absolute summation before truncation. We give the construction that transports these identities across a block. For a two-curvature atom a, let qa be its coefficient with its plaquette and color types fixed, and let δa ∈Z4 be the displacement from its first leg to its second. Its length bounds |δa|. A charged exponential bound implies X a |qa| < ∞, X a |qaδi a| < ∞, (LM1) because s ≤Cηeηs. Thus all changes of summation below are absolute, including those with a displacement factor. For one atom translate its first leg to the origin in the finite torus sum. Along a fixed route from zero to δa, telescope the second leg. Subtract its value and its first differences at the origin. The remaining discrete second-order Taylor difference can be written as a finite sum of products of differences, one on each curvature factor, by summation by parts along successive route edges. Explicitly the removed terms are qa X x Fa(x)Ga(x), qaδi a X x Fa(x)∇iGa(x). (LM2) The discrete Taylor identity is exact on the torus: each link telescope is a translated finite difference and the sum of a translated product can be shifted by the same lattice translation. Repeating this along the two route indices generates at most the square of the route length in remainder terms. The exponential budget absorbs this square after any fixed positive width discount. If the full family has zero zeroth and first displacement moments, summing (LM2) gives zero separately for each type and color. Its total action therefore equals the sum of the two-difference remainder patterns. This proves the representation used to inherit a quiet loop; the subtraction is performed on the whole summable family, not on an arbitrary finite prefix. Let ψi(r, b; z) be the installed fine-gradient column for a coarse residue r, curvature type b, direction i, and unwrapped displacement z. These columns are the product of the fine frame inverse, the differentiated linear tower, and the coarse frame. Whenever the stationary-column construction uses two curvature legs, its dependent-momentum join is precisely (TEL1)–(TEL3), with the retained Taylor orders stated there. On harmonic input the unmodified tower has residue- independent total columns. Install it on a one-site coarse torus. Its output is then constant on the fine torus; a fine difference of that output is zero. The intertwining identity between the differentiated column and the fine difference consequently gives X z ψi(r, b; z) = 0. (LM3) The frame factors preserve the constant-output statement by their translation convolution identities. Odd blocking is the hypothesis that makes this centered-gradient readout agree with the installed fine difference. Exponential column bounds justify passing from the installed periodic identity to the absolutely summed cover columns. After substituting these columns in a two-difference pattern, one fixed choice of its two out- put types has coefficient uf0(z0)f1(z1), where P f0 = P f1 = 0 by (LM3). Its zeroth total is 159 ## Source page 162 A Wilson Yang–Mills construction J. Rodgers u(P f0)(P f1) = 0. Its displacement is (c1 + z1) −(c0 + z0), with fixed carried offsets c0, c1. Split- ting its first moment gives u ( ( X f0) X z (c1 + z)if1(z) − X z (c0 + z)if0(z)( X f1) ) = 0. (LM4) The absolute first moments exist by the column width bounds. Summing over residues, types and patterns is justified by (LM1) and the charged bound on the full substituted family. Thus inherited loops have the exact two moment identities, with their block price L−2, rather than merely small moments. Joining two families means enumerating their disjoint union. Absolute sums add, so both mo- ment identities are preserved. Blank initial loops have zero coefficients and satisfy them. The coefficientwise limiting argument (SH1)–(SH3) preserves them as well: first pay the common ex- ponentially weighted tail, then pass to the limit on a finite list. This supplies the full-family justification used to turn retained moments into omitted-tail moments in (TF18). Any fresh loop must satisfy the same identities before it is joined; they cannot be inferred from gauge invariance or the quadratic-potential normalization alone. 8.7 The one-loop drift used by the finite histories The loop calculation evaluates the drift of the very finite histories used later in shooting. The fresh and inherited moments are kept separate until their transport identities are combined. The coefficient s = 11CA/(24π2) refers to the physical Lie-algebra and Fourier normalizations fixed earlier, and the error allowance is bought before the final cutoff. The subsequent calendar turns these local drift estimates into uniform running for the actual represented couplings. The finite coupling drift comes from the committed path coefficient and the bounded scalar reset, not directly from a formal beta function. At a transition with block length L, put μ = L−1. The determinant calculation on the retained physical path defines a coefficient αN,i with sign appropriate to descent from level i + 1 to i. We explain the logarithm and its uniform remainder. For a fixed color direction, the continuum pole of the relevant second external-momentum coefficient is pG(k) = 4CA k2 1 |k|6 −4k2 1k2 0 |k|8 − 1 |k|4 ! . (DR1) The coefficient includes the vector determinant with weight 1/2 and the gauge determinant with weight −1. These are the same two determinant weights kept in the massive-symbol calculation of the trace. The finite symbol coefficient fμ(k), on the angular cube [−π, π]4, satisfies |fμ(k)| ≤ A (|k| + μ)4 , |fμ(k) −pG(k)| ≤D |k|3 + Dμ2 |k|6 (|k| ≥μ), (DR2) where A, D ≥0 depend on the fixed analytic widths, coefficient and ellipticity bounds, and not on L, the torus period or its terminal cutoff. The input paths have their fixed quadratic normalization and the allowed higher soft jets; the latter preserve the continuum pole. Here is the matrix estimate yielding (DR2). At scale T = |k| + μ, the zero-field massive matrix, one-field vertex, and two-field vertex have sizes T 2, T, 1. Coercivity bounds the inverse by CT −2. Differentiate the identity AA−1 = 1 to obtain D(A−1) = −A−1(DA)A−1 and its second derivative, the sum of the two products with two first vertices minus the product with one second vertex. Along a momentum direction scaled by T, all jets of each inverse still have size T −2. The lower determinant bubble therefore has size T −2, and its second unscaled external-momentum coefficient 160 ## Source page 163 A Wilson Yang–Mills construction J. Rodgers has size T −4. The upper insertion obeys the same rough bound. Summing the finitely many matrix entries gives the first bound in (DR2). For |k| ≥μ, take T = |k|. The zero-field symbol differs from its continuum quadratic part by O(T 3 + μ2), and the first vertex differs from its continuum linear part by O(T 2 + μ2/T). Use A−1−B−1 = A−1(B−A)B−1 and telescope the factors of the determinant bubbles. Their relative er- ror has size T +μ2/T 2; their second momentum coefficient consequently has error C(T +μ2/T 2)T −4, which is the second bound in (DR2). The upper insertion contributes bounds of these same or smaller orders. The origin-jet cancellations, analytic derivative bounds, inverse formulas and ex- plicit continuum vertex contraction are developed in the weighted massive-symbol and logarithmic momentum calculations below. This argument uses them before integration, so its error bound is absolute. To integrate, split the angular cube into |k| ≤μ, μ < |k| ≤1, and its part outside the unit ball. The first bound in (DR2) gives Z |k|≤μ fμ(k) dk ≤Aμ−4 π2μ4 2 = π2A 2 . Outside the unit ball it gives a bound A times the volume of the angular cube. On the annulus the second bound gives Z μ<|k|≤1 D |k|3 dk = 2π2D(1 −μ) ≤2π2D, Z μ<|k|≤1 Dμ2 |k|6 dk = π2D(1 −μ2) ≤π2D. (DR3) All these errors are bounded independently of 0 < μ ≤1. The pole integral is exact. Rotational symmetry gives, on this annulus, Z k2 1 |k|6 dk = 1 4Iμ, Z k2 1k2 0 |k|8 dk = 1 24Iμ, Iμ = Z |k|−4 dk = 2π2 log(1/μ). (DR4) For the mixed fourth moment, symmetry makes all off-diagonal moments equal. Expanding the fourth powers of k0 + k1 and k0 −k1, and rotating these directions to a coordinate direction, shows that the diagonal fourth moment is three times the mixed one. Summing over the four coordinates then gives the factor 1/24. Substitution of (DR4) into (DR1), with Fourier weight (2π)−4, gives (2π)−4 Z μ<|k|≤1 pG(k) dk = −11CA 24π2 log(1/μ) = −s log L. (DR5) Equations (DR2)–(DR5) bound the difference between the infinite-volume coefficient and this log- arithm by a pre-block constant. We now connect this integral to the coefficient of the actual stationary path. It is important to distinguish three determinant films: Mact, the film of the actual stationary integration; Mcore, its core film before the stationary carrier change; and Mrank, the auxiliary rank film. Each film is a matrix of analytic functions of the external momentum, with indices consisting of a direction and a color. Their scalar reading is αa(M) = 2 Re leapTip(profitable M)(e0, 1, (a, a)). (DP1) The operation leapTip is the coefficient of degree two in that external-momentum direction, with its Taylor factorial; profitable passes from the determinant film to its curvature tensor using its 161 ## Source page 164 A Wilson Yang–Mills construction J. Rodgers analytic contraction identities. Thus αa is linear on films satisfying the common bounds, and is the specific normalized kinetic coefficient, not the full film norm. Here are the quantitative comparisons used in the transition. Fix an analytic width h > 0, with h ≤σ ≤1/128, a block patch of stage J, its penalty k, a positive stationary width, and all the common metric, covariance and path bounds. The required hypotheses are: the input path is attached to its quadratic metric and has its prescribed analytic bounds; that metric has the uniform floor on every torus; the coarse metric has its analytic strip bound; the stationary graph is critical, guessed by the installed tower, and satisfies the gap and heat identities; the same forest and tag are used in its determinant realization; and the relevant torus sides form an auxiliary sequence tending to infinity. These are precisely the linear-horizon and stationary-graph data already constructed. The input loop-tensor norm may have its own bound Λ, whereas the comparison constants below depend only on the fixed analytic and matrix bounds specified before the block length. They do not grow with Λ. Let F ≥0 bound the inverse matrices in the determinant calculation, let M∗≥0 bound the common zero-, one- and two-field film factors, and let K∗≥0 bound their carrier difference. All three are obtained from the fixed stage, kernel, penalty and common analytic bounds before L is chosen. More explicitly the common factor bound is the sum of the absolute bounds for the leg, core, rank and frozen metric factors; the carrier bound is the common bound for the difference between the core and rank factors. Put D∗= FK∗F, E∗ = D∗M∗+ FK∗, R∗= E∗+ (E∗F + FM∗D∗)M∗+ FM∗FK∗. (DP2) This is the source’s exact reserve polynomial. It arises by subtracting inverse factors with A−1 − B−1 = A−1(B −A)B−1, then telescoping the zero-, one- and two-field determinant products. For example the first inverse difference costs D∗; multiplying by the common first factor and adding the factor difference costs E∗; the remaining two placements and their cross product give the last line. The trace estimate is a relative rank estimate, so this subtraction introduces no count of all L4 block sites. In detail, if r = | Loc0 | + | Loc1 |, the individual films are bounded on width h/2 by L4 2r(FM∗+ FM∗FM∗). (DP3) The separate difference estimate is Mcore −Mmrank bounded on width h/2 by R∗. (DP4) Here is why (DP4) has no block-volume factor. The matrix weights are invertible analytic frames SX(p), SY (q). A bound for a matrix U means a Euclidean operator-norm bound for eU = SX(p)−1USY (q). For the difference estimates the stronger bound is used: | tr( eUV )| ≤C∥V ∥2→2 for every compatible V. (DP10) It is subadditive, is invariant under transposition with dual frames, and multiplies with an ordinary operator bound on either side. At equal source and target frames, taking V = 1 bounds the actual trace by C. A general matrix with operator norm C only has price dim(X)C in (DP10); using that conversion would incur the unwanted L4. The exact difference between the core and rank quadratic films cancels their common sector and pure Gram terms. Write S for the shear row, T for its localized emitter correction, and B for the fixed block inclusion. Expanding (S +T)∗(S +T)−S∗S = T ∗(S +T)+S∗T gives its reflection part. The complete remaining difference is the sum of the local house film and this reflection part, minus 162 ## Source page 165 A Wilson Yang–Mills construction J. Rodgers B∗OB and the finite motive film; O is the localized orbit correction. This identity holds separately for the zero-, one- and two-field matrices, with all their Leibniz placements. The common sector cancels before any trace is bounded. We derive the uniform trace-ideal estimates before using them in the difference. For a finite matrix U, set ∥U∥tr,∗= sup ∥V ∥2→2≤1 | tr(UV )|. For U = uv∗, the trace is v∗V u, so its size is at most ∥u∥∥v∥. Decomposition into columns bounds the size by the sum of their norms. A matrix with a fixed n0-dimensional source consequently has price n0∥U∥2→2. Multiplication on either side by a bounded matrix multiplies this price by its operator norm, by trace cyclicity. Apply these statements after the analytic changes of frame in (DP10). One further factor has a source dimension growing with L, and needs an actual summability argument. Put μ = L−1, let ω be the lattice Laplacian symbol, and define dμ(k) = μ2 ω(k) + μ2 , J = 16 ∞ X n=0 (n + 1)−2 < ∞. On the strip of width 1/128, choose the centered real-momentum representative before enumerating aliases. The massive lower bound gives |dμ(kl)| ≤ 2 (1 + pdist(l))2 . At l = 0 this is the origin massive bound. At l ̸= 0, centered alias separation bounds the symbol scale below by (1 + pdist(l))μ; strip ellipticity bounds the denominator below by half the square of that scale. Division gives the inequality. Changing the centered representative permutes the aliases. There are at most (2n + 1)4 aliases at distance n, because their lifts inject into [−n, n]4. Therefore X l |dμ(kl)3| ≤8 X n≥0 (2n + 1)4 (n + 1)6 ≤8J. The alias-diagonal matrix d3 μI has trace-ideal price 8n0J, with fixed n0 = | Loc0 |. The Fourier frame and inverse frame each cost at most e4/128, leaving the block-independent price Q = e4/128(8n0J)e4/128. This origin estimate controls the zero-, one- and two-field matrices of the regularizing ninth power. Write P = μ2(A+μ2I)−1, where A is the sector Gram matrix, and U = P 3. The preceding estimate applies to U at zero field. If C3 bounds its three ordinary film matrices, the corresponding three matrices of U3 have trace-ideal price at most 20C2 3Q. In fact every zero-field product has an unmarked U, every first-field product has two, and every second-field product has at least one. Put one unmarked factor in the trace ideal and the other two in operator norm. The second-field Leibniz formula has three placements of a second-field matrix and six ordered placements of two first-field matrices. Their sum and the lower orders are covered by 20. This never converts an L4-dimensional bound into a trace bound. The regularizer is μ2P 9, so its price is μ2B, where B = 20C2 3Q. We now price each localized composition. We specify how the derivative prices used here are obtained. If an analytic finite-dimensional map F is evaluated at z, define, for a derivative bound c ≥0, Em(F, z, c) = m X n=0 X π∈Pn ∥D|π|F(z)∥c|π|, (DP12) 163 ## Source page 166 A Wilson Yang–Mills construction J. Rodgers where Pn is the finite set of partitions in the iterated chain rule, with its multiplicities, and P0 contributes the value bound. If an inner map takes the value z at the origin and its derivatives through order m have the prescribed tested bound c, its composite is bounded by (DP12): differen- tiate successively, grouping the directions according to the block that enters each derivative of the inner map, and sum their norm products. The source’s apollo and encore are precisely these finite sums. For a product use F(u, v) = uv and inner price c1 + c2; inversion of a group matrix is its adjoint and preserves these derivative bounds. The same construction applies to the matrix expo- nential at zero, the chosen analytic group logarithm at the identity, and the analytic nearest-group projection in its fixed neighborhood. These rules generate the charter and emission prices from the actual stencil. A link perturbation has its explicit factor L−1. On a length-L path, the product rule chooses the differentiated links. If q distinct links are differentiated, there are at most Lq choices and their derivative bounds contribute L−q; all undifferentiated group factors have norm one. The remaining assignments are the finite partitions of the derivative directions. Thus every fixed derivative order has a common bound independent of L. Patch averages are probability averages. Moving from the padded patch to its translated subpatch costs at most its fixed stage-dependent cardinality ratio, since both counts are bounded multiples of L4. The frame estimates already proved give the corresponding finite derivative bounds for the recentered gauge transformations. Form the connector by multiplying the two endpoint frames and the path transport, average it, and apply the nearest-group projection; (DP12) pays all these operations. Multiply by the coarse-link factors and the endpoint frames, and apply the logarithm. This constructs the finite positive prices denoted springs for the charter and rocketf are for the emission in FrameEst/Stencils. Every logarithm is evaluated at the identity in this centered calculation. None of these steps takes a supremum over a number of arbitrary independent site values. Here is the last conversion into row norms, including its block powers. Let n ≤2 be the number of inserted field directions and let c bound their fixed Fourier-frame readings. Put c′ = muralRoom(J)c, including the fixed stage enlargement of those readings. The scalar directional estimate for the charter is at most 2n+1| springs(J, n + 1, c′)|; the corresponding emission estimate, after changing from the patch probability average to the block-site probability average, is at most portrait(J)2n+1| rocket_fare(J, n + 1, c′)|. The factor 2n+1 pays the real-and-imaginary expansions of the n inserted directions and the one tested input direction. Conversion of the finitely many color coordinates to their Euclidean norm multiplies these by √n1 and √n0, respectively. For the emission the probability-average estimate is a sum of squares divided by L4; removing that denominator and taking a square root contributes exactly L2. Taking the maximum, or the larger finite sum, over n = 0, 1, 2 gives prices a and h independent of L, with charter price a and emission price L2h. These are RealGap.CharterR.costs and Chromatic.costs. The same finite derivative and Fourier-frame bounds give the sector price p; the positive massive inverse and its differentiated inverse identities give the regularizer price μ2g. The house kernel has fixed dimension and the already constructed inverse bound, giving its ordinary price b. Each of a, h, p, g, b thus bounds the zero-, one- and two-field matrices on the same strip. The house kernel has fixed source dimension n1 = | Loc1 |, so its trace-ideal price is n1b. Film multi- plication costs at most four times the product, covering its four second-field Leibniz terms. The 164 ## Source page 167 A Wilson Yang–Mills construction J. Rodgers house price is consequently CH = 4(4an1b)a. Composition of the regularizer with the site-to-color row cancels μ2L2 = 1. Its trace-ideal price is r = 4Bh and its ordinary price 4gh. The shear row has ordinary price 4p, so the corrected row has price t = 4p + 4gh. The two reflection products have combined trace-ideal price CR = 4rt + 4(4p)r. For the orbit term let si bound the fixed stage row in direction i, and put S(z) = P3 i=0 4(4siz)si. Its unweighted trace price is μ2S(B). The two changes to the dual massive frames cost at most e4μ−1 each, yielding CO = e4S(B)e4. The motive film factors through the fixed n0-dimensional space, with inclusions of ordinary price one; its trace-ideal price is 4n0. All widths and derivative orders have been fixed before L, and each cancellation of a block power has been displayed. These four estimates supply (DP10). Composing a difference factor with an ordinary bounded factor keeps (DP10). In particular the house film has price CH; the two reflection products have combined price CR; B∗OB has price 16CO, including its two film-composition factors; and the motive film has price 4| Loc0 |. Thus one may use exactly K∗= CH + CR + 16CO + 4| Loc0 |. (DP11) Each price here is the absolute weighted column sum of its stated local factor with its zero-, one- and two-field placements, not a norm of a full trace. The analytic row and localized column estimates bound those sums by the fixed stage and strip prices. They use the same σ ≤1/128, branch bounds and common stationary rows as (DP3), and are uniform in L. The inverse identity gives difference price FK∗F = D∗. For a two-field determinant frame, the direct term difference costs D∗M∗+ FK∗= E∗. Telescoping its three-factor bubble gives (E∗F + FM∗D∗)M∗+ FM∗FK∗. Taking its trace through (DP10) gives precisely (DP2) and (DP4). The factor 1/2 in the core- minus-rank determinant may be discarded when increasing this bound. This proves the relative projection estimate while retaining the L4 factor in the unrelated individual estimates (DP3). The stationary carrier change has zero kinetic reading: αa(Mact −Mcore) = 0. (DP5) To see this, write its diagonal film as 1 2(W(p) + W(−p) −V (p)/2). The stationary tax row has zero jets below order two. Its mixed carrier row has zero jets below order one, by the stationary graph identity. Hence each summand of W, a product of these two rows, vanishes to order at least three. Every summand of V contains two tax rows and vanishes to order at least four. These films are analytic on the common strip, so their external degree-two coefficient is zero. Linearity proves (DP5). For an analytic scalar function bounded by A on width r0, two successive Cauchy derivative estimates give ∥D2f(0)∥≤ A r0/2 1 r0/4. For the normalized reading (DP1), the source keeps the slightly weaker explicit price B(r0, A) = A r0/4 1 r0/8∥axes(e0)∥2. (DP6) 165 ## Source page 168 A Wilson Yang–Mills construction J. Rodgers The factor two in (DP1) cancels the Taylor factorial. Apply this estimate to (DP4), with r0 = h/2, and use (DP5). The exact projection bound is |αa(Mact) −αa(Mmrank)| ≤B(h/2, R∗). (DP7) No inverse power of the lattice period and no L4 multiplier occurs in (DP7). There is one more comparison, which is an equality rather than a bounded error. Let Πμ be the analytic film of the massive pole test on the fine lattice, with μ = 1/L. Its finite-period analytic tensors approximate that test with error cents(n)en, where en →0, and the rank film approximates the same test pulled back through the stationary uniform map with error cents(m)ym, where ym = Ce−cm for fixed L. The constants and analytic strip in these approximation statements may depend on L. They are used for identification only, not as the uniform bound in (DP7). Fourier evaluation of the stationary pullback gives the exact alias formula H(p) = L4 X l S (p + 2πl)/L [nl −(p), nl +(p)]. (DP8) Here S = Πμ, and nl ± are the incoming and outgoing stationary response columns. The analytic contraction identities of S remove longitudinal columns. For l̸= 0, the remaining response columns each vanish to order two at p = 0; their product therefore has no degree-two coefficient. For l= 0, the primal and second-order stationary identities identify its contribution, through degree two, with μ2S(μp). The discrepancy vanishes to order three. Taking two momentum derivatives introduces another μ2, so the coefficient multiplier is L4μ2μ2 = 1. Consequently αa(Mmrank) = αa(Πμ). (DP9) The finite-period approximation just described proves that the analytic pullback in (DP8) is the rank film: let the auxiliary coarse and fine sides tend to infinity at fixed L, keeping the same stationary graph and tags. The vanishing approximation errors identify their Fourier tensors and derivatives. The reality, gauge and signed symmetries of the actual pole test give its contraction identities and zero lower jets. No assumption about a momentum continuation between different lattice meshes is used. Combining (DP7), (DP9), and (DR2)–(DR5) now gives the exact allowance used in the transi- tion: d = |B(h/2, R∗)| + Dann + 1, |αN,i + s log Li| + 1 ≤d, (DR6) where Dann ≥0 is the bound for the massive-symbol integral remainder. The stage construction fixes h, F, M∗, K∗, Dann, and hence d, before the block length. Afterwards it imposes one lower bound on odd L, and post-block bounds on the analytic norm and loop allowance of the output path. In the transition notation these latter bounds have the forms Θ ≥β∗(L), c′ ≤γ∗, and Λ′ ≥Λ0(L), Λ′ ≥Λ/2. They may depend on L; they do not enter d. The stationary tensor is produced using arbitrary growing torus sequences satisfying the stated floor and stationary identities; after coefficient identification it is one cover path and works on every realized torus satisfying the construction’s patch and support margins. Thus the auxiliary large-volume limit imposes no extra period-dependent error on (DR6), and the estimate is uniform over the terminal cutoffs whose paths have those same pre-block bounds. The scalar reset has |∆| ≤1. Since its actual coarse coupling is βN,p,i = βN,p,i+1 + αN,i + ∆, we obtain |βN,p,i+1 −βN,p,i −s log Li| = |αN,i + ∆+ s log Li| ≤|αN,i + s log Li| + 1 ≤di+1. (DR7) 166 ## Source page 169 A Wilson Yang–Mills construction J. Rodgers Thus the one-step drift used in the shooting and running arguments includes both the calculated path coefficient and the actual nonlinear reset. The choices of large block lengths and late coupling entry below pay this drift simultaneously with all the local activity and insertion bounds. 8.8 Massive auxiliary symbols and the uniform alias trace bound The massive comparison used in the preceding determinant argument is fixed as follows. With the lattice exterior symbol qa(p) = eipa −1, put s(p) = X a qa(−p)qa(p), dμ(p) = μ2 s(p) + μ2 , cμ(p) = μ2dμ(p)9, μ = L−1. (MA1) On real momentum s(p) is the nonnegative squared curl frequency. The scalar massive covariant- site operator at a group field is D∗ UDU + μ2; its normalized inverse is μ2(D∗ UDU + μ2)−1. The auxiliary regulator is μ2 times its ninth power, and the comparison site operator is D∗ UDU plus that regulator. These are ordinary finite positive matrices when μ > 0. Their symbols at the identity are respectively s + μ2, dμ, cμ, s + cμ. The ninth power is part of the construction, not a variable exponent sent to infinity. For link fields, copy the site regulator into each of the four link directions at the link’s initial site. Add it to the physical quadratic Hessian and to DUD∗ U. This defines the positive auxiliary link matrix. The associated scalar functional is one half of its logarithmic determinant minus the logarithmic determinant of the site comparison operator. The copying operation respects gauges and unsigned permutations of axes. Reversed links move the initial site, so this auxiliary functional is not assigned a signed symmetry by that copying argument. The physical signed laws are established separately. The symbol at the identity is the physical transverse quadratic symbol plus the longitudinal symbol qq(−p)T plus cμ(p) times the identity. On real momentum the gap estimate bounds its quadratic form below by min(ε, 1) 210 (s(p) + μ2) ∥v∥2 2, (MA2) where ε > 0 is the physical transverse floor. The scalar companion has the same bound without the transverse factor. Indeed the physical and longitudinal pieces bound their respective curl and gradient components; in the small-frequency region s ≤μ2, dμ ≥1/2, so cμ ≥μ2/29; in the other region the frequency term already pays a fixed multiple of s+μ2. This proves the required positive floor including zero momentum. To see the uniform trace summation, aliases are pl = L−1p + 2πL−1el, l ∈(Z/LZ)4, (MA3) using the fixed centered integer lift el. Write r(l) = maxa |ela|. There are at most (2n + 1)4 indices with r(l) = n, by injection into the integer box. Consequently, uniformly in L, X l (1 + r(l))−6 ≤J0, J0 = 16 X n≥0 (n + 1)−2 < ∞. (MA4) This bound uses a box estimate for each shell; its excess over the actual shell size is harmless. For odd L, on the strip of imaginary width 1/128, first choose the real part of the coarse momentum in its fixed reciprocal fundamental cell. The lattice sine lower estimate on every nonzero alias and the complex perturbation estimate for s + μ2 give |dμ(pl)| ≤ 2 (1 + r(l))2 . (MA5) 167 ## Source page 170 A Wilson Yang–Mills construction J. Rodgers At the zero alias the massive denominator directly gives the bound two. For a nonzero alias, its real frequency is at least (r(l) + 1)μ in the norm used by the sine estimate, and 1 2((r(l) + 1)μ)2 ≤ |s(pl) + μ2|; divide μ2 by this lower bound. A general coarse momentum differs by a reciprocal integer translate; this permutes the alias set. Thus summing (MA5) cubed gives, on that whole strip, X l |dμ(pl)3| ≤8J0. (MA6) In particular the diagonal alias matrix d3 μ ⊗1Ch has nuclear norm at most 8hJ0. This is the three-massive-arc trace bound used inside the ninth-power regulator. Here are the frames in which it is used. Let TL(p) have entries L−2eipl·exδαβ, with row x, α and column l, β. The finite character sum shows that it is unitary at real p; its explicit inverse is the opposite-phase character matrix. On the fixed complex strip the two matrices have operator norms at most e4/128, by separating their real unitary Fourier factor from the diagonal complex phases. Furnishing an alias symbol means conjugating its diagonal alias blocks by these two matrices. Thus its three-arc nuclear bound is at most e4/128 (8hJ0) e4/128. (MA7) The weighted frames additionally use hl(p) = q pebWeight(pl, μ), AL(p) = TL(p) diag(hl(p)−1)TL(0)−1. (MA8) Here pebWeight(p, μ) = P a |ei Re pa −1|2 + μ2. Thus the positive weight uses the real part of momentum even when p is complex. Such a weight is a norm-comparison device; it is not asserted holomorphic. For complex differentiation, freeze the weights at the real anchor and use the analytic TL(p) factors. Comparing the original and frozen weights at nearby momenta gives the fixed height allowance. The link-plus-site frame is the direct sum of (MA8) on link components and the unweighted site frame. This separation justifies extending the real floor to a complex strip without a hidden dimension factor. Test the conjugated matrix against two fixed vectors and apply the one-variable Cauchy estimate to the resulting scalar analytic function. If its operator norm is bounded by M on a ball of radius R, its derivative is bounded by 2M/R on the half ball. Integrate along the segment from the real anchor to a nearby complex momentum. Choose that distance so that the perturbation norm is less than half the real lower bound in (MA2). The Neumann series then inverts the perturbed matrix with at most twice the anchored inverse bound. Apply this in the frozen frames, then restore the compared positive weights. All constants are fixed before L; none comes from summing matrix entries to estimate an operator norm. The derivative films used above are the actual zeroth, first and second coefficient matrices of the local analytic rows. A row’s real directional derivative bound is first converted componentwise, and then its color-coordinate square sum is taken. For the link row this costs exactly p | Loc(G, 1)|; for the site row it costs p | Loc(G, 0)| and the displayed local sampling-density factor. Pulling the site row into the block adds L2, since the normalized site square sum has L4 terms. These are the link and site row allowances used in the weighted operator bounds. The rows are differentiated through third order when constructing a second-order film of their first derivatives; that shift of order is retained. For the regulator film, form three consecutive cubes of the normalized inverse film in (MA1), giving d3 μd3 μd3 μ = d9 μ, and then multiply by μ2. At the common center this is exactly cμ. Its trace- ideal bound uses one unsplit cube as (MA7), while the other factors receive operator bounds. In 168 ## Source page 171 A Wilson Yang–Mills construction J. Rodgers a first or second product derivative, choose the corresponding three-arc occurrence after applying the exact inverse and product rules; the column placements in (DP10)–(DP12) give the listed alternatives. The resulting finite sum is bounded by the positive sum of their tariffs. Finally the house term is the physical Hessian film sandwiched between its two row films; the orbit term is the sum over four initial-site copies of the regulator film. Sandwiching a trace- ideal factor by operator-bounded factors preserves its nuclear bound, with the product of the two operator allowances. Sum the four copied terms and conjugate to the common frame; the fixed Fourier norms cost at most e4 on each side at the wider intermediate allowance. The reflection correction is the sum of its two cross products with the regulated site row, and the fixed extra block contributes 4| Loc(G, 0)|. Addition and the same two-sided multiplication rule give the full trace toll in (DP10)–(DP12). Thus the cancellation there is backed by the massive symbol (MA1), the alias sum (MA6), the real and complex floors, and the actual local row derivative bounds. It does not require treating an arbitrary large block matrix as uniformly trace class. The complex denominator estimate used in (MA5) can be proved before alias summation. Put x = Re p, u = ∥q(x)∥2, and t = ∥p −x∥∞≤1. Integrating the derivative of the exponential gives ∥q(p) −q(x)∥2 ≤4t, and the same estimate at −p. Expanding their scalar product yields |s(p) −s(x)| ≤4t(u + 4t) + 4tu = 8tu + 16t2. For t ≤μ/128, the right side is at most (u2 + μ2)/2: use 2uμ ≤u2 + μ2. Since s(x) = u2, the reverse triangle inequality proves |s(p) + μ2| ≥1 2(u2 + μ2) > 0, |dμ(p)| ≤2, |cμ(p)| ≤29μ2. (MA9) The coarse strip has width 1/128; each alias divides its imaginary part by L, so its fine momentum has exactly the width μ/128 required here. In particular the inverse denominator is analytic throughout the strip. For completeness the weights in (MA8) obey hl(p) ≥μ, ∥q(pl)∥2 ≤5hl(p) on the unit coarse strip. At coarse points of distance at most one, the preceding exponential estimate gives ∥q(Re pl) −q(Re rl)∥2 ≤4μ. Using hl ≥μ shows that the two positive weights have ratio bounded by a fixed constant in both directions. Freeze hl at a real anchor while retaining TL(p) and its explicit analytic inverse. These are the analytic frames used in the Cauchy estimate. In ordinary unweighted norm their bounds are ∥AL(p)∥≤e4σμ−1, ∥AL(p)−1∥≤e4σ sup l hl(p). (MA10) For a real anchor, h2 l ≤16 + μ2 ≤17. These estimates display the mass powers; they must not be silently replaced by two mass-independent unweighted bounds. Here is an explicit form of the strip-inverse argument. Suppose a matrix K, in the two specified weighted frames, has norm at most A, its real lower singular bound is a > 0, and the changes to frozen analytic frames cost at most C on a ball of radius σ. Test the analytic matrix on two unit vectors and apply the scalar Cauchy estimate. This gives derivative bound k = CAC/(σ/2) on the half ball. On the strip η = min{σ/2, (a/2)/(1 + k)} the displacement from its real anchor changes the matrix by at most a/2. The lower singular bound is therefore a/2, so its weighted inverse has norm at most 2/a. This also proves invertibility 169 ## Source page 172 A Wilson Yang–Mills construction J. Rodgers and analyticity of the inverse by the determinant formula. The constants are independent of the dimension of the block. In the site application the real lower form bound is proportional to h2 l , and the two inverse weights cancel it; it is that weighted matrix to which the assertion applies. The remaining mass bookkeeping can now be stated exactly. The normalized inverse film has bounded ordinary operator coefficients because its factor μ2 cancels the two inverse-weight losses. The regulator film has both operator and trace-ideal price μ2 times a fixed price, by its definition as μ2(d3 μ)3. Converting it back to the weighted form costs at most μ−1 on each side, hence cancels this μ2. This accounts for the mass-independent regulator toll in the link determinant comparison. It does not remove the explicit μ2 in its ordinary frame. Finally the actual matrices behind the last paragraph of this chapter are as follows. Let B be the fixed inclusion of the link block and let P be the complementary site projection; let Si read the initial-site slot of direction i. Write C for the regulator film, R for the differentiated row, and H for the local second variation of the scalar correction. With the rear operation denoting the transpose at reversed momenta, including the reversed external legs, the relevant products are House = RrearHR, Orbit = 3 X i=0 Srear i CSi, Motive = P rearP. (MA11) These are exact products of films, so their first and second coefficients are their ordinary ordered product derivatives. The motive is constant and its finite site rank is the reason its trace allowance is 4| Loc(G, 0)|. The orbit allowance is the regulator trace allowance multiplied on each side by the Fourier-frame bound. The house allowance follows by first assigning the finite local trace allowance to H, then multiplying by the two row bounds. The two reflection cross products are bounded in the same order. The total toll is the positive sum payLedger + reflectLedger + 4(4 orbitLedger) + popLedger. (MA12) Each factor in this expression has thus been paid either by a local finite rank, the alias trace sum, or a weighted operator norm; no trace norm of a general extensive Hessian is assumed. 8.9 Constructing the common asymptotic calendar Every finite stage contains only finitely many requests of its admitted degree and derivative order. Their maximum is finite. A countable family is met by letting the stage order increase, so that each fixed request is eventually present. This is weaker than a uniform estimate at all orders at one stage, and is exactly what the diagonal construction needs. We give the numerical schedule construction explicitly, since both finite shooting and uniform running use it. For each stage j ≥j0 ≥4, all pre-block parameters have already been chosen: its derivative order, support exponent, analytic width, path bound, column bound and drift allowance d(j). Let A(j) = log  1 + paths(j) + width(j)−1 + columns(j) + relaxation(j) + supportExp(j)  . Choose an odd integer block length Lj ≥3, at least every previously required lower bound, such that d(j) log Lj ≤min(s/4, j−2), A(j) log Lj ≤j−2. (CAL1) This is possible because the numerators are fixed before this choice and 1/ log n →0. To make the block lengths nondecreasing, take for each j the maximum of the finitely many lower bounds 170 ## Source page 173 A Wilson Yang–Mills construction J. Rodgers through j, then an odd integer larger than that maximum. These choices prove the drift and logarithmic-cost inequalities without making the drift allowance depend on the selected terminal cutoff. After fixing the block lengths choose the positive protection exponents, decreasing with j, small enough for all post-block requests and at most 1/j. This can be done by taking successive finite minima of positive admissible upper bounds. Next choose the stage’s coupling-entry and scale- entry thresholds; every analytic estimate proved with these parameters has one eventual threshold, so a finite maximum handles all its requests. Later increases of these thresholds preserve their inequalities. Choose the initial coupling ̄β > 1 above the first-stage entry threshold, and sufficiently large that x−1/2 exp((log log x)2) ≤1 (x ≥ ̄β). (CAL2) The ratio (log log x)2/ log x →0 proves this. For fixed j, the required lattice-size bound at logarith- mic scale u has the explicit form Vj(u) = Sj 1 + c(u) + c(u + log Lj) qj100(c(u) + 1) + 100 (j + 3)Lj + 1 , c(u) = l log2( ̄β + su) m . (CAL3) Here Sj and qj are the fixed shell-count and growth requests from the local estimates. The last two terms include the small-field and patch buffers. For each fixed stage, c(u) and its fixed translate are O(u); hence Vj(u) = O((1 + u)qj + u + 1) = o(eu). Choose an integer entrance bound Bj such that m ≥Bj and u ≥m log 3 imply simultaneously ̄β + su ≥betaEntry(j), u ≥scaleEntry(j), u > 0, log Lj ≤u/j, Vj(u) ≤eu. (CAL4) Each condition holds eventually as u →∞; m log 3 →∞gives one common integer Bj. There are also stagewise comparison-window requests wj(u) satisfying wj(u)/u →0, and finite readiness thresholds rj. Choose integers Cj so that for every k ≥Cj and k log 3 ≤u ≤k log Lj, rj ≤u, wj(u) ≤k/2. (CAL5) Indeed eventually wj(u)/u ≤(2 log Lj)−1; multiplication by the upper bound on u proves the second inequality, and the lower bound on u supplies readiness. Integer rounding only strengthens the resulting bound by ⌊k/2⌋. We now build the stage clock, rather than presupposing one meeting all these requests. Use an unshifted stage number h ≥0, corresponding to j = j0 + h, and write Bh, Ch for the chosen integer requests after this shift. Define plateau starting indices recursively by a0 = 0, kh = Ch + 2ah + 1, ah+1 = ah + Bh+1 + 2kh + 1. (CAL6) Set E(i) = h when ah ≤i < ah+1. The sequence ah is strictly increasing, so E is defined everywhere, nondecreasing, increases by at most one in a step, and tends to infinity. Also E(0) = E(1) = 0. Its recursion gives two useful properties: E(i + 1) > 0 =⇒BE(i+1) ≤i, (CAL7) and, for every h, kh ≥Ch and E(i) = h whenever kh −⌊kh/2⌋≤i ≤kh + ⌊kh/2⌋. (CAL8) 171 ## Source page 174 A Wilson Yang–Mills construction J. Rodgers For (CAL7), the first index of plateau h > 0 is ah = ah−1 + Bh + 2kh−1 + 1 ≥Bh + 1. For (CAL8), kh = Ch + 2ah + 1 places its lower endpoint above ah, while ah+1 > 2kh places its upper endpoint strictly before ah+1. Thus every stage receives a sufficiently long central comparison window, however large its requests. Define J(i) = j0 + E(i), Li = LJ(i+1), τ0 = 0, τi+1 = τi + log Li. (CAL9) Then i log 3 ≤τi ≤i log LJ(i), by monotonicity of the blocks. The first inequality proves τi →∞. For every noninitial plateau (CAL7) lets us apply (CAL4) at u = τi. In particular 0 ≤log Li τi ≤J(i + 1)−1 −→0. (CAL10) This is the gentle-mesh condition. Equations (CAL1) and (CAL9) give (FCA4) and (RUN3). The stage order, power and support exponent stay nondecreasing, the protection exponent decreases, and (CAL2) gives the small-field threshold at every scale. These properties all belong to the same calendar. For volume, let di = Q r d, and a jet order k, and is admitted only when 2(D +k +1) is at most the incoming smoothing grade. At outgoing grade j, there are consequently only finitely many such triples and adjacent predecessors. The maximum of their local length thresholds is the integer floor used in (CAL12). On the fixed patch, take a second finite maximum, at least one, of their polynomial estimate constants, the local stationary constants, and all reciprocal widths ρ−1 E,i with E, i ≤j. Call it Rj. Its post-length request is eRj; payment by bβm is exactly Rj ≤log bβm. Every auxiliary derivative index in a fixed triple ranges over finite intervals, so it introduces no supremum over arbitrarily high orders at one stage. This proves the finiteness of the native stationary and comparison prices used above, at their actual chosen lengths. This includes the finite jet, local coordinate and path readouts together, with one selected system of their dictionaries and bases. The inverse-map request has zero precondition. Its actual cost is the inverse estimate of the specified local patch map at its selected analytic width and axis forest. It can therefore be paid from step zero: at each j, maximize its finitely many adjacent predecessor costs and use (CAL17). One does not need to assume a growth law for this inverse price. For a reporting window based at level k, the affine running reference coefficient gives z( bβm) ≤z( bβk) + s|tm −tk| ≤(1 + s){2 + Zk + |tm −tk|}, (CAL18) where Zk includes z( bβk). To obtain the first inequality, differentiate z on v > 1: z′(v) = 1/[v(2 + log v)] ≤1, and use | bβm −bβk| = s|tm −tk|. Thus the paid inverse cost has the same polynomial reporting-age control required by the local comparison estimates, on the actual windows of the actual history. The common reference-period registry is fixed at the same pre-length stage. At outgoing grade j its finite list consists of horizons e ≤j and the allowed adjacent incoming grades. Their integer size requirements are included in (CAL16). With the resulting lengths fixed, its uniform finite-patch, chart and normalization prices give a real entry gate for that same list, which is included among the later delay maxima. This retains one registry for every eventual horizon and every reference torus used in the periodic-copy argument; a later numerical enlargement does not reselect it. The physical periods themselves are still chosen as in (CAL11). We explain the identification with the actual run, since it is needed for the order of quantifiers in the theorem. The run’s stage profile is the original profile, its kernel is the original kernel, and its factor at profile j is the length just selected. Hence (CAL12) at J(m + 1) is the actual step’s local length inequality. Its stationary horizon satisfies the original primary price list. Projecting the two maxima in (CAL15) gives each local patch payment, and therefore its stationary admissibility and the delay required for the flat covariance limit. The preselected flat supplier may now be applied with entrance index zero to this very horizon; the curvature and dual suppliers apply to it by the other two floors. Their local results concern the same levels, patches and retained horizon as the run. No new family is chosen when these numerical payments are consumed. The finite volume and token comparisons are established afterwards on this fixed run, and only then is a requested extraction order refined. 174 ## Source page 177 A Wilson Yang–Mills construction J. Rodgers 8.11 Paying the actual compulsory-emission forecast The compulsory-emission estimate must retain its quadratic monitor exponent until it has been compared with the output forecast. Here is that comparison. Fix the block, the positive local emission parameter μ, and the finite output order. Put y = βF (x, L), p = M(y), with M(y) = exp{(log log y)2}. The integrated emission bound obtained from the packet and connected-cluster estimates is 2E(x) exp{−μp2/4}, E(x) −→0. (EF1) This is the bound before weakening its exponent in the dictionary estimate. The requested strong forecast has form C exp{−at2}, C ≥1, t = y−1/2p β−1/2 , β ∈[y/2, 2y]. (EF2) In particular 0 ≤t ≤2p. After μ has been selected at this block and chart width, choose the positive forecast rate a ≤μ/32, and, if necessary, decrease its requested rate to at most μ/64 so that its strict positivity margin is retained. This is a choice of the numerical forecast, made before choosing its entry delay. It gives at2 ≤4ap2 ≤μp2/8 ≤μp2/4. (EF3) For any prescribed c > 0, choose the entry delay so that 2E(x) ≤c. Equations (EF1)–(EF3) then put the actual emission bound below c times (EF2), uniformly for every coupling in the band. The source uses the weaker sufficient condition a ≤μ/16; the chosen μ/32 leaves room for subsequent attachments. These choices are made separately for each of the finitely many legal adjacent stage pairs and combined by their maximum. Thus the forced forecast used by the step, and later by local grafting, is purchased with its actual quadratic decay and on its actual chart. A previously weakened subquadratic estimate is not needed for this payment. 8.12 Uniform reserve at every step, including startup The numerical purchases also leave a squared reserve for errors that will later be divided by local normalization factors. We derive that reserve, since eventual smallness at each fixed stage alone would not supply a bound at every step of the chosen history. Put M(x) = e(log log x)2, κ(x) = e−M(x)1/5, x > e. (CAL19) Thus κ( bβm) is exactly the forced allowance at step m. Fix a positive stage rate q. Eventually qM(x) ≥2. Since M(y) ≥M(x) ≥1 for y ≥x > e, qM(y)2 ≥qM(x)2 ≥2M(x) ≥2M(x)1/5, e−qM(y)2 ≤κ(x)2. (CAL20) The stage rate in this formula is the positive rate of the actual already chosen post-length estimates. Choose its threshold separately for each stage and include these thresholds among the entry-delay maxima. At step m, use the outgoing stage J(m + 1), x = bβm, and y = bβm+1 ≥x. This gives the squared-reserve bound simultaneously for every m, with the same post-length data. The same delay can enforce log bβm ≥1. Startup has a corresponding reserve without requiring a preceding block. In the startup formulas write s = log x, so their large parameter is Ms = e(log s)2 = M(x) and their small remainder exponent is γs = e−Ms/2. The startup estimates established above provide its local expansion with 175 ## Source page 178 A Wilson Yang–Mills construction J. Rodgers packet-energy charge σ = Ms, extracted suppression charge κ0 = Ms, and remainder charge γs, at coupling x/2. The connected support budget for that remainder is e−Mse eγs−1 −1  ≤e−Ms(e −1) ≤e−2M1/5 s = κ(x)2 (CAL21) for sufficiently large s. The first inequality uses eγs −1 ≤1, and the second follows from Ms − 2M1/5 s →∞. In particular it pays the full connected budget, including its exponential conversion from the local remainder bound. For each fixed stage the startup threshold is finite. Include it in the same entry-delay maximum. At level zero the calendar identifies the initial stage with its first entry stage, so this threshold is paid there too; at a successor level it follows from the previous entry payment and the positive coupling increment. Consequently the chosen calendar gives (CAL21) at every possible finest cutoff. For any actual coupling in [x/2, 2x], the startup expansion is still valid: its lower-coupling hypothesis follows from the left endpoint, and its vacuum price is bounded using the right endpoint. The exact startup density identity then turns (CAL21) into the required forced-activity bound. These choices enlarge delays before the final calendar and history are selected, and preserve all earlier purchases. They add no assumption to the theorem. 8.13 Inverse curvature coordinates and the startup fourth coefficient The fourth-degree correction used in startup is determined by the Wilson plaquette price in curva- ture coordinates. It is separate from the earlier inequality bounding the difference between chord squared and curvature squared. Write c(U) ∈g for the projected curvature, and define k(a) = c(exp a). (SQ1) This real analytic map has k(0) = 0 and Dk(0) = I. Indeed the exponential has derivative a 7→a at zero, its inverse group factor has derivative −a, and taking the projected skew part of their difference divided by two returns a. The finite-dimensional analytic inverse theorem gives open neighborhoods of zero and an analytic inverse u with u(k(a)) = a, k(u(z)) = z, u(0) = 0, Du(0) = I. Set R(z) = exp(u(z)). On a group neighborhood of the identity and a color neighborhood of zero, c(R(z)) = z, R(c(U)) = U. (SQ2) For the second identity use the group logarithm chart constructed earlier: write U = exp a, shrink so that a is in the inverse neighborhood, and substitute c(U) = k(a). Thus no global group logarithm or connectedness assumption is involved. Define the real analytic scalar function f(z) = price(R(z)). (SQ3) Since R(0) = 1, the price is zero there. Its first derivative is zero because the Wilson price is a squared chord. For R(tz), its matrix derivative at zero is the matrix of z, so the physical color normalization gives f(tz) = 1 2t2|z|2 + o(t2). The second derivative is symmetric; polarization of this identity gives D2f(0)[z, w] = ⟨z, w⟩. (SQ4) 176 ## Source page 179 A Wilson Yang–Mills construction J. Rodgers Both inversion and conjugation preserve the price, while c(U−1) = −c(U) and c(hUh−1) = Adh c(U). Apply (SQ2) to the nearby group elements R(z)−1 and hR(z)h−1. Uniqueness of the inverse curvature coordinates gives f(−z) = f(z), f(Adh z) = f(z) (SQ5) as germs at zero, for every fixed h ∈G. Differentiating these germ identities proves that every odd Taylor coefficient vanishes and every Taylor coefficient is adjoint invariant. The positive physical inner product is unchanged by the adjoint action. Use Taylor coefficients, including the factorial, to set Q4(z) = 1 4!D4f(0)[z, z, z, z]. (SQ6) This is the fourth-degree invariant polynomial called Q4 in (FST1). By (SQ4)–(SQ5), the Taylor polynomial through order four is exactly 1 2|z|2 + Q4(z). Consequently f(z) −1 2|z|2 −Q4(z) has all derivatives of orders 0, . . . , 4 zero at 0. (SQ7) For a smooth family of actual group elements U(v) with U(0) = 1, (SQ2) identifies its Wilson residual R(U) = price(U) −1 2|c(U)|2 −Q4(c(U)) (SQ8) with the composition of the remainder in (SQ7) and c(U(v)), near zero. The chain rule shows that all derivatives below order five vanish: each term contains a derivative below order five of the outer remainder. This proves the exact flat-jet assertion underlying the startup estimate. We next explain the uniform derivative bound, including arbitrary plaquette links and threshold equality. Let U = (U0, U1, U2, U3) ∈G4 and P(U) = U0U1U−1 2 U−1 3 . For a list of four-link directions A1, . . . , Aq, let Jq(A; U) be the ordered left derivative of R(P(U)), in the convention (AB3). It is jointly real analytic in the direction variables and the four matrix entries on their actual group locus. This follows by induction: differentiating the analytic expression along the analytic left increments preserves analyticity. It is homogeneous of degree one in every direction; in particular Jq(tA; U) = tqJq(A; U). (SQ9) Keep U1, U2, U3 arbitrary and replace the first link by eU0(z) = exp(z)U3U2U−1 1 . (SQ10) The resulting plaquette is exactly exp z. At z = 0, the residual has the vanishing property (SQ7). Each ordered link derivative lowers this vanishing order by at most one: introduce one extra real variable for its left increment, apply the vanishing composition to the joint variables, and then differentiate in that variable at zero. Induction gives vanishing through order strictly below max(5 −q, 0) for Jq(A; eU(z)). For directions in a fixed ball |A| ≤r, the external parameters consisting of these directions and the four group entries lie in a compact finite-dimensional set. On that set times |z| ≤1, the required derivatives of the analytic expression are bounded. Taylor’s estimate, obtained by integrating successive derivatives along the segment from zero to z, therefore gives one Cq,r > 0 such that |Jq(A; U)| ≤Cq,r|z|max(5−q,0) if |A| ≤r, |z| ≤1, P(U) = exp z. (SQ11) This argument bounds actual left derivatives; it is not a differentiation of a healthy-set mask. The three unaltered links need not be small. 177 ## Source page 180 A Wilson Yang–Mills construction J. Rodgers The local logarithm estimate gives constants c0, b0 > 0 with | log V | ≤c0 chord(V ) for chord(V ) ≤b0. Choose δ ≤min(b0, c−1 0 , 1). If chord(P(U)) ≤θ ≤δ, and |A| ≤rθ, apply (SQ11) to A/θ, then (SQ9). It gives |Jq(A; U)| ≤Cq,rcmax(5−q,0) 0 θq+max(5−q,0) ≤C′ q,rθ5. (SQ12) For q ≥5 the last inequality uses θq ≤θ5; for q ≤5 its exponent is exactly five. All inequalities allow equality in the plaquette threshold. At a fixed derivative horizon take a maximum over its finitely many q’s. At a plaquette anchored at o, an allowed direction (AB4) restricted to its four links is at most rθ, with r depending only on the fixed power p and the bounded plaquette distances from o. Pullback to these four links therefore converts (SQ12) into the derivative envelope used in (FST8). A plaquette wrapping around the torus still has these same four stored group entries and bounded periodic distances, so the estimate is unchanged. Summing over the fixed finite number of signed plaquette types and multiplying by β gives the anchored bound Cβθ5. The constants are chosen before coupling, cutoffand torus period. This completes the startup Taylor and derivative input from Extra/ WILSONS TARTUP/Curvature and Residual. 8.14 The actual finest-scale Wilson representative The Wilson startup is an exact identity for the whole density, although its activity dictionary is nonzero. The regular and forced entries cancel the added principal path terms configuration by configuration. For the allowance estimates, the forced suppression needs only β ≥x/2. The regular term C′βθ5 also needs β ≤2x. Thus both allowances are asserted together on the shooting band x/2 ≤β ≤2x; the exact density identity itself does not acquire this restriction. The startup used in finite shooting has nonzero activities; its total density is nevertheless exactly the bare Wilson density. We describe this cancellation. Fix a finest-level clip c with cdirect ≤1, threshold θ > 0, and support width B ≥2. Put b = 1−cdirect ≥0. For a plaquette holonomy u, let R(u) = price(u) −1 2| curvature(u)|2 −Q4(curvature(u)), (FST1) where Q4 is the fixed fourth-degree invariant curvature correction. The finest path has row √ b times the elementary plaquette curvature, zero loop and cubic tensors, and quartic tensor bQ4 on the same plaquette. Thus its quadratic-plus-quartic part is the corresponding local Taylor polynomial. It uses the entire plaquette footprint, including a wrapping plaquette. For each plaquette P, write uP for its holonomy, rP = chord(uP ), and kP = b| curvature(uP )|2/2. Define its unpaid nonnegative part by dP (U) =        price(uP ) −clinearθrP , rP > θ, kP if the footprint is healthy, else b price(uP )  − kP if its anchor is kept, else 0 , rP ≤θ. (FST2) The kept predicate is the exponentially relaxed plaquette test of the clip. On a kept anchor its own plaquette satisfies rP ≤θ. The identities price(u) = chord(u)2/2, | curvature(u)| ≤chord(u), and clinear ≤1/4 show 0 ≤dP ≤2. If rP > θ, they give the stronger dP ≥θ2/4. Let Atop(U) denote the actual principal path action, including its clipped direct part and its masked quartic term. Checking separately the healthy case, the unhealthy case with rP > θ, and the unhealthy case with rP ≤θ, gives the pointwise identity price(uP ) = baseTermP (U) + maskθ,P bQ4(curvature(uP ))  + b maskθ,P [R(uP )] + dP (U). (FST3) 178 ## Source page 181 A Wilson Yang–Mills construction J. Rodgers The first two terms here are precisely the plaquette terms in Atop. In the healthy case substitute (FST1) and cancel. In the large-plaquette case both masks and the kept quadratic term vanish, and the clipped direct term is clinearθrP , canceled by (FST2). In the remaining case both masks vanish and the two alternatives for the kept quadratic term cancel those in (FST2). These are exhaustive cases, including threshold equality. Average (FST3) over all signed coordinate frames with weight ω = | Signed |−1, and sum over plaquettes. Index these oriented terms by i, with footprint Si, anchor xi, residual Ri and unpaid part di. Signed-frame invariance of the Wilson energy gives βS(U) = βAtop(U) + βωb X i maskθ,Si Ri(U) + βω X i di(U). (FST4) Take the middle sum as the regular activity action. To represent the final sum by forced activities, partition the actual flag set into its connected components for the relation that two flags’ radius-2B neighborhoods overlap. If D is a nonempty component, let R(D) be the union of those neighborhoods and set qD(U) = ω X i: xi∈haloB(D) di(U). (FST5) The extended regions of distinct components are disjoint. Every nonzero di lies within distance B of some flag, and hence belongs to exactly one component’s sum. The assertion about distance follows from the relaxed test: outside that distance the clip condition θ exp((B −2)/crelaxation) ≥2 and the universal chord bound 2 force all its tests to be kept and healthy. Therefore P D qD = ω P i di. Assign the forced entry e−βqD(U) to the support-inventory pair (R(D), D), eligible precisely when R(D) ∩flagsθ(U) = D, and assign zero to all other active entries. In the gas sum, a nonzero admissible collection must have disjoint supports and inventories partitioning the flags. Its invento- ries are consequently exactly the connected components just described. Conversely that collection is admissible. The gas sum thus has one nonzero term and equals gas(U) = exp −βω X i di(U) ! . (FST6) Combining (FST4) with (FST6) proves the exact startup density e−βS(U), with scalar normalization zero. These explicit entries also satisfy the needed bounds. The normalized derivatives of R on a healthy plaquette, through each fixed required order, are at most Cθ5. This follows by Taylor expansion of the smooth four-link product and (FST1): its curvature correction cancels the jets below degree five; the permitted derivative directions have size O(θ), so differentiating does not lose the total factor θ5. The constants are chosen for a fixed order before the coupling or period. There are only the fixed number of signed plaquette types at an anchor, and their cover cost is bounded. Consequently the anchored regular bound is at most C′βθ5. For a forced component D, every flag is incident to a plaquette above threshold and each plaquette touches at most four sites. Using dP ≥θ2/4 and keeping one signed frame gives qD(U) ≥|D| ωθ2/16. (FST7) This exponential suppression pays the connected-component support count and its support weights. To see its size in the actual schedule, set x = bβN, z = log x, M(z) = e(log z)2, and θ = e−z/2M(z). For the forced estimate, the lower bound β ≥x/2 is sufficient. On the actual shooting band 179 ## Source page 182 A Wilson Yang–Mills construction J. Rodgers x/2 ≤β ≤2x, (FST7) has strength proportional to M(z)2|D|. The neighborhood count (4⌈z2⌉+1)4 and each fixed stage support cost are eventually at most M(z)/4. The component expansion may therefore use local allowance e−M(z)/2; its summed connected-activity bound is at most e−M(z)+1, which is eventually at most e−M(z)1/5, the required forced allowance. The regular bound satisfies C′βθ5 ≤2C′e−3z/2e5(log z)2 ≤e−11z/10, (FST8) the required regular allowance. The same finite thresholds also ensure the clip condition preceding (FST6), since B = ⌈z2⌉, and θ →0. These thresholds are uniform in N, p once the stage is fixed, and have already been included in its entry delay. The dictionaries of supports and flag predicates are independent of β. Regular entries are linear in β, and forced entries are exponentials of bounded finite-volume functions times fixed eligibility indicators. Their weighted sums obey the preceding majorants. Thus the startup family is continuous in precisely the activity budgets used for finite shooting. Countably many unused addresses can be appended with value zero, without changing density or bounds; they furnish the marker slots required later. This constructs (FCA2) with its actual representatives. 8.15 Finite coupling arcs and calibrated finite histories Finite shooting selects a coupling on a continuous finite arc with Wilson startup. The estimates control that arc in the specified band, and the common calibration condition fixes the retained endpoint. Retirement removes only terms whose exact density identity and purchased error have already been recorded. The limiting history is unique within this fixed recipe and calendar; unique- ness across arbitrary choices of recipe is not part of the construction. We construct the finite histories before using their compactness or uniqueness. Fix the common calendar, a terminal depth N, and a period p. Put τi = log(a−1 i ), s = slopeConstant(G) > 0, bβi = ̄β + sτi, Bi = [ bβi/2, 2 bβi]. (FCA1) Here τ0 = 0, ̄β > 0, and τi+1−τi = log Li > 0. The path tensors at fixed N, i are fixed independently of p and of the bare-coupling shooting parameter. A represented state consists of those tensors, its prescribed clip and budget, its actual coupling β, scalar normalization v, and regular and forced activities. Admissibility means the local regularity, symmetry and absolute activity bounds proved in the nonlinear step, together with the retained startup and transition conditions. It includes β ∈Bi. Continuity of a family of such states has a specific meaning. Its β and v are continuous; its regular and forced address dictionaries are constant; and differences of its activity values tend to zero in the absolute weighted derivative budgets. For regular terms derivatives are tested only on the common healthy domain, through the prescribed order and in the prescribed directions. For forced terms the bound is on every configuration. On a fixed period a global address sum is legitimate for this continuity assertion. It is not substituted for the period-independent anchored bounds in admissibility. The nonlinear construction provides a forward family at every finite step on this one calendar. At the finest level its states FN,p(b), b ∈BN, have β(FN,p(b)) = b, v(FN,p(b)) = 0, FN,p(b). eval = bareb . (FCA2) They have the fixed startup representatives and insertion slots. For a continuous family Q(b) of admissible level-i + 1 states on a compact parameter interval, the same primitive integration and 180 ## Source page 183 A Wilson Yang–Mills construction J. Rodgers reset construction produces continuous coarse states T(b), with the same retained choices, such that T(b). eval = P β(Q(b)) i (Q(b). eval), |β(Q(b)) −β(T(b)) −s log Li| ≤di+1. (FCA3) The number di+1 is the fixed drift allowance of this stage. The output satisfies all level-i local bounds before imposing its coupling band. This distinction permits the shooting argument to detect crossings of that band. These forward families are obtained from the actual primitive formulas, rather than a choice of unrelated admissible outputs. At a step, keep one primitive construction, its ordinal and tree choice, and its fixed support dictionaries. Its output is the moment state produced by that primitive. The local integral comparison and the absolute activity estimates above prove continuity of the output in the stated budgets. The reset identity gives the density equality in (FCA3); its coupling bound gives the second inequality. For a continuous input curve these same formulas therefore define a continuous output curve. For two occurrences of the identical input state they give the identical output state: equality is equality of the represented moment states, including the activities. Choosing this rule once at each N, i, p, and the startup rule (FCA2) once at N, p, gives a single deterministic forward recipe. Its underlying path, clip, normalizers, source bounds and calendar remain those of the nonlinear construction. Restricting a parameter interval preserves every one of these properties. To make the normalization in this construction explicit, write Hb(U) = P β(Q(b)) i (Q(b). eval)(U). The positive push kernel and the retained positive vacuum give Hb(1) > 0. If αN,i is the already committed path correction, the output coupling and scalar are β(T(b)) = β(Q(b)) + αN,i + ∆(b), v(T(b)) = log Hb(1) | Sitei | . (FCA9) The scalar reset produces |∆(b)| ≤ri and θ2 i |∆(b)| ≤ei, where ri is its fixed correction radius and ei its regular allowance. In the exact normalized splitting, the output trunk density times its attached- log exponential is Hb(U)/Hb(1). Installing (FCA9) multiplies that density by exp(| Sitei |v) = Hb(1), proving (FCA3) at every configuration, including flag boundaries. The site count, rather than the bond count, is essential in this cancellation. The correction in this construction is the source Hessian projection: ∆(b) = Averaging. delta X. read(a(b)) . (FCA10) Here X is the actual fixed source listing and a(b) its parameterized source. The affine-curvature Hessian projection, rebates, faults and low- and high-path loan identities in the coupling-reset exponent section prove the exact all-configuration representation with this value of ∆. In particular the correction is not defined by solving a two-probe contrast equation. The projection’s diagonal bound gives θ2 i |∆(b)| ≤H(b), where H(b) is the paid regular source bound. The schedule pays H(b) ≤fi/4 and the reset fees, giving the stated correction radius and the unit bound used in (DR7). The regular and forced bounds after the projection and the two dictionary fusions are paid independently by those same fees. To see the normalization without an extra characterization of the correction, let τ(b) be the primitive scalar before reset and let lj(b, 1) be its attached logarithms at the identity. The exact exponent calculation gives the installed scalar v(b) = τ(b) −| Sitei |−1 X j lj(b, 1). (FCA11) 181 ## Source page 184 A Wilson Yang–Mills construction J. Rodgers The resulting state has regular entries zero at the identity, no identity flags, and kinetic and loop actions zero there. Its density at the identity is therefore e| Sitei |v(b). The already proved all- configuration density identity makes this equal Hb(1), proving the logarithmic formula in (FCA9). Thus (FCA9) and (FCA11) are the same scalar normalization. The interval estimate for this same projection value is obtained by the probe argument below. We must still show that this projected correction lies in the purchased correction interval. The source uses a nonflat probe for this purpose, on the very same trial family. Its activity terms are independent of the correction dial z: they have already been computed with the Hessian projection value ∆. Only the scalar coupling β + α + z varies in that trial. Thus there is no second reset algorithm. Here is the probe construction, uniform in the even torus side. Fix the clip’s positive direct coefficient a and its common cubic/quartic path bound C ≥0. A nonzero Lie-algebra direction gives a group curve through the identity with nonzero curvature derivative. Its chord is therefore positive somewhere; continuity and the intermediate value theorem give ε > 0 such that every chord in [0, ε] is attained. Put t = min{ε, 1/4, a/[512(1 + C)]}, η = at2/4 > 0. (RSP1) Choose u ∈G with chord tθ. Set all links to the identity except direction-zero links whose direction- one coordinate is even; set those links to u. This is periodic on every even side. Every (0, 1) plaquette has holonomy u or u−1, and every plaquette has chord at most 4tθ ≤θ. Thus the field V has no flags. Its direct plaquette charge in every signed-frame sum is at least na(tθ)2/2, where n is the number of sites. The cubic and quartic terms have absolute sum per site at most C(4tθ)3 + C(4tθ)4 ≤a(tθ)2/4, by (RSP1) and θ ≤1. Averaging over the signed frames leaves K(V ) ≥nηθ2 > 0, (RSP2) where K is the actual kinetic action. No smallness of the loop tensor enters this lower bound. Write H(U) for the actual transferred density at a fixed input parameter, and set B = β + α. On any flag-free configuration its early representation is H(U) = exp{nτ −[β main(U) + α second(U) + loop(U)] −E(U)}. (RSP3) The source’s anchored regular budget is at most F/4, with F = fundsi. Summing its anchors bounds |E(U)| ≤nF/4 at every flag-free configuration, including 1 and V . Formula (RSP3) proves positivity at both probes and gives | log H(1)/n −τ| ≤F/4. Subtracting its two logarithms gives contrast error at most nF/2 relative to β main +α second + loop. On flag-free fields the main and loop actions equal their unmasked path expressions. The higher-order kinetic correction obeys |K(V ) −second(V )| ≤2nCθ3. The stage’s cubic payment is |α|, 2Cθ3 ≤F/2. Therefore, with L0(U) the zero-coupling loop background, |BK(V ) + L0(V ) −[log H(1) −log H(V )]| ≤nF. (RSP4) This is the actual contrast estimate, with its α and loop terms retained. Let e be the final regular allowance, r ∈(0, 1] the correction radius. The reset price is chosen before history selection so that 2(F + e) ≤ηθ2r. (RSP5) This is paid by the coupling-entry delay: F = ce, e = g11/5, θ = gM, and g ≤1, whereas the monitor M →∞; it suffices to impose 2(c + 1) ≤ηrM2. This same condition is imposed for both permitted adjacent-stage choices. 182 ## Source page 185 A Wilson Yang–Mills construction J. Rodgers For the fixed trial activities, let AV be their regular action at V . Define R(z) = (B + z)K(V ) + L0(V ) + AV −[log H(1) −log H(V )]. (RSP6) Their budget gives |AV | ≤ne. Equations (RSP2), (RSP4), and (RSP5) imply |R(0)| ≤n(F + e) ≤ K(V )r/2. Since R(z) −R(z′) = (z −z′)K(V ), this affine function has a unique root in [−r, r]. Its root varies continuously with the input: the contrast and coupling are continuous, the fixed-trial activity series are continuous by their common deviation bounds, and K(V ) > 0 is fixed. The projected state already satisfies the full density identity, so evaluating it at 1, V gives R(∆) = 0. More explicitly the exact fixed-trial identity is R(z) = (z −∆)K(V ). Thus the unique interval root is exactly the previously defined Hessian projection ∆. This proves its radius bound and transfers the interval service’s continuous curve to the actual represented state. The independent projection estimate θ2|∆| ≤H retains the sharper paid bound. The all-configuration density equality is still supplied by the exponent and gas identities, not inferred from equality at two probes. In the source the corresponding shape comparison is also explicit: for any configuration U, the trial shape is its value at ∆times e(∆−z)K(U); the inequality |ex −1| ≤e|x||x| bounds its discrepancy by a finite configuration-dependent constant times |R(z)|, using K(V ) > 0. Hence the interval and projection formulations are fully compatible on this same trial family. For a compact input curve, the connected-expansion tails can be made small at one common finite retirement index. Their absolute tail bounds and the finitely many local bounds permit that choice throughout the curve. In the deterministic recipe the index is chosen as a function of the incoming address layout. A curve’s layout is constant, so that index is constant on the curve. The identities for the primitive expressions show that changing the incidental local guide or the point at which the curve was presented does not change the resulting state: all such expressions have the same tensors, canonical matching coordinates, source list and scalar reset. This supplies the continuous forward rule used above without silently choosing new dictionaries at different parameter values. The source-level continuity and admissibility statement can now be written without suppressing its series hypotheses. Fix a closed parameter interval I and an incoming curve Q(t) in its admissible band. Use one point t0 ∈I as reference. The incoming path equals the prescribed path at every point, and the regular and forced layouts equal those at t0. The matching maps therefore use identity label correspondence on these dictionaries. Local physical guides may be selected separately at each point; the matched primitive expression is independent of that selection. Transport all entries to the reference layout. This fixes the unretired source dictionary, its support windows and the finite debris predicate used for retirement. For its source a(t), the actual continuity assertions are the following. Each regular entry, with all permitted derivatives on its fixed healthy support, and each forced entry on all configurations, converges as t →t0 within I. Their absolute support-weighted envelopes have uniformly small tails near t0. These two facts imply the deviation budget tends to zero: given ε > 0, choose a finite set of addresses leaving total tail less than ε/2; then choose a neighborhood on which the finitely many remaining differences cost less than ε/2. This uses no continuity of flag indicators as functions of the gauge field: their support and threshold predicates are held fixed in the compared evaluation. On a compact input interval one retirement index can be used throughout. The uniform tail estimate for the unretired source and compactness give an eventual regular tail bound g > 0, while the local finite-volume comparison gives an eventual bound for all the retained local beacon entries. Intersect these eventual conditions, choose one integer above their thresholds, and retain it at every parameter value. The resulting source obeys the explicit regular estimate BudgetPi(a(t)) ≤(1/16 + 2/16)Ei + turnFee(Pi)g + g, (FCC1) 183 ## Source page 186 A Wilson Yang–Mills construction J. Rodgers where Ei = forecastEi. Each beacon on its fixed neighbor support has the derivative bound gleami. The source graft combines the retired regular entries, beacon entries and the fixed local profile. Its forced ingredient estimate is paid by its original forced allowance: the primitive forced part costs at most forecastKi(β)/8, while the added local profile has size at most the prescribed β−1/8 F reserve after its cover weight. The local gluing estimate sums these contributions into forecastKi(β). The turn fee times the local beacon bound is included in the regular price. The stage delays make the right side of (FCC1), with that additional beacon price, at most Ei. This proves the regular and forced local estimates for the grafted source simultaneously at every parameter value. The uniformity over all presentations of one layout requires a separate argument. A layout is the regular and forced address lists, including supports, without coefficient functions. If no admissible state has a layout, its assertion is empty. Otherwise choose one such state U. Form the canonical curve Uβ on the entire closed incoming coupling band by retaining its path, cutoffand address lists, setting every regular and forced coefficient function and the vacuum scalar to zero, and assigning coupling β. Keep the address symmetry permutations. Zero functions satisfy every derivative and hit bound, so this curve remains admissible and is continuous. Choose local guides along this canonical curve and match their expressions to its reference layout. At a fixed coupling the weighted connected series has a summable majorant. For a target tail bound g > 0, take a finite set of terms outside which its sum is below g/2. Each of these finitely many terms eventually leaves the debris set: a term is either kept or eventually appears in the finite retirement roster. Hence one depth removes them all. Continuity in the deviation budget bounds the change of the entire weighted series by g/2 in a neighborhood of that coupling. The retired part is a subfamily of the same majorant, so the same neighborhood works for every larger retirement depth. Finitely many such neighborhoods cover the closed coupling band. Their maximum depth supplies a common tail bound g. Intersect it with the eventual finite beacon bounds, taking g = gold > 0. Now let Q(t) be any other admissible curve with this layout and any allowed guide choices. At a point t, compare its guide with the canonical guide at β = Q(t).β. The paths, lists, cutoffs, budgets and couplings agree. The matching map is the identity on the lists, and its induced relabeling preserves the finite retirement roster and debris predicate. For a debris address the primitive term does not read the incoming coefficient functions: it is an omitted foreground cluster formed from the common geometric and Gaussian factors, or an unretired scalar path term. Matching preserves its connected coefficient, integrand and scalar coupling. Thus the two debris functions and beacon contractions agree: 1debris(d,i)Wi(Q(t)) = 1debris(d,i)Wi(Uβ). This is equality of local functions, so it preserves every derivative envelope tested in the budget. Transfer the canonical tail and beacon bounds through it. Adding the principal unretired contribu- tions gives (FCC1) for the arbitrary presentation. The threshold therefore depends on the layout, stage, depth, period and fixed prices, but not on the curve, its coefficients or its guide choices. Apply this construction both with standard allowances 1/16, 1/16 and with the allowances needed for the contraction. Intersect the two eventual sets with d ≥N, and choose one integer d(L) for each layout. Along every admissible curve L is constant, hence so is d(L). The matching identities identify the deterministic rule with the fixed-index continuous formula on that curve. No continuity of this integer-valued selection as the layout changes is needed. For the deterministic operation, the retirement index is selected once from the incoming layout, at least the terminal depth N, to satisfy these bounds for every compatible presentation of that layout. The universal retirement estimate supplies such an index. Because a curve has one layout, this selected index is constant on the curve. The primitive changes of local guide leave the matched source unchanged, 184 ## Source page 187 A Wilson Yang–Mills construction J. Rodgers including its finite debris predicate and beacon contraction. Thus the deterministic moment state agrees with the continuous curve construction just described. Restricting the interval in the shooting argument changes neither index nor dictionary. The Hessian projection is continuous in the resulting deviation budget. For two source values a, a′, its bound is θ2 i |∆(a) −∆(a′)| ≤deviationPi(a, a′). (FCC2) The threshold is a fixed positive number at this step. The rebate, fault, freshness and loan difference identities are linear in the source difference and in ∆(a) −∆(a′), as proved in the reset chapter. Their summable quote envelopes therefore pass continuity and uniform tails to the reset entries. The finite-support dictionary fusions preserve these properties by their absolutely convergent product bounds. In particular both the final coefficient families and their actual fixed address layouts form a curve in the stated sense. Finally the local primitive identity identifies the transferred density with its early-value expres- sion on every configuration. The reset’s exact exponent and gas identities then give its installed state with that same transferred density; its regular and forced closure estimates give all the coarse admissibility conditions. The scalar vacuum term is continuous by the same source-series estimate. Equivalently, evaluate the positive identity at the flat configuration to obtain its logarithmic nor- malization (FCA9). Together with continuity of the incoming coupling and (FCC2), this proves continuity of the output coupling as well. These are precisely the continuity, density, drift and band-free output conditions used in (FCA3); the subsequent interval restriction is what imposes the coarse coupling band. The calendar makes the following drift condition hold at every step: di+1 ≤s 4 log Li. (FCA4) Define narrower target intervals li = 3 4 ̄β + 5 8sτi, ui = 5 4 ̄β + 3 2sτi. (FCA5) Since τi ≥0, they obey li ≤ui and [li, ui] ⊂Bi. If the fine coupling in (FCA3) equals li+1, then β(T) ≤li+1 −s log Li + di+1 ≤li −1 8s log Li < li. If it equals ui+1, then β(T) ≥ui+1 −s log Li −di+1 ≥ui + 1 4s log Li > ui. (FCA6) These strict crossings require no monotonicity or derivative bound on the coarse coupling as a function of the bare coupling. We use the following elementary interval restriction. Suppose f is continuous on [a, b], f(a) ≤ l≤u ≤f(b). Let b′ be the first point at which f = u, and let a′ be the last point in [a, b′] at which f = l. These points exist by the intermediate value theorem and compactness of the two level sets. Then a′ ≤b′ and f([a′, b′]) ⊂[l, u]. A value below lwould force a later crossing of lbefore b′; a value above u would force an earlier crossing of u. Both contradict their definitions. This argument also covers l= u. Starting at N, restrict the bare parameter to [lN, uN] in (FCA2). Descend by finite induction. If an interval already carries trajectories through levels i + 1, . . . , N and its endpoint level-i + 1 couplings are li+1, ui+1, apply (FCA3) to their level-i + 1 curve. Equations (FCA6) and the preceding interval restriction give a closed subinterval on which its level-i coupling stays in [li, ui] 185 ## Source page 188 A Wilson Yang–Mills construction J. Rodgers and takes these exact endpoint values. Retain the already constructed higher states and insert this new coarse state. Every higher density identity, startup condition and continuity estimate survives the restriction. At level zero we have a nonempty closed bare-parameter interval [aN,p, bN,p] and complete forward trajectories b 7−→(R b N,p,i)0≤i≤N. (FCA7) Their finest coupling is exactly b; each level is continuous in the stated sense; all states are admissible; and every adjacent pair satisfies (FCA3). Their bottom couplings take the endpoint values 3 ̄β/4 and 5 ̄β/4. Thus every bottom value in this interval is attained by a trajectory of the same recipe. In particular choose b ∗ N,p with bottom coupling ̄β, and set RN,p,i = R b ∗ N,p N,p,i. This constructs the calibrated finite histories for every N, p. The same retained arc supplies the auxiliary trajectories at nearby prescribed bottom couplings used in the confluence argument. It supplies whole trajectories, not just scalar sequences satisfying the drift inequality. No uniqueness of an inverse shooting map has been assumed. Determinism as- serts only that equal finest states give identical forward trajectories; the later contraction argument establishes the required uniqueness of their limiting calibrated data. There is also a uniform interior margin. If the bottom value of any successful trajectory lies in [3 ̄β/4, 5 ̄β/4], then 3 4 bβi ≤β(Ri) ≤5 4 bβi (0 ≤i ≤N). (FCA8) At i = 0 this is the given bottom interval. If it holds at i, solve (FCA3) for the fine coupling and use (FCA4): its increment lies between (3s/4) log Li and (5s/4) log Li. Adding these bounds proves (FCA8) at i + 1. This margin justifies the nearby-trajectory comparisons while retaining the original full bands. Finally, assembling these states into the history preserves the common path at fixed N, i, its period independence, the prescribed budget and clip, the finest bare Wilson law, and all exact push identities. Induction in the push integral identity gives the same finest Wilson partition function at every coarser level. It is this calibrated history, rather than a separately selected sequence of couplings, that enters the limits below. 8.16 Exact density identity after retirement The retirement used above is an exact decomposition before it is an estimate. Keep one anchored listing for the whole input curve. For its full regular source write Afull(V ) = P j aj(V ). Partition its indices by the chosen retirement predicate and write aj = akeep j + atail j , including their original healthy masks. The same predicate is used throughout the compatible curve. The local absolute envelopes make both subseries summable, hence Afull = Akeep + Atail. (RET1) The forced gas is retained in the kept source exactly; it is not split into two multiplicative copies. If the density of the full source is ev|Λ|−H(V )−Afull(V ) G(V ), then the identical integrated density after retirement is e−Atail(V )ev|Λ|−H(V )−Akeep(V ) G(V ). (RET2) 186 ## Source page 189 A Wilson Yang–Mills construction J. Rodgers This follows by substituting (RET1) in the exponential; in particular the tail has a minus sign. Summability is needed here to identify the sums in (RET1), rather than merely cancelling finitely many formal letters. Replacing the old listing by the anchored common listing preserves each masked occurrence, so it preserves this equality on all configurations. The full-source density identity follows from the connected expansion already established. Its empty-flag pool is strictly positive and equals the exponential of its absolutely convergent connected logarithm. The scalar normalization contains the positive Gaussian and coordinate prefactors, the incoming scalar volume term, and the reference-forest normalization. Taking their logarithm per coarse site defines v. The cubic, quartic, loop and local correction scores have the exact totals displayed above. Their signs cancel those terms in H + Afull, leaving precisely the logarithm of the positive pool. Multiplying by the forced pool then recovers the original conditional integral, including its prefactors. Thus (RET2) holds before either the kept source or the tail has its sharper output budget. The kept-source estimate and the tail estimate are purchased separately on this same listing. The former gives the desired small regular price together with the original forced forecast; the latter is the remainder used by the finite reference beacons. Reattaching that remainder restores (RET2). Consequently neither continuity nor projection replaces the exact density by a merely close one. 8.17 Finite reference beacons and weighted retirement continuity The finite beacons used in (FCC1) have an explicit five-volume construction. For integer half-width w, use reference torus sides 2(w + j), j = 0, . . . , 4, and coefficients ρj = (1, −4, 6, −4, 1)j 24 4! . (RB1) Fold each coordinate of a reference torus by reflection into its interval of length w + j. To pull a configuration to this folded torus, a positively traversed original edge is read directly, a negatively traversed edge is inverted at its lower endpoint, and a collapsed edge is the identity. Translate this folded view to a chosen coarse anchor o. Denote the resulting group configuration by Sj,oV . This finite word map defines the views even at folded faces. Fix the local read radius r, with 2r ≤w. A position in a folded coordinate has a type: within r of the left face, within r of the right face, or in the interior. A boundary type records the face and its distance 0, . . . , r −1. Its number of preimages under folding is independent of j; an interior type has 2(w −2r + 1) + 2j preimages. In four coordinates a full type therefore has a multiplicity which is a product of four constant or affine functions of j. The fourth difference (RB1) kills every such product having a constant factor, and gives one on the product of four interior factors: its fourth difference is 4! 24. Hence for any function f of the four-coordinate type, 4 X j=0 ρj X x∈torus2(w+j) f(typej(x)) = f(all-interior type). (RB2) This is a finite counting identity, including faces and edges of every codimension. Let fj(x, V ) be the compatible anchored local reference densities. Center them by f0 j (x, V ) = fj(x, V ) −fj(x, 1). Compatibility means that on healthy configurations two such readings agree whenever their radius-r cover inputs agree, and they obey the exact signed and translation laws. Define the beacon at o by bo(V ) = 4 X j=0 ρj X x f0 j (x, Sj,oV ). (RB3) 187 ## Source page 190 A Wilson Yang–Mills construction J. Rodgers Choose its support radius z ≥w + 4. Introduce also the shifted expression which uses anchor o − stagej(x) for the x-summand and masks that summand on its radius-z neighborhood. Summing this expression over o gives exactly the sum of the masked beacons, by the finite translation permutation of the torus. On a healthy radius-2z neighborhood all these masks are one. Their folded local readings depend only on their four-coordinate type; compatibility and the signed laws identify equal types. Equation (RB2) reduces the shifted expression to the original centered local density at o. The all-interior type reads its unchanged cover neighborhood. Thus the difference between the assigned retained density, its identity value and this shifted expression vanishes on that healthy neighborhood. This is the local remainder used in the forced part of the graft. Its global identity follows by summing over o; there is no discarded face correction. The derivative allowance of (RB3) is equally concrete. If the centered full reference sum on side 2(w +j) has left-derivative bound qj on the small reference chart through order k, and the ambient budget has direction power p, then ∥bo∥budget ≤ 4 X j=0 |ρj| 2k(1 + z)pk qj. (RB4) On a healthy neighborhood the folded reference configuration can be gauge transformed into that small chart, by the local axial chart construction; its displacement leaves the fixed margin required by the reference estimate. Gauge invariance transfers that estimate back. Each pulled-back direc- tion reads one original edge or its reversed adjoint transform. Its norm is bounded by (1 + z)pθ; reversal preserves the color norm. Repeated left derivatives of the finite word substitution produce at most the 2k factor in (RB4). Finite summation over the five centered reference sums proves the displayed allowance. These are the actual finite offers used for the beacon, with their common threshold and derivative order. The qj’s in (RB4) are supplied by the finite reference construction. Put k = B.order, s = turnFee(B), h = gold(G, L, x) > 0, and Vj = |Site(2(w + j))|. Its actual allowance is qj = k! h 3k + 2Vj{s(h + h) + h}. (RB4a) The centered chart expression has bounds ∥Dlf∥≤hp−l, l≤k, at every admitted point of the chart of radius p = eaves(G, L, x), with margin 2 furl(G, L, x). The logarithm of each varying link recovers its local coordinate. If J bounds that logarithm and C is the size of a permitted left variation, the scheduled inequalities are (1 + royalty(G))C ≤1, k!J(1 + royalty(G))C ≤p. The chain rule bounds the coordinate map through order k by p, with slope one. Composition with the chart expression costs at most k!h(1 + (1 + 1))k = k!h3k. On the strict healthy chart the exact decomposition expresses this chart function as a constant minus the retained full reference sum minus the summable activity source. The source budget bounds the latter by Vj{s(h + h) + h}, through the same left-derivative order. Centering at the identity costs at most twice this number; derivatives of positive order kill the centering constant. The triangle inequality proves (RB4a). To include a healthy endpoint on the boundary of the strict chart, approach it along tP, t ↑1. This path keeps the prescribed chart margin and is eventually strictly healthy. Regularity of the retained reference function gives continuity of each left derivative at the endpoint, and hence the same bound there. No threshold indicator is differentiated. The reference estimates hold eventually in their approximation depth uniformly on the compact coupling band. The five reference volumes therefore have a common depth. The actual result after 188 ## Source page 191 A Wilson Yang–Mills construction J. Rodgers folding is gleam = 4 X j=0 |ρj| 2k(1 + z)B.power k qj. (RB4b) The radii and reference cardinalities have rate zero, and h has rate −23/20; so does this allowance. The small chart required here is supplied on any healthy neighborhood by axial gauge: its link distances are bounded by a fixed multiple of (2z + 1)5 times the threshold. The interior extension and its inverse chart place it inside radius p −2 furl. The radius has rate zero and the threshold rate −1/2, whereas p decays more slowly; the scheduling thus gives the strict margin eventually. The reference offers are indexed by the coupling itself: at a curve parameter t, the selected offer is the one at β(t). It is therefore the same offer for two presentations with the same coupling, independently of how that point was reached. The fixed-reference finite-volume estimates hold eventually in their approximation depth simultaneously on the compact coupling band; (RB4b) gives the number denoted by gleami in (FCC1). The compatibility needed above comes from the same local cylinder formulas in the different reference views. Here is also the weighted-tail step behind the retirement argument. At one finite torus all supports have cover number bounded by its site count. Thus multiplying any summable envelope by its support exponential, even the early exponent s + 3, costs only a finite constant depending on that torus. Changing its derivative anchor also costs a finite constant: K = (1 + D)pk, (RB5) where D is the maximal torus separation; the direction triangle inequality gives this bound. Conse- quently the local source estimate supplies one summable envelope in the common anchored listing. If the debris indicator of each fixed address is eventually zero, dominated convergence for series makes its weighted debris sum tend to zero at that parameter. This eventual removal is literal: a nonkept address eventually belongs to the finite retirement roster, whereas a kept address never belongs to the debris. Continuity in the deviation budget extends that small-tail bound to a parameter neighborhood. More explicitly, spend half a requested allowance on the tail at the center and half on the full source difference to a neighboring parameter, converting the latter to the common anchored weighted norm by the finite constant (RB5) and its support-weight counterpart. Restricting the difference to the debris subfamily cannot increase its positive majorant. Thus the neighborhood works for every sufficiently large depth. A finite subcover of the compact band gives a single depth. Combin- ing it with the uniform beacon depth proves the simultaneous retirement used in (FCC1). The canonical fixed-layout curve argument already given then transfers this depth to every compatible presentation; the construction here supplies its weighted continuity and actual finite beacons. Here are the reference and continuity inputs with their finite-volume dependence exposed. In this section alone n is a fixed reference side, V = n4, B is the fixed budget, k = B.order, p = B.power, and e = B.supportExponent. The five signed fourth-difference coefficients in (RB1) are denoted throughout by ρj; the λj formerly used for them in (RB4b) mean the same coefficients. They are unrelated to the parameter λ in the local fusion argument. The representation constant used here is royalty(G) = √m, where m is the dimension of the fixed faithful unitary representation. In particular it is at least one. It is distinct from the Fourier aliasing price, which depends on a strip width and an analytic norm. The reference-chart radius and margin have the explicit values p0 = (x + sG log L)−7/16, d0 = (x + sG log L)−1, h = (x + sG log L)−23/20. (RB6) 189 ## Source page 192 A Wilson Yang–Mills construction J. Rodgers The logarithm of the normalized finite Gaussian integral is centered by subtracting its value at the identity. On the reference chart ∥P∥+ 2d0 < p0, with no flags, its derivatives through the purchased order satisfy ∥Dlf(P)∥≤hp−l 0 . The lower bound on its unnormalized integral is strictly positive. These statements are obtained on the actual reference forest and its Gaussian marginal, with the same stationary section and coupling, by the Gaussian differentiation and connected estimates (ASP2a)–(ASP2c), the local integral normalization (PRI2)–(PRI5), and the logarithm series (ASP1)–(ASP2). In particular the bound concerns the centered logarithm, not the integral itself. The reference side must exceed the prescribed patch and covariance read radius. The path width, path norm, loop allowance, correction parameter and derivative horizon are fixed first; the odd block length is then above their finite lower bounds; finally x is above their common delay. Bounds for the two adjacent loop allowances and the correction parameter put both sides in these same estimates. The exact logarithmic density identity on the strict healthy chart has the form f(P) = c − X y∈Site(n) Fy(actual P) − X α Aα(actual P). (RB7) Here Fy is the retained finite ordinary reference density and Aα the emitted regular source. Their actual occurrence indices and supports are retained in the sum. The nonretained source consists of the omitted ordinary occurrences and omitted short stationary paths, not of a difference between Gaussian integrals on different tori. Its anchored early allowance is at most h once its approximation depth is large. The retained-source part has allowance at most turnFee(B)(h + h). Summing these two bounds gives turnFee(B)(h + h) + h per site. Summation over V sites and centering (RB7) gives the second term of (RB4a); the reference logarithm gives its first term after the logarithmic chart chain rule already proved there. This explains every summand in qj. We give the tail argument that supplies the omitted-source allowance used in (RB7). A regular address is a pair (oα, Xα), its anchor and finite support. For nonnegative numbers aα, its early bound u means sup o X α:oα=o aα exp{(e + 3) covB(Xα)} ≤u. (RB8) Sum over the V anchors. Since each exponential is at least one, P α aα ≤V u. This proves absolute summability of the derivative envelopes, including derivatives in any fixed finite list of directions: divide those directions by one common scalar until they satisfy the threshold bound, apply (RB8), then multiply by that scalar to the derivative order. The same scalar works for every address. Termwise differentiation on the strict healthy open set therefore follows from uniform absolute convergence of the derivative series through order k. Changing every anchor to the one in the common listing multiplies its envelope by at most Kanc = ((1 + D)p)k, as in (RB5). On this fixed torus, covB(X) ≤V, Ksup = exp{(e + 3)V } bounds every early support weight. Thus an unweighted summable envelope aα gives the summable weighted envelope gα = Kancaαe(e+3) covB(Xα). For the debris predicate Pm(α) = {α is not kept and does not belong to the depth-m finite roster} one has 1Pm(α) →0 at every fixed address: ordinary occurrences exhaust their finite shelves and short paths exhaust their length cutoffs. Dominated convergence for the nonnegative series yields P α 1Pm(α)gα →0. This proves a small early bound, rather than merely a small unweighted sum. 190 ## Source page 193 A Wilson Yang–Mills construction J. Rodgers For clarity, source deviation has the following meaning. Two sources have identical regular and forced address layouts. There are nonnegative envelopes cα, dα such that on each regular healthy domain every left derivative through order k of their difference is bounded by cα, and on every configuration every forced coefficient difference is bounded by dα. Both total weighted sums X α ee covB(Xα)cα, X α ee covB(Yα)dα are at most the deviation allowance δ. Regular directions use the anchor-dependent budget (AB1)– (AB4). Forced coefficients use their full configuration domain. The dependence on the coupling and input activities is continuous in this deviation sense. On a finite set of occurrence addresses, the local integrands and their derivatives vary continuously: the Gaussian covariance and mean, finite Haar factors, cutoffprofiles and stationary coordinates are the actual ones already constructed. Their local integral bounds (ASP2b)–(ASP2c) provide a common integrable majorant. The normalized denominator stays positive in a sufficiently small neighborhood. Differentiation and dominated convergence hence give continuity of each regular derivative supremum on its compact direction and healthy test sets; the full-domain forced bounds use the corresponding full-domain envelopes. For the infinite family, the signed connected expansion gives a common summable majorant on that neighborhood, by (ASP3)–(ASP5). Given δ > 0, first choose a finite address set whose common tail is below δ/4, then use finite continuity to make its sum below δ/2. The two tails contribute at most δ/2. This proves deviation continuity with the actual support weights. More explicitly, the dependence of the stationary path-only terms on the coupling is affine: if Aα(β) denotes one such term, then Aα(β) −Aα(β′) = (β −β′){Aα(1) −Aα(0)}. The two early envelopes at couplings zero and one give a summable Lipschitz envelope for this difference. The other connected terms are compared by the same marked interpolation before summation. If the input source discrepancy is t, their regular and forced absolute envelopes have sums bounded by tM and tQ for fixed finite M, Q, whenever 0 < t ≤t0. Signed translations cost Z = |Seat(n)| Rim(n) turnFee(B)M. Reindexing does not change a sum; multiplication by the symmetry normalization and the direction- change allowance gives exactly these factors. Converting a total unweighted envelope to a weighted one costs at most A = eeV . Thus the outgoing source deviation is bounded by t{1 + AZ + AQ}. All these constants may depend on this reference torus, budget, fixed path bounds and derivative horizon, but are independent of the approximation depth. Finally fix a parameter value and an early target u > 0. Pay u/2 for its debris tail using (RB8). Choose a parameter neighborhood with full source deviation below u/(2Ksup); then its restricted debris differences have early allowance at most u/2. The triangle inequality gives the same tail bound u throughout that neighborhood for all sufficiently large depths. A finite subcover of the compact coupling band makes the depth uniform. Applying this to the five fixed reference tori proves the common h allowance in (RB7), then (RB4a) and (RB4b), and hence the beacon term in (FCC1). No torus-uniform version of Kanc or Ksup is used. 8.18 Derivation and transfer of the reference chart estimate The reference chart comparison uses a stationary expansion on fixed reference tori. Constants at this point may depend on the chosen reference torus and finite horizon. The subsequent periodic-copy 191 ## Source page 194 A Wilson Yang–Mills construction J. Rodgers and activity arguments identify where that dependence is admissible. A finite reference calibration should therefore not be read as a uniform inverse estimate at every level. We first derive the protected-logarithm jet input. Fix the stage and derivative order before the delay. Put y = βF , u = y−7/16, m = y−1, δ = β−1/2, and D = d fineMonitor(t). Work at centers with ∥v∥+ 2m < u; eventually δ < u and the stationary radius exceeds u. Every z ∈B(v, m) then satisfies ∥z∥+m < u, which is the margin needed for the Gaussian logarithm estimates throughout this ball. Let I be the actual protected integral, V (δ, ·) its truncated Gaussian approximation with fluctuation cutoffD, and p = Ωexp(−coverLog) > 0. The Gaussian comparison on the whole closed ball B(v, m), using the vertical gap g > 0, is 0 ≤I(z) −pV (δ, z) ≤pe−gD2/8, V (δ, z) ≥a = e−1/2. (PL1) The positive omitted tail is bounded by the quadratic Gaussian tail estimate; the local comparison includes the chart and Haar factors. Here Ωis field independent, whereas p(z) = Ωe−coverLog(z) is a positive function of the center. The inequalities and the following logarithm argument are pointwise in that center. The cutoffs equal one near zero fluctuation, where the density is strictly positive, so I > 0. Its Gaussian determinant is positive by the vertical gap. Writing I = pV (1 + q) gives 0 ≤q ≤e−gD2/8/a. Hence the logarithmic discrepancy f = log I −log Ω+ coverLog −log V satisfies |f| ≤a−1e−gD2/8, (PL2) by 0 ≤log(1 + q) ≤q. This is only a value estimate. We record how the actual integral comparison in (PL1) is obtained, so that its positive error and its domain are explicit. On the fixed forest fiber let Uz be the set represented by the fluctuation chart U = draw(Vz, μ(z) + δi), ∥i∥≤D, where μ is the stationary graph. The chart radius inequalities are (1 + trailCost)u + δD < initial and the corresponding color-scaled displacement is below the group-chart radius. On this set all protecting flags vanish. The exact change of variables, including its Haar Jacobian and the vertical Gaussian determinant, gives Z Uz Iz(U) dνF (U) = p(z)V (δ, z), I(z) = p(z)V (δ, z) + Z Ucz Iz(U) dνF (U). (PL1a) Here Iz is the nonnegative summed fiber density used in the exact push. This is why the comparison has a one-sided error. The outside-field energy bound is paid in the same chart. Denote by k the number of free bonds, by wg(β) > 0 the single-bond chart normalization, by vC the common finite size bound, and by K0, T0 respectively the local derivative and determinant expenses. With RD = gD2/4, its normalized logarithmic expense is YD = −k log wg(β) −RD + vC loopsPrice + K0 + T0. (PL1b) The fiber complement estimate is R Ucz Iz ≤p(z)eYD. It uses the local link reconstruction to put every small-toll, small-wire configuration in the safety chart. Otherwise its fine or prediction penalty pays at least the threshold energy. The parameter inequality β meter(vC)u + RD ≤βppenaltym2 0/2 makes that alternative at least as expensive as the fluctuation tail; m0 is a fixed positive safety threshold, not the shrinking radius m. The quadratic lower bound within the chart pays the other alternative. Both comparisons use the same actual field and forest fiber. 192 ## Source page 195 A Wilson Yang–Mills construction J. Rodgers The delay makes YD ≤−RD/2, and also the stronger inequality YD + E + 2 ≤−RD/2. Here the extra expense used to locate a clean chart point is E = β{vCCcl∥z∥5 + vCC′(θ′3 + θ′4)} + K0 + T0 + vCAcoarse/2. The two terms multiplied by β and the loop expense are eventually bounded by one: the field radius has rate −7/16, the threshold has rate −1/2, and the remaining size factors have rate zero. The nondecaying expense K0 is affine in the fine monitor, K0 = a(t) + b(t)fineMonitor(t), with a, b bounded by the already fixed polynomial size budgets. The normalization and determinant expenses have the same type of bound. The square RD = (gd2/4)fineMonitor(t)2 dominates this fixed finite list, after the stage parameters and positive d are fixed. This proves both inequalities. The stronger one, together with the positive lower bound for the total push, forces a clean point inside Uz; the local chord and displacement estimates then keep all chart flags zero on the surrounding radius-m ball. Thus (PL1a) applies throughout that ball. The first inequality and (PL1b) give its error e−RD/2 = e−gD2/8, as claimed. The truncated Gaussian lower bound and its jet estimates use the same size and radius choices. Their derivation is the local Gaussian expansion above; no full-field lower bound for an inverse density is used here. The Gaussian approximation uses the actual functions constructed above. Keep the retained maximum-norm real coordinates x, and give the vertical real coordinate array its Euclidean norm, denoting it by h. The linear map L(x, h) that joins these arrays into the joint maximum-norm array has norm at most one. In this notation set φ(x, h) = F(L(x, h)), ψ(x, h) = loop(U(x, h)) − X e free log JG(tint(h)e). (GL0) Here F is the main-action-plus-prediction score (AS1), U is its actual fine configuration, loop is the retained quadratic-loop contribution, and JG is the local Haar density of the chart in the earlier primitive integral. The minus sign makes e−ψ contain the positive Haar factor. Let ρ(x) be the vertical Euclidean version of the unique graph (US7), and let M(x) be its Hessian in these vertical coordinates. The equality of counting and Euclidean color squared norms proves Dhφ(x, ρ(x)) = 0, D2 hφ(x, ρ(x))[z, z] = zTM(x)z, M(x) ≥(g/2)I. (GL0a) Thus the inverse covariance bound used below is M(x)−1 ≤(2/g)I; it is the retained half-gap of the actual stationary Hessian. We explain the common derivative hypotheses for these functions. The joined points used below satisfy ∥L(x, ρ(x) + tbz)∥≤Rpath∥x∥+ |b|D, 0 ≤t ≤1, ∥z∥≤D, where Rpath is the stationary Lipschitz price, enlarged to include the retained identity coordinate. The radius inequalities before (PL1a) put this whole set inside the purchased stationary domain and put every color logarithm inside half the Haar-chart radius. The graph jets are (US8). The score jets follow from (AS6) after the norm-at-most-one map L; converting coordinate multilinear bounds to operator bounds costs at most the number of joint coordinates to the derivative order. The loop jets follow by summing the absolutely weighted path jets, multiplied by the fine-site count. A Haar summand is the fixed analytic log JG composed with a color projection of norm at most 1+dim g, so its order-k price is at most its purchased chart price times (2+dim g)k; summing costs the free-bond count. The centered determinant jets required below are controlled on this same domain. Differentiate D log det M = tr(M−1DM) and D(M−1) = −M−1(DM)M−1. Repeating gives a finite sum of 193 ## Source page 196 A Wilson Yang–Mills construction J. Rodgers traces of products of inverse factors and derivatives of M, indexed by ordered derivative partitions. Each trace is bounded by the vertical dimension times the product of the operator norms. The inverse factor has bound 2/g, and derivatives of M are obtained from two score derivatives and the graph chain rule, all at the already purchased higher orders. This gives a finite polynomial price in those jet bounds, the coordinate counts and 2/g. The value of the centered determinant logarithm follows by integrating its first derivative from zero. These operations give the common price C for the graph, classical score, quantum contribution and centered half-log-determinant used below. Its dependence on the auxiliary size bounds is polynomial at fixed derivative order; those sizes have the assigned polynomial logarithmic bounds. Consequently C has rate zero. This establishes the hypotheses of the local Gaussian calculation on the entire displaced set, using the actual score, chart and graph. Define the Taylor-integral expression A(b, x, z) = Z 1 0 (1 −t)D2 hφ(x, ρ(x) + tbz)[z, z], dt. (GL1) Stationarity and twice integrating a line derivative give φ(x, ρ(x) + bz) −φ(x, ρ(x)) = b2A(b, x, z), including b = 0 by the displayed integral. In particular A(0, x, z) = 1 2zTM(x)z. Put G(b, x, z) = 1 2 log det M(x) −1 2 log det M(0) + A(0, 0, z) −A(b, x, z) + ψ(x, ρ(x)) −ψ(x, ρ(x) + bz). (GL2) Then the same truncated approximation in (PL1) is V (b, x) = Z ∥z∥≤D eG(b,x,z)dγM(0)−1(z). (GL3) At b = 0, multiplying its fixed Gaussian density by the exponential in (GL2) gives exactly the density of γM(x)−1. Hence V (0, x) = γM(x)−1(B(0, D)). (GL4) This identity retains the moving determinant and covariance; replacing either by its value at zero would lose the derivative argument. Here is a uniform estimate for the mass in (GL4). If a centered Gaussian covariance Q satisfies Q ≤cI, its one-coordinate exponential moment gives P(|Zi| ≥v) ≤2e−sv+cs2/2 for s ≥0: apply Markov’s inequality to e±sZi and add. If the number of coordinates is at most P ≥1, and Pv ≤D, the complement of the radius-D ball is contained in the union of those coordinate events. Indeed, if every |Zi| < v, then ∥Z∥≤ √ Pv ≤Pv ≤D. Thus its probability is at most 2Pe−sv+cs2/2. Take P = 1 + Q0(t), where Q0 is the fixed polynomial size budget, and write H = fineMonitor(t). Eventually P ≤ √ H. For D = dH, choose v = d √ H and s = v/c. The error is at most 2P exp(−d2H/(2c)). (GL5) The monitor is chosen to dominate these polynomial size budgets and logarithmic coupling powers, so (GL5) is smaller than ya for every fixed exponent a, after increasing the delay. This estimate is uniform for the moving covariances, since the inverse-Hessian bound gives one common c. For completeness the derivative price in (GL3) is finite without any derivative of a moving integration set. At order s, put Us = 2(1 + C + D), As = D2Bs(Us, C), Fs = 2C + 2As + 2Bs(Us, C). 194 ## Source page 197 A Wilson Yang–Mills construction J. Rodgers Here C bounds the purchased section, score, quantum and centered determinant jets, with two extra score and section orders for (GL1). The affine displacement map in (GL1) has price Us; evaluating its second score derivative on two copies of z gives the factor D2. Integration against 1 −t, followed by the sum in (GL2), bounds all jets of G by Fs. Since G(0, 0, z) = 0, its first derivative also gives |G(b, x, z)| ≤Fs max(|b|, ∥x∥). The shrinking chart makes this at most one. Consequently e−1 ≤eG ≤e on the fixed cutoffball. Its base Gaussian mass is eventually at least one half by (GL5), so e−1/2 ≤V (b, x) ≤e. The exponential composition rule bounds its jets by Bs(Fs, Es); the fixed Gaussian measure is a probability, so integration preserves this bound. The logarithm on the compact interval [e−1/2, e] has a finite fixed jet price Ls, giving the joint (b, x)-jet bound B(t) = Bs(Bs(Fs, Es), Ls). All these prices have rate zero. Finally (GL2) is unchanged by simultaneously replacing (b, z) with (−b, −z). Symmetry of the cutoffball and base Gaussian makes V (−b, x) = V (b, x), and therefore its logarithm is even in b. Each of its x-derivatives has zero first b-derivative at zero. Integrating its second b-derivative bounds the difference from its b = 0 value by B(t)b2, through order j, using the purchased joint order j + 2. At b = 0, (GL4)–(GL5) and | log v| ≤2(1 −v) for 1/2 ≤v ≤1 give an arbitrarily small polynomial value error on the surrounding ball. Apply the interpolation (PL4) below with a rough joint bound B(t) ≤y and a sufficiently small chosen value exponent. This bounds the x-jets of log V (0, x) by y−2. Combining the even difference and this zero-parameter estimate gives the asserted Gaussian logarithm jet price B(t)δ2 +y−2 at b = δ, as well as its lower bound. This proves the Gaussian input used in (PL1) and (RBG2). The separate rough derivative estimate for the discrepancy on that closed ball is, for a requested order j, sup z∈B(v,m) ∥Dif(z)∥≤yS0 (0 ≤i ≤j + 1), S0 = (j + 1)2 + 2. (PL3) For this rough estimate one can avoid estimating an inverse integral on a whole chart. We give the relative derivative bound used in the construction. Write the exact push as an integral over a fixed forest fiber of a finite sum of nonnegative functions I(x) = Z X α aα(U)e−βNα(x,U) dνF (U), aα(U) ≥0. (PL3a) The coefficients include the fine density and all factors independent of the retained coordinate x. The only x-dependence here is the prediction penalty Nα, with the fiber configuration fixed. In particular this argument does not require a nearby point to retain a selected protecting flag. Let r bound the number of coarse bonds. The link product and trace estimates in (AS3) give the penalty jet price AN = ppenalty r P(2non colorScale). The predictor at fixed U is a constant group element of matrix norm colorScale; hence this price is uniform in U and the finite choice α. For clarity write the composition polynomial as Bn(a, b) = n!b(2 + a)n. It bounds the finite partition sum for composition when the inner and outer jets have respective prices a, b at scale one. Subtracting the value of −βNα at the center costs a factor two in its jet bound. If En bounds the exponential jets at zero, each integrand in (PL3a) therefore has jets through order n bounded by its own value times Kn = Bn(2|β|AN, En). 195 ## Source page 198 A Wilson Yang–Mills construction J. Rodgers Integrating preserves this relative bound. Differentiation under the integral follows from the same estimate on a neighborhood together with the bounded fine-density envelope and the finite Haar fiber measure. Consequently ∥DiI(x)∥≤KnI(x) for i ≤n. At each center where (PL1) gives I(x) > 0, normalize by this single positive value: Jx(z) = I(z)/I(x). Its jets at z = x are bounded by Kn, and Jx(x) = 1. If Ln bounds the logarithm jets at one, the chain rule gives jets of log I(z) −log I(x) = log Jx(z) bounded by Bn(Kn, Ln). The size of I(x)−1 has canceled before estimating. This is the normalization that supplies rough logarithm derivatives uniformly over the surrounding ball. The other terms of f have explicit prices as well. If C(t) is the stationary chart-jet price, set Tn(t) = Bn(2 + 2C(t), C(t)), On(β, t) = βTn(t) + Tn(t) + C(t). Composition of the classical action, quantum action and Gaussian determinant on the stationary graph bounds the centered cover logarithm by 2On. Let w(t) be the rate-zero Gaussian-log coeffi- cient and put Q(t) = |w(t)| + y−2; since δ ≤1, the Gaussian logarithm jet price is at most Q(t). Combining the centered logarithms and then restoring the value bound (PL2), which is eventually at most one, gives the rough price M(t) = Bn(Kn(2y, r(t)), Ln) + 2On(2y, t) + 2Q(t) + 1, n = j + 1. (PL3b) The actual coupling lies below 2y, so positivity and monotonicity of these polynomials justify that substitution. The bond-count bound r(t), chart coefficient C(t) and w(t) have rate zero. Thus AN has rate zero, Kn has rate n, and Bn(Kn, Ln) has rate n2. The cover term has rate one and the Gaussian term rate zero. Their sum has rate at most S0 −1 = n2 + 1, and hence is eventually at most yS0. This proves (PL3), including the zeroth derivative by (PL2). It uses actual integral derivatives and the surrounding ball, rather than differentiating the small value error. Here is the interpolation and its order of choices. If ∥Dih∥≤vi on a ball of radius s and ∥Di+2h∥≤C, integration along unit directions gives ∥Di+1h(0)∥≤2vi/s + Cs. (PL4) For multilinear derivatives one tests unit vectors before applying this line estimate. Suppose rough derivatives through order j + 1 are bounded by yd on a ball of radius y−a, a > 0. For any target exponent b, a sufficiently small value bound yB on that ball implies every jet through order j at its center is bounded by yb. At order zero take B = b. For the step from order i to i + 1, choose c = a + max(0, d −b) + 1, b′ = b −1 −c, then use the order-i assertion with target b′ and ball exponent c to determine B. Eventually 2y−c ≤y−a, so the smaller balls around points within y−c of the center lie in the original ball. The induction gives ∥Dih∥≤yb′ there. Formula (PL4) gives 2yb′+c + yd−c, whose two exponents are strictly below b; their sum is eventually at most yb. Lower jets already satisfy the stronger bound. Apply this recursion with a = 1, d = S0, b = −2, j = n. It fixes B first; then choose the tail accuracy and delay so that (PL2) is at most yB. Thus the discrepancy jets are bounded by y−2. Gaussian differentiation bounds the separate local Gaussian logarithm jets by H(t)δ2 + y−2, with H of rate zero. Keeping that term in L0 leaves the discrepancy Y in (RBG1). Since u ≤1, max i≤n ui∥DiY (v)∥≤y−2. (PL5) This includes v = 0. The critical extension and actual chart composition use the already purchased higher jets and finite composition prices. Those prices are fixed before the tail and reference delays. 196 ## Source page 199 A Wilson Yang–Mills construction J. Rodgers This supplies the input to the rate calculation below and hence (RB6); the retirement argument then uses the resulting bound with its existing delay order. We supply the quantitative calculation behind (RB6). Put y = x + sG log L, A = y−7/16, and ε = y−2. All derivative orders and stage parameters in this calculation are fixed before x tends to infinity. A function of rate r is eventually bounded in absolute value by yr+η for each η > 0. Products add rates, a finite sum has the largest of its rates, and multiplication by any fixed constant preserves a rate. Thus a rate strictly smaller than −23/20 gives the bound in (RB6), with coefficient one, after increasing the delay. Write the actual reference logarithm, before centering, as g = log Ω+ L0 + Y + βf1 + W. (RBG1) Here Ωis the field-independent Gaussian normalization; Y is the logarithm of the normalized protected local integral after its Gaussian main part is removed; L0 is the remaining Gaussian normalization term; f1 is the difference between the prescribed coarse classical action and the fine classical action evaluated on the stationary graph; and W is the difference between the prescribed coarse quantum action and the thermal determinant and Haar contribution on that graph. These are functions on the same real primary-coordinate space. In particular (RBG1) is an equality of the actual densities before any estimate is taken. For a function F, use the scaled derivative bound ∥F∥n,A;z = max 0≤q≤n Aq∥DqF(z)∥. The protected-integral comparison and interpolation (PL1)–(PL5) give ∥Y ∥n,A;z ≤ε, including at the origin. The derivative price of L0 on the radial segment from zero to z is d = X(x)gstep(β)2 + ε, gstep(β) = β−1/2, (RBG2) with X of rate zero. The integral itself is positive: the Gaussian measure is positive, the protected integrand is nonnegative, and on a nonempty open neighborhood of the zero fluctuation all pro- tecting cutoffs are one and the exponential density is strictly positive. The Gaussian has positive density there. Its normalizing Gaussian determinant is positive by the vertical Hessian gap. Hence its real logarithm is defined on the chart. The estimates for its derivatives follow by differentiating its absolutely dominated integral and using the logarithm expansion established in (ASP1)–(ASP2). Let M ≥1 bound the reference coordinate and site counts, let H be the stationary-response price, and let Kn be the sum of the purchased bare-action jet price and the stationary-composition price. More explicitly, the coordinate conversion multiplies an order-q multilinear price by Mq; the latter composition price is the fixed polynomial clumpn(1 + H + (1 + H))H appearing in the ordered-partition derivative bound. Thus Kn has rate zero. Classical matching removes the jets through order four, giving ∥β(f1 −f1(0))∥n,A;z ≤βKnA5. (RBG3) The higher jets of W, of orders three through n, have a price Cn of rate zero. This price is the bare jet bound multiplied by Mn(1+|α|+Λ), together with the same stationary-composition polynomial and H. Here α is the bounded kinetic correction and Λ is the path allowance. The finite-period thermal comparison bounds its second jet at zero by ∥D2W(0)∥≤A0, A0 = M2{M4m(g)M} Θp e−γc. (RBG4) In this formula m(g) = (1 + dim g)2 is the fixed color-coordinate conversion factor. The reference side is c, γ, Θ > 0 are the purchased decay constants, and p ≥0 is bounded by the adjacent loop allowance. Only the displayed exponential dependence on c is used below. 197 ## Source page 200 A Wilson Yang–Mills construction J. Rodgers The first jet is smaller than a generic extensive-action bound would suggest. The signed symme- tries give Dg(0) = 0, while classical matching gives Df1(0) = 0. Differentiating (RBG1) therefore yields DW(0) = −DL0(0) −DY (0), ∥DW(0)∥≤d + ε. (RBG5) Taylor’s formula along segments in the chart, with these first and second jets and the third-and- higher jet prices, gives ∥W −W(0)∥n,A;z ≤CnA3 + 3{(ε + d)A + A0A2}. (RBG6) For orders zero, one and two this follows by integrating the next derivative and using (RBG4)– (RBG5). For an order q ≥3, multiply the unscaled bound CnA3−q by Aq. Taking the maximum and enlarging the three low-order contributions gives the factor three. Centering Y costs at most 2ε; centering L0 costs dA. Equations (RBG1), (RBG3), and (RBG6) now give ∥g −g(0)∥n,A;z ≤dA + 2ε + βKnA5 + CnA3 + 3{(ε + d)A + A0A2}. (RBG7) Here y/2 ≤β ≤2y. Consequently d has rate −1. The lower bound on the reference side makes Θpe−γc, and hence A0, have rate −1. All remaining prices have rate zero. The respective rates on the right of (RBG7) are at most −23/16, −2, −19/16, −21/16, −23/16, −15/8. Every one is strictly below −37/32, and −37/32 < −23/20. The defining rate inequalities, with a sufficiently small positive slack and then a common delay, therefore make the whole finite sum at most y−23/20. This derives the quantitative gain rather than assuming that a small integral automatically has a small logarithmic jet. It remains to explain why the estimate applies to the five reference tori, whose sides need not be physical periods. Choose a physical coarse side divisible by the given reference side; the divisibility requirement was included in the block calendar. Copy the reference forest periodically to this larger torus. The uniform stationary construction gives a critical section on each torus. On the larger torus the newly constructed section and the retained physical section have the same zero value and solve the same critical equation in the same uniqueness corridor. They agree near zero. Their tested classical jets and their thermal second jets therefore agree there. The periodic-copy comparison transfers classical matching and the finite-period Hessian error to the reference section. Its decay exponent can be replaced by the minimum of the two positive exponents, and its price by the sum of the two prices. This is sufficient for (RBG4). No equality of the Gaussian integrals on different tori is required. On a fixed reference torus, differentiation along the logarithmic link chart costs the factorial and direction factors already displayed in (RB4a)–(RB4b). The omitted-source allowance comes from (RB8), uniformly in the compact coupling band. The same normalized offers depend on the coupling value, so substitution of the actual coupling curve preserves these estimates and their continuity. Summing the finite-site logarithmic and omitted-source bounds gives the reference offer bound used in (FCC1). This completes the passage from the actual stationary integral to the retirement beacons. The uniformity required in this last assembly can be made explicit. For a fixed transition between stage profiles, the local argument is purchased at derivative order n + 6 for the stationary score and at the indicated higher orders for the bare-action and Haar jets. Its conclusion uses order n ≥2. The thermal comparison is first known for the retained critical section. A newly selected uniform analytic stationary graph is an extension of this section: the two solve the same 198 ## Source page 201 A Wilson Yang–Mills construction J. Rodgers vertical equation, and uniqueness near zero identifies them. Therefore their classical matching and finite-period second-jet comparison are the same tests, with the enlarged common price and the smaller common positive width. This establishes the hypotheses of (RBG3)–(RBG5) for the graph used in (PL1), rather than replacing that graph by a different critical solution. For each fixed incoming stage only finitely many outgoing transition tests are required. Take the maximum of their coupling delays. Enlarging the calendar’s delay preserves all the estimates, since they hold for every coupling above that delay. At an actual calendar step the identities for the block length and next coupling identify its local parameter with y = βF (t, L). The two adjacent loop caps are bounded by the same local loop allowance, and the kinetic correction satisfies |α| ≤drift + |sG log L| + 1. The assumed stage bounds on paths and loops therefore place both sides of this actual step in the local comparison just proved. This accounts for the order of choices and for the fact that the estimates hold for all allowed coupling values in the compact band. Finally put s = y−23/20, let q be the derivative order of the current budget, and let T be its fixed conversion price for left derivatives. After retiring the source tail to budget s, the centered retained source on any one reference torus of side ci has left derivatives bounded by Fi = q! 3qs + 2c4 i {T(s + s) + s}. (RBG8) The first term is the finite composition bound from the logarithmic coordinates to the prescribed left derivatives. The other terms pay the two source differences and the finite sum over sites; their factors are positive, so restriction to the omitted subfamily does not enlarge them. The retirement depth is common over the compact coupling band by (RB5)–(RB8), and common over the finite reference tori by taking its maximum. Each reference side is chosen above the read radius, and its finite joint-coordinate, fiber and site counts have the same preassigned size bound. No constant in this step is chosen from the later macroscopic volume subsequence. If the fixed interpolation stencil has signed weights ai, its support reach is e, and the budget support power is p, transporting these local estimates to one insertion gives X i |ai| 2q(1 + e)pqFi. (RBG9) Indeed the directional substitution costs 2q, the support enlargement costs (1 + e)pq, and the finite sum is bounded with absolute weights. Smooth dependence of each reference offer on its coupling allows substitution of the actual continuous coupling curve. Formula (RBG9) is the bound for the reference offer in (FCC1). It is on the same ordinary source and common finite stencil as the retirement construction, and supplies the needed passage from the local protected integral to its assembled source bound. 8.19 The local activity norm and analytic comparison estimates The norms in finite calibration are the paid activity norms, not the uniform norm of the resulting density. Fix the current budget P and a common address layout. Let bP , kP > 0 be its regular and forced normalization allowances. For an activity pair A = (E, K), define ∥A∥P = inf{t ≥0 : E has anchored regular derivative budget tbP , K has forced hit budget tkP }. (LCN1) The regular budget tests every left derivative through P’s order, in its permitted directions, on the healthy domain of that address, with its exponential support weight. The forced hit budget takes the 199 ## Source page 202 A Wilson Yang–Mills construction J. Rodgers absolute bound on every configuration and sums over addresses hitting each site. Both include the fixed locality and invariance conditions. The infimum is attained: take the least supremum envelope for each regular derivative family and each forced coefficient, and pass every finite nonnegative partial-sum inequality to the infimum. Hence ∥A∥P ≤t is equivalent to the two displayed budgets with that exact t. Triangle inequalities for derivatives and coefficients give subadditivity; scaling gives ∥cA∥P = |c|∥A∥P . Fix, for each signed translation, one permutation implementing that symmetry on the layout. The activity pairs obeying all these fixed permutations form a linear subspace HP . Its seminorm is (LCN1). Use its separated completion bHP ; the map from the original presentation is an isometry after identifying seminorm-zero differences. This retains the off-mask regular germs required by the derivative budget, rather than identifying them merely by their evaluated values outside the healthy domain. The scalar-plus-activity space has norm ∥(λ, A)∥= max(|λ|, ∥A∥P ). The actual retained activities have norm at most 1/8 in these coordinates. We explain how the local primitive estimates give an analytic estimate in this norm. For a finitely supported signed measure w = P j cjδtj on [−1, 1], say it has moment price r at radius p ≥1 if X j cj(tj/p)n ≤r for every n ≥0. (LCN2) A family f(t) has comparison price C at radius p if every such measure satisfies ∥P j cjf(tj)∥≤rC. This is the precise signed-combination bound used here. If f(t) = P n(t/p)nan with P n ∥an∥≤C, the bound follows by interchanging a finite sum and an absolutely convergent series. Conversely, define the moment vector ((t/p)n)n in l∞. The signed-combination bound defines a bounded linear map on their span and then its closure. Coordinate vectors belong to that closure: subtract the already recovered lower coordinates from the moment vector at a small nonzero t, divide by (t/p)n, and let t →0. Its image supplies the coefficients. Restricting from radius p to p′ < p sums the geometric coefficient majorant and gives an absolutely convergent series. This is the recovery step used to turn signed-combination bounds into analytic line estimates. On an affine activity line A + tB, regular occurrences and forced coefficients are affine in t on the same configuration: their masks and inventories are fixed. The finite calibrated coupling along the line is β(t) = Ω[Ω−1/32(λ + tu)]2, with |u| ≤1. The comparison center has Ω−1/32λ ∈ [33/40, 47/40], and the climate bounds ensure the whole line stays in the allowed coupling band. Assume ∥A + tB∥P ≤1 for |t| ≤1. The primitive’s signed regular and forced estimates, followed by the Hessian projection and the fixed dictionary fusions, give prices output activity: d at radius q/2, scalar correction: fi at radius q, fi = choresi /θ2 i . (LCN3) The regular signed bound is tested in the anchored derivative budget. The forced signed bound has one nonnegative summable hit-envelope common to all line parameters and configurations. For any weights satisfying (LCN2), these two envelopes give budgets rd bP ′, rd kP ′ on the actual output layout, so (LCN1) gives the activity price in (LCN3). The scalar price follows by applying the linear Hessian reading to the same signed regular source estimate and dividing its paid bound by θ2 i . This explains both the norm and the threshold denominator in (LCN3). The primitive locality, derivative and signed-sum bounds used in this step are the marked integral and source projection estimates of the preceding chapters; no assertion about analytic dependence of a chosen forest or local guide is required. We make explicit the gain that pays (LCN3), since an individual bound on each line value would not imply the signed-combination bound. Use the 301 distinct interpolation nodes rj = j/301, 200 ## Source page 203 A Wilson Yang–Mills construction J. Rodgers 0 ≤j ≤300, in [−1, 1], and their Lagrange polynomials Lj. If w has moment price η at radius q, let cj = P t wtLj(t) and w0 = P j cjδrj. The coefficient bounds of these finitely many polynomials give P j |cj| ≤Cqη for one fixed Cq ≥0. Every polynomial of degree at most 300 is interpolated exactly, so w −w0 annihilates its moments through that degree. If V ≥q, its remaining moment prices at radius V are bounded by η(1 + Cq)(q/V )301: factor (q/V )n from the degree-n moment and use n ≥301. Thus a family with signed price B at radius V and individual value bound b has signed price at radius q at most Cqb + (1 + Cq)(q/V )301B. (LCN8) This is valid for the regular derivative budget as well as a scalar norm, because finite sums, absolute scalar multiplication and envelope sums preserve that budget. We derive the preliminary large-radius signed price used in (LCN8). Fix the stage, block ratio and address layout. Write y = βF (x, L), φ = y1/256, B = B(x), and let A > 0 be the fixed turning price. We first explain the parameter estimate on each vertex. A function polynomial in the line parameter t, of degree at most two, is reconstructed exactly from its values at −1, 0, 1: f(t) = 1 2t(t −1)f(−1) + (1 −t2)f(0) + 1 2t(t + 1)f(1). (LCN11) The sum of the absolute coefficients of these three Lagrange polynomials is 4. If each nodal function has all required spatial derivatives bounded by vdm at derivative order m, (LCN2), applied to each parameter power, bounds its signed interpolation by 4p2vdm at radius p ≥1. The same identity holds after spatial differentiation: its parameter coefficients are fixed scalars. Thus this is a signed derivative bound, with no factor P t |wt|. Logarithmic vertices enter as ef(t) −1, rather than f(t). If their nodal derivative budget is at most v ≤1/2, first recover the budget of f using f = log(1 + (ef −1)). The scalar value bound is |f| ≤2v; all derivatives of log(1 + z) through the fixed order are bounded on [−1/2, 1/2]. Its finite chain-rule sums therefore give a bound bjv with the same spatial derivative scale, for a constant bj depending only on that order. Apply (LCN11), and then the absolutely convergent exponential series in the signed moment norm. Multiplication in that norm multiplies prices, so a log price s ≤1 gives es −1 ≤(e −1)s; the differentiated exponential is controlled by the finite chain-rule sums at the same scale. Consequently there are constants Aj ≥0, Bj ≥1 such that the signed price of this vertex is at most Ajp2v, provided Bjp2v ≤1. These are the estimates FullVolume/Ray.Badge.starpath and White.faded, applied at the three displayed nodes. They apply to the actual vertices because the line dependence is placed in the couplings and occurrence amplitudes, not in a newly chosen forest. With the spatial configuration held fixed, an old vacuum logarithm is constant or is multiplied by the square of the affine coupling-amplitude line, hence is quadratic in t. A regular occurrence has the prescribed degree-at-most-two ampli- tude representation on this line; multiplication by its fixed local enforcement factor preserves that degree. The forced occurrence has the same representation. Warning vertices, including their com- mon steering and course variables, have the retained parameter-independent derivative envelope and therefore satisfy the constant-function signed estimate. In the differentiated formula the back- ground, steering and Gaussian-coordinate variables are varied together as specified in the common chart. The spatial chain-rule prices are included in the nodal bounds. These are the four cases in FullVolume/Sector/Grant.mural, lured, loglift and alerts. In particular polynomial interpolation is applied to the log before exponentiation, not to the exponential. The preliminary signed price now follows by the connected expansion with these vertex prices. Three nodal log envelopes sum to at most 3q, where the old target q has rate at most −7/16. At p = φ, their enlarged price has rate at most −7/16 + 2/256 = −55/128 < −2/5. 201 ## Source page 204 A Wilson Yang–Mills construction J. Rodgers Thus both 3q ≤1/2 and Bjφ2(3q) ≤1 hold eventually, and the reweighted regular and log prices fit the target of rate −2/5. This is the explicit target enlargement in FullVolume/Sector/Reign.shows. The finite-order product rules multiply these prices at each connected tree. The support exponent exceeds the final exponent by more than three; split this excess into three positive reserves to pay the tree-degree factors, collar enlargement and log attachments. The positive tree sums have ratios at most 1/2, hence cost at most 2 each. All remaining fixed-order spatial and collar prices have rate zero. Multiplying them by the enlarged forecast of rate −2/5 is eventually at most y−3/8. Restricting the moment radius to φ/2 improves the bound. The forced entries retain a common nonnegative hit envelope by the same positive majorants. This establishes the preliminary signed estimate for every finite combination satisfying (LCN2); it does not infer it from individual activity bounds. Here are the product and summation steps in that signed estimate. A radius-R series certificate means an actual representation on [−1, 1], f(t) = X n≥0 (t/R)nbn, X n ∥bn∥≤A, R ≥1. (ASP1) For a finite signed combination w = (wt), put Λwf = P t wtf(t). If |P t wt(t/R)n| ≤η for every n, absolute convergence gives ∥Λwf∥≤ηA. This estimate uses cancellation of the combination; replacing wt by |wt| would lose it. Products preserve (ASP1) with the product allowance, by the absolutely summable Cauchy product. Summable families preserve it by the double-series theorem. Exponentiation gives allowance eA−1 for ef −1, by summing the positive scalar majorant P q≥1 Aq/q!. For the simultaneous spatial derivative estimates, fix a base coarse field x. Suppose each diagonal derivative through order j, in a unit spatial direction v, has (ASP1) allowance Abq at order q. If another factor has allowance Cdq, the derivative of their product is q X r=0 q r ! Dr vf Dq−r v g. The product rule for the series certificates bounds its allowance by q X r=0 q r ! AbrCdq−r = AC(b + d)q. (ASP2) Induction gives, for a finite product of vertices, allowance Q Ai and slope P bi. Thus the signed cancellation is retained before multiplying the vertices. After applying Λw, polarization bounds the full symmetric q-linear derivative by polarCost(q)ηAbq. Taking P q≤j polarCost(q) as one common constant controls all orders through j. Finite linearity commutes Λw with these derivatives. Here the polarization constant used by the source is explicitly Pq = 2q(q + 1)q, P≤j = j X q=0 Pq. (ASP2a) For a symmetric q-linear form A and vectors ∥vi∥≤1, expand X S⊆{1,...,q} (−1)q−|S|A X i∈S vi, . . . , X i∈S vi ! = q! A(v1, . . . , vq). 202 ## Source page 205 A Wilson Yang–Mills construction J. Rodgers Every term whose list omits an index cancels by the alternating subset sum; a surviving list contains every index exactly once, giving q! identical terms. If the unit diagonal bound is C, each subset term is bounded by (q + 1)qC, by homogeneity. There are 2q terms and q! ≥1. This proves the stated constant, including q = 0, and normalization of the nonzero vectors proves the operator norm bound. Applying it after the signed parameter combination gives the allowance P≤jηA at all orders through j. It uses no bound depending on the dimension of the spatial chart. The Gaussian integration prices entering these allowances also have a direct finite-dimensional proof. Let γΣ be the centered Gaussian actually used for a packet, with Σ ≥0 and |Σii| ≤B. The one-coordinate marginal gives Z etzi dγΣ = et2Σii/2 ≤eBt2/2. Thus (1 + |zi|)q ≤eqzi + e−qzi implies R (1 + |zi|)q dγΣ ≤2eBq2/2. Independence of the coordinates is unnecessary. For a local edge set I, let J be its colored coordinate set, a = |I|, b = |J|, and FI(z) = 1 + a + b + X i∈J |zi|, Y = 1 + a + 2b. Choose a coordinate attaining the maximum when J ̸= ∅; then FI(z) ≤Y (1 + |zi|). The empty set case follows directly. Consequently Z FI(z)q dγΣ ≤Y q{1 + 2beBq2/2}. (ASP2b) This proves integrability as well as the bound, by domination by a finite sum of the two exponential moments. A tilt changes only the marginal on J. Its covariance ΣJ is positive definite. With c ≥1, its exact density multiplier is Mc(z) = cb exp 1 −c2 2 zT J Σ−1 J zJ ! , 0 ≤Mc(z) ≤cb, FI(cz) ≤cFI(z). (ASP2c) This is obtained by the substitution yJ = czJ in the Gaussian density; the determinant con- tributes cb. The quadratic form is nonnegative, proving the bound. The remaining scalp factor and this multiplier combine exactly into the original scalp times cb times the exponential of the tilt’s quadratic cost. Applying the product and exponential estimates to that equality gives derivative price charge(j)(1 + δ)2b, with spatial slope bounded by L′2KFI(z)K. Formula (ASP2b) pays its moments. Finally a warning event E with probability at most r2m and a polynomial factor with second moment at most A2 contribute at most Arm, by Cauchy–Schwarz. Haar variables have total mass one and do not change this bound. All polynomial moment prices are included by taking q twice the required derivative degree. If the colored coordinate count is bounded by 2b ≤2 covB(Y ) ballCost(B) density(L) dim g and the tilt margin satisfies 2δ ballCost(B)density(L) dim g ≤α, then log(1+δ) ≤δ gives (1+δ)2b ≤ eαcovB(Y ). This is precisely the reserved cover weight; the Gaussian determinant factor is paid before the final summation. The interpolation step preceding this argument supplies those certificates at the real nodes. Explicitly, at k + 1 distinct nodes ti, let li be the Lagrange basis and Bk = P i Pk r=0 |[tr]li|. A 203 ## Source page 206 A Wilson Yang–Mills construction J. Rodgers degree-k family with nodal derivative allowance Abq has (ASP1) allowance BkRkAbq: expand li(Rz) and bound Rr ≤Rk. For optional exponential vertices the nodal hypothesis is on ef(ti) −1. When its allowance is at most 1/2, log(1 + y) has bounded derivatives on [−1/2, 1/2] and | log(1 + y)| ≤ 2|y|. The finite chain rule therefore bounds the nodal logarithm through the same spatial order. Interpolate this logarithm, and only then exponentiate by (ASP1). The condition BjR2(3q) ≤1, with its fixed derivative constants included in Bj, pays this last exponential step. This explains why the source can use polynomial dependence of the log without asserting polynomial dependence of the exponential. The remaining tree summation is positive. If vertex prices have hit allowance γ at support exponent s + δ, δ > 0, the rooted forest estimate of the connected expansion gives for the terms with m nonroot vertices the bound eδ covB(Y ) 4 ballCost(B)γ δ m . (ASP3) This estimate is proved by summing the leaf label at its first overlap with its parent: a radius-B cover of the parent has at most its cover number times ballCost(B) possible hit sites. The hit bound pays that sum; the reserve eδ covB pays the parent cover factors and the enumeration of parent assignments. The rooted occurrence enumeration and factorial weights give the factor 4m already included in (ASP3), as in the absolute tree estimate used earlier. A designated marked root instead pays its own weighted allowance, multiplied by the same geometric factor. For ratio at most 1/2, summing m ≥0 costs at most two. Polynomial factors from the spatial direction growth are paid by the collar reserve: collarj,p = (shift(x))pj sup r≥0 rpje−δr. (ASP4) The supremum is finite (with the usual value one when pj = 0); it is just the elementary absorption of a fixed polynomial by a positive exponential. The shift is bounded by a fixed multiple of the local support width plus one. The packet’s additional factor (1 + ε)nA, 0 ≤ε ≤1, is also paid on the whole packet. The number nA of relevant off-tree color coordinates satisfies nA ≤C shares(A) Delay(x)5, by the link census and the 10B visit bound. Since 1 + ε ≤e, and any fixed positive derivative constant is at most its exponential, these factors are absorbed by increasing the coefficient of Delay(x)5 shares(A) in the packet income. They do not change its gain exponent or replace a whole-packet cost by a single-site hit. In the final scheduling set α = (Qexp −e)/4 > 0, where e < Qexp is the requested output exponent. Then e + 3α ≤Qexp. Spend the three α’s separately in the tree bound, collar bound and attached-log bound. The enlarged root price has rate −2/5; the degree ratio in (ASP3) has that rate times a width factor of rate zero, so it is eventually at most 1/2. The fixed derivative and collar factors also have rate zero. Their product with the root price is eventually below bβ−3/8, since −2/5 < −3/8. The parameter radius is R = bβ1/256, eventually at least two; restrict to R/2 for the final bound. These are the simultaneous choices giving the common nonnegative hit envelopes for the signed forced entries and the series certificates for the attached logs. Retirement selects a fixed subset of this same nonnegative majorant. Adding the boundary correction and turning operation gives the common signed price w(x) = A{y−3/8 + 4y−7/16}. Every reference torus and comparison ball in the graft has size polynomial in B(x) at a fixed stage. Their site counts Ni(x) and N∗(x), and their derivative factors, have rate zero. Summing the 204 ## Source page 207 A Wilson Yang–Mills construction J. Rodgers two compared anchored expressions over a reference torus gives signed price pi(x) = 2Ni(x)w(x). Summing the unrestricted value bound over the comparison ball gives price N∗(x)w(x). Both retain rate −3/8. Let z be the graft radius, let c∗bound the cover number of its radius-z support, and let λi be the fixed finite interpolation weights. If the derivative order and spatial derivative power of the budget are j and p, put Tz = 2j(1 + z)pj, O(x) = X i |λi|Tzpi(x). For a signed measure with moment price η at radius φ/2, the finite interpolation derivative formula gives beacon envelope ηO(x) at each site. There is one anchored beacon per site. Its support weight therefore gives anchored budget at most ηO(x)e(supportExponent +3)c∗. The retired part has budget ηw(x). Since the regular graft is linear in these parts, the complete regular signed price is w(x) + AO(x)e(supportExponent +3)c∗. (LCN10) The cover bound c∗is fixed before x, and all radius factors have rate zero. Thus (LCN10) has rate −3/8, and is eventually at most y−1/4. This is the rough price at radius V = φ/2 required in (LCN8). It applies to the deterministic source because the retirement index depends only on the layout and the matching identities identify the actual source with this graft. The forced half uses separate all-configuration envelopes. Its primitive entries retain their exponential flag suppression. The attached local insight entries have a signed scalar price obtained from the comparison-ball and reference-torus value prices, multiplied by their ball count and support weight. Denote the resulting primitive and attached hit prices by a and b. The delay purchases κ ≥0, b ≤1, and γ = eκa + 2b, (2B + 1)4γ ≤1, e−κ(eγ −1) ≤forecastK . The local contribution b has rate −3/8 and is eventually below the comparison ceiling y−1/8. The forced forecast uses the primitive exponential suppression as well; it does not follow from comparing two power rates. To fuse the two families, inflate each old forced atom by eκ, bound each attached exponential minus one by twice its logarithm envelope, and sum the connected components using the displayed support reserve. Each component contains an old forced atom, so undoing the inflation saves at least e−κ. The resulting hit price is at most e−κ(eγ −1), as proved in the attached-log fusion argument. This gives the forced forecast at radius φ/4, with one support-exponent unit reserved. Shrinking later to q/2 improves the moment inequalities and supplies the common forced envelope in (LCN3). The sharper individual price in (LCN9) comes from universal retirement. Its remaining beacon envelope is a finite sum, with fixed coefficients, of gold(x), Ni(x) gold(x), 2j(1 + z)pjNi(x) gold(x), where gold(x) = y−23/20. Every multiplier has rate zero. The fixed support exponential and turning price preserve rate −23/20, proving the claimed rate of εj. Thus finite interpolation combines a large-radius signed estimate and a sharper real individual estimate. For the primitive source, the individual bound b is its regular price (FCC1) plus the beacon contribution. With x = bβi, it has form b = 3t Cjx−11/10 + εj(x), εj has rate −23/20, (LCN9) where t > 0 is the freely chosen small primitive allowance. The three principal payments are the coefficient t + 2t. The other terms are the turning fee times the retired tail, the tail itself, and the 205 ## Source page 208 A Wilson Yang–Mills construction J. Rodgers beacon price times its fixed support exponential. The local estimates give them the stated −23/20 rate at fixed block and stage. The larger signed radius is V = fire(x)/2, where fire(x) = βF (x)1/256; the rough signed price has rate −1/4. Consequently the last term of (LCN8) has rate at most −301/256 −1/4 < −23/20. Choose t so that 6Cqt ≤v, and then the coupling-entry delay so that the two remaining terms cost at most (v/2)Cjx−11/10. Equations (LCN8)–(LCN9) give signed regular source price at most vCjx−11/10 = choresi. This is the numerator of the scalar price fi. The Hessian reading of this source yields the scalar correction price fi by linearity and its bound θ2 i |∆| ≤Budget. Applying the regular reset price to that same signed source, including the committed coefficient bound |αN,i| ≤di+1 + s log Li, gives regular output price dbP ′ once the small allowance and delay have been chosen. The forced reset entries use their common all-configuration envelopes; the forced and attached-log fusion estimates retain one unit of support-weight reserve and yield hit price dkP ′ at radius q/2. The expenses and the forced cutoffare the paid local reset inequalities, not a consequence of the scalar reading alone. Combining these two prices through (LCN1) proves the activity assertion in (LCN3). Thus the analytic recovery uses a signed estimate obtained by a finite interpolation cancellation, with the exact exponent 301, rather than assuming analyticity of arbitrarily selected primitive witnesses. In fact the output values are independent of such incidental choices. To compare two representatives with identical coupling and activity data, use the same constant affine input line but select the first representative at t = 0 and the second at t ̸= 0. The bounds (LCN3) still hold. For q ≥4, the signed measure δt −δ0 has moment price at most |t| at either radius. Therefore its two output differences are bounded by d|t| and fi|t|; letting t →0 proves equality. This supplies the single-valued activity and scalar maps on the common normed presentation. Choose q ≥4 so large that 2e (1/4)(q/4) ≤1, 2d ≤γ = 1/128. (LCN4) At any center (λ, A) with Ω−1/32λ ∈[17/20, 23/20] and ∥A∥P ≤1/8, every neighboring center within distance 1/8, followed by any direction of norm at most 1/4, stays in the enlarged scalar window and activity ball used in (LCN3). Recover the line series at radius q/4. Polarization makes its degree-n coefficient a symmetric multilinear map: the finite difference formula divides by n!, has 2n summands, and each sum of n unit vectors has norm at most n. Since nn/n! ≤en, the resulting multilinear coefficient is bounded by  2e (1/4)(q/4) n C, C = 2d or 2fi. (LCN5) The formulas agree on overlaps because they recover the same scalar line coefficients. They extend through the isometric dense inclusion into the completion, preserving their operator norms. By (LCN4), the activity coefficients are bounded by γ, and the scalar coefficients by 2fi, at every order. For increments z, z′ of norm at most 1/4, telescope each multilinear diagonal and sum: X n≥1 n(1/4)n−1∥z −z′∥C = 16 9 C∥z −z′∥≤2C∥z −z′∥. (LCN6) Thus the local Lipschitz constants are 2γ for activities and 4fi for the scalar correction. To get the estimates for any two centers in the whole stated window and activity ball, split into distances at most 1/4, where (LCN6) applies, and distances greater than 1/4. In the latter case (LCN3) applied 206 ## Source page 209 A Wilson Yang–Mills construction J. Rodgers to singleton measures bounds the two values separately by d and fi; their differences are at most 2d ≤γ and 2fi. Both cases imply ∥Φi(y) −Φi(x)∥≤8γ∥y −x∥, |∆i(y) −∆i(x)| ≤8fi∥y −x∥. (LCN7) With si = 2fi, this is exactly the scalar constant 4si and activity constant 8γ used in (FCB). All prices are for the fixed step of the common calendar and are uniform in the actual cutoffand period when its paid stage and support margins hold. The local analytic constants are chosen before the final history is selected; the completion introduces no new choice of primitive data. The common root envelope in this argument uses the energy gain proved before the expansion, in its actual order of quantifiers. For every sufficiently large odd block side L, all sufficiently large fixed window widths W admit a number κ > 0 and a coupling threshold such that κ tsf(β, L)2 nseed ≤Eassessed (ASP5) for every admitted tube, sector, tree and packet store above that threshold. Here nseed is its number of charged seeds, tsf its small-field threshold and Eassessed the sum of the actual fine, coarse and mismatch departure costs. The assignment of those costs to packet cores, the paid annuli and the three clipping ranges was proved in the energy chapter. Formula (ASP5) is a per-seed energy statement; the expansion additionally uses the support census, normalization expenses and reserves (ASP3)–(ASP4). For adjacent scheduled profiles use the same angular covers for both copies of (ASP5). They provide one positive coefficient for each of the two requests; replace both by their smaller value and increase the common coupling threshold. The geometric block factor is fixed before these choices. The resulting calendar delays are chosen above both thresholds, and its actual coupling update identifies the next small-field threshold with the one in (ASP5). Every admitted step therefore has the claimed bound with the same charged seeds and the same angular read sets. The pointwise domination obtained this way gives one nonnegative envelope for all the signed entries, including when a parameter path changes which sector is active. Combining it with the tree and collar sums proves the all-root envelope used above. 8.20 Stability of finite calibration and its comparison weights The retained forward recipe also gives quantitative stability of the complete finite trajectory under changes of its bottom coupling. This is the finite statement behind the auxiliary trajectories in the uniqueness argument. Write xi = bβi and define wi = x−1/32 i , bi = xiwi = x31/32 i , λi = p βi/xi wi , βi = xi(wiλi)2. (FCB1) The positive square root is used. By (FCA8), wiλi ∈[17/20, 23/20]; this wider interval contains [ p 3/4, p 5/4]. The activity coordinate Xi is its image in the common completed activity chart. At a fixed finite horizon all these coordinates live in one finite product of their scalar and activity spaces. Let γ = 1/128. The actual local analytic comparison, including the initial state at N, bounds differences of activity outputs by 8γ times the maximum input-coordinate difference. Its scalar correction output has Lipschitz bound 4si times that maximum. The weights are explicitly si = 2 choresi θ2 i , choresi = v Cstage(i)x−11/10 i , (FCB2) 207 ## Source page 210 A Wilson Yang–Mills construction J. Rodgers where v > 0 is the fixed comparison cost and Cj > 0 the stage’s local analytic price. The common small-field threshold is θi = x−1/2 i exp((log log xi)2). Thus si ≤2vCstage(i)x−1/10 i . These comparison bounds apply to the same retained primitive operation at both points, as ensured by its fixed layout and choices. The calendar pays the cumulative weight inequality X j 0. The mean value theorem, for x ≥a, gives (x + a)31/32 −x31/32 ≥a 31 32 2−1/32x−1/32. Since x−1/10/x−1/32 →0, eventually the upper bound in (FCB2) is at most γ times this increment. Include this one eventual threshold in the coupling-entry delay for each of the two allowed adjacent- stage possibilities. It follows at every realized step that si ≤γ(bi+1−bi). Summing gives P j 1. Equations (FCB3)–(FCB4) imply |λ′ i −λi| ≤|δ| bi + 4γU ≤|δ| b0 + 4γU. Taking the maximum over all levels and both coordinates proves U ≤|δ| b0 + 8γU, U ≤2|δ|/b0. (FCB5) Only finite products and finite sums occur at this stage. If δ = 0, (FCB5) gives identical finest normalized couplings, hence identical finest actual cou- plings. Their startup states are equal by their fixed startup rule. Descending determinism of the primitive recipe then gives equality of all represented states. Thus the successful finite trajectory with a prescribed bottom value is unique within this retained recipe, although monotonicity of a scalar coarse readout was unnecessary for existence. The arcs prove existence for every bottom value in the nearby interval. Combining them with (FCB5) gives continuous dependence of its normalized scalar and activity coordinates on that value, 208 ## Source page 211 A Wilson Yang–Mills construction J. Rodgers on the same finite calendar. For example, choose a bottom perturbation |δ| ≤(l/4)b0, with the fixed small comparison radius l> 0, and small enough to stay in [3 ̄β/4, 5 ̄β/4]. Then U ≤l/2, placing the entire perturbed trajectory inside the common analytic comparison neighborhood. This constructs the nearby finite trajectories used in passing local comparison estimates to banked limits. Equations (FCB1)–(FCB4) also identify explicitly the positive scalar weight and summable correction weights in that argument. 8.21 Uniform running of the actual coupling The drift estimate also proves the uniform running bound needed in the public theorem. Define D0 = 0, Di+1 = Di + di+1. (RUN1) For the calibrated finite history, (FCA3), βN,p,0 = ̄β, and the exact recursion for bβi imply |βN,p,i −bβi| ≤Di (i ≤N). (RUN2) Indeed subtract the two recursions and apply the triangle inequality at each step. The right side depends only on the calendar, so this bound is uniform in both N and p. The stronger asymptotic drift condition of that calendar is di+1 log Li ≤J(i + 1)−2, J(i) −→∞. (RUN3) Here J(i) is the stage index, not a cutoffor period. Given ε > 0, choose j so that J(i + 1)−2 ≤ε/2 for every i ≥j. Summing (RUN3) gives, for m ≥j, Dm ≤Dj + ε 2(τm −τj). Since τm →∞, eventually Dj ≤(ε/2)τm. Together with (RUN2) this proves ∀ε > 0 ∃m0 ∀m ≥m0 ∀N ≥m ∀p, |βN,p,m −bβm| ≤ετm. (RUN4) This is uniform control of the actual finite-cutoffcoupling, including m = N. It is stronger than merely writing down the reference line bβm = ̄β + sτm. Dividing by τm > 0, and then absorbing the fixed intercept ̄β/τm, yields sup N≥m, p βN,p,m τm −s −→0. (RUN5) Explicitly apply (RUN4) with error ε/4, and take τm ≥4(| ̄β| + 1)/ε; the triangle inequality gives an error smaller than ε. Taking m = N proves the bare coupling slope in the theorem, uniformly in the physical period. Set b0 = 11CA/(48π2), so 2b0 = s. For arbitrary cutoffand period sequences Ni ≥i eventually and pi, let gi = β−1/2 Ni,pi,i. Positivity of the coupling and (RUN5), followed by continuity of the square root and reciprocal at s > 0, give gi q log(a−1 i ) −→(2b0)−1/2. (RUN6) The equality used here is β−1/2√τ = (β/τ)−1/2 for β, τ > 0. This regulator coupling is defined from the inverse action coefficient. Its numerical leading logarithm does not by itself identify it with a coupling defined by a physical position-space observable. That separate identification, and its normalization, are proved in the trace and Ward calculation. 209 ## Source page 212 A Wilson Yang–Mills construction J. Rodgers 8.22 Comparing omitted interactions at a common normalization The fine-to-intermediate comparison uses one fixed denominator throughout. If N is the finest level and p is the period, define, for a polynomial-valued integrable insertion P at level m, Im(P; t) = R [P(U)]t dμm(U) ZN,p , where t is a polynomial mode, μm is the product Haar prior, and ZN,p > 0 is the original finest- level Wilson partition function. In particular the denominator is not replaced by a new partition function when m changes. The root estimate, for insertion weights wv > 0, is |Ik(P; t)| ≤Bk(t) := exp{|Λk| τk + |V |(x + y + 2)} toll(w, t) . Here τk is the vacuum treasury, (x,y) bound the two insertion envelopes, and the positive toll is the coefficient-extraction weight. This bound follows by applying the small-field integral bound to the represented density, bounding its vacuum bill by the treasury, and using the exact identity between the integrated represented density and ZN,p. At an intermediate level j > k, separate the two omitted interaction polynomials P0, P1 from the retained insertion data. Multiply both omitted parts by a complex parameter z, restore the resulting density, and call it (P(z)). Then (P(1)) is the full insertion and (P(0)) is the insertion with those parts omitted. The paid analytic estimate makes Y (z) = Ij(P(z); t) holomorphic for |z| < Sj, where Sj = brisk(Bj) > 2. For every such z, transport (P(z)) through the actual retained Mark steps from j to k. The retention estimate preserves the served-insertion bounds with root weights wk; integration of the transported polynomial equals integration of the original polynomial. Coefficient extraction commutes with this integral because it is a continuous linear map on the finite-dimensional polynomial space. The denominator remains ZN,p, so the root estimate gives |Y (z)| ≤Bk(t) uniformly throughout this disk. For any holomorphic Y bounded by B on |z| < S, S > 2, every disk of radius (S/2) centered on the segment ([0,1]) lies inside that disk. Cauchy’s derivative estimate gives |Y ′(z)| ≤2B/S on the segment. Integrating along the segment proves |Y (1) −Y (0)| ≤2B S . Consequently the omitted interactions change the normalized coefficient by at most 2Bk(t)/Sj. The root cost depends on k, not on the intermediate volume at j. The analytic paid bound, the retained transport estimate, and the root small-field bound are the substantive inputs to this comparison and must be established below. The holomorphy under the integral in this comparison can be justified directly. The polynomial algebra has finitely many modes; its truncated exponential is a finite sum of powers. For a fixed configuration the restored polynomial is a finite sum of products of this truncated exponential and the local field factors. Its masks and flags depend on the configuration, not on z. Replacing the omitted terms by zP0, zP1 therefore makes each mode an entire function of z. All masks are measurable: a prescribed flag pattern is a measurable event, and restoration is a finite sum of products on these events. The weighted polynomial norm is equivalent to the ordinary finite-dimensional coefficient norm when all insertion weights are positive. The served-insertion estimate for the scaled omitted terms, 210 ## Source page 213 A Wilson Yang–Mills construction J. Rodgers together with the envelope bounds and nonnegative coupling, gives a single finite constant Y such that ∥P(z, U)∥≤Y (|z| < Sj, U ∈Cj). Since the Haar prior is a probability measure, this is an integrable majorant. It proves integrabil- ity at every z. Dominated integration of Cauchy’s formula on small circles, or equivalently Morera’s theorem with Fubini, proves holomorphy of each integrated coefficient. Thus no differentiability of the configuration masks is needed. The served-insertion envelope proved in the marked-integration chapter gives this uniform majorant. The retained transport statement has a descending induction with no limiting step. At the top j, scaling the omitted parts spends two units of support exponent. Suppose the transported polynomial has the required served bound at (m+1). Its first-degree coefficient is the prescribed insertion manifest there, by the relay identity. Integration through the Mark kernel gives the prescribed manifest at m, by the insertion feed identity. The parent map sends the chosen pins at (m+1) to those at m. The matching assumptions supply their occupancy bound and the allowed reweighting of the insertion weights; the wake data supply the local finite transport tales and their estimates. The one-step duty estimate now gives the served bound at m with the full support exponent at that level. This proves the assertion down to k. Integrability at each stage follows from integrability of the Mark push. This induction explains why the two-unit expenditure occurs only at the top; it does not accumulate once per step. The finite inventory and one-step duty estimates in the marked-integration chapter establish the inputs to this induction. 8.23 Comparing the varying Gaussian integrals A change of protected variables also changes the Gaussian law. If z′ = z′(z) has Jacobian JV (z), the general density ratio is JV (z) ρCV (z′(z)) ρC0(z) . For positive covariances on the same d-dimensional real space this is JV (z)  det C0 det CV 1/2 exp  −1 2z′TC−1 V z′ + 1 2zTC−1 0 z  . The fixed-covariance expression follows only when CV = C0. The determinant and inverse deriva- tives in the protected-variable estimates below are precisely what control the genuinely varying case. The fluctuation space at a fixed step is one finite-dimensional Euclidean space of dimension D. A prepared datum has Gaussian density ρC(z), mean displacement tX in physical coordinates, and scale g = β−1/2 > 0. Use the common physical variable w = tX + gz. Then z = g−1(w −tX), cX(w) = g−DρC(g−1(w −tX)), and for every integrable vector-valued function A, Z A(z) dνC(z) = Z cX(w)A(g−1(w −tX)) dw. This is ordinary translation invariance of Lebesgue measure followed by scalar change of variables, including its determinant g−D. It also proves integrability of the transformed integrand. 211 ## Source page 214 A Wilson Yang–Mills construction J. Rodgers The common local covariance bounds give ρC(z) ≤C0e−τ∥z∥2 with C0, τ > 0 independent of the history index and gauge-slice variable. The protected mean gives ∥tX∥≤B0: sum its finitely many squared coordinate bounds and take the square root. The coupling window gives fixed 0 < u ≤g−1 ≤v. For r = g−1 these imply 1 + ∥r(w −tX)∥≤(1 + v + vB0)(1 + ∥w∥), e−τ∥r(w−tX)∥2 ≤eτu2B2 0e−τu2∥w∥2/2. The second inequality uses ∥w∥2 ≤2∥w−tX∥2+2B2 0. For any fixed set of tested bonds its face-scale function is at least one and at most K0(1+∥z∥), since each coordinate is bounded by the Euclidean norm. Consequently every transformed integrand whose original norm is bounded by a fixed power of that face scale has a common majorant C1e−τu2∥w∥2/2(1 + ∥w∥)k. This is Lebesgue integrable: in finite dimension Gaussian decay dominates any polynomial. Its constant is independent of the gauge-slice variable. The integral convergence argument is applied twice. At each gauge-slice point, the common majorant and pointwise convergence of the difference of the two transformed integrands give con- vergence of their Lebesgue integrals by dominated convergence. Fubini identifies these with their original Gaussian integrals for almost every gauge-slice point, simultaneously for the countably many history indices. Their norms are bounded by the integrals of the two majorants. The outer gauge-slice measure is a probability measure, so this constant is integrable. A second dominated- convergence application proves convergence of the full Gaussian–gauge integrals. For ordinary primitive derivatives, differentiation under the integral identifies the derivative of the local activity with the integral of its differentiated integrand. Its paid bound is C 1 + (p facescaleK + 1) j ≤C(p + 2)j facescaleKj, because the face scale is at least one. Thus it has the required polynomial majorant. Marked primitive derivatives have the same form of estimate from their common marked budget. The undifferentiated marked amplitude has a uniform constant bound, which is the case k = 0. Pointwise comparison of the transformed integrands is taken with one common amended draft. At each fixed gauge-slice point, the draft data have only finitely many discrete possibilities; an ultrafilter selects one possibility eventually. The same viability, threshold and choice data are then used in both histories. Their density factors and local vertex products compare in these common physical coordinates; products retain both real and imaginary marked ingredients. The preced- ing two-stage integral argument converts this comparison into convergence of ordinary derivatives, marked derivatives and marked amplitudes. The finite reserve products and boundary-choice aver- ages in the actual primitive definitions preserve it. Their healthy-support local identities identify these formulas with the actual leaflet and marked-grove germs near the tested base configuration. This is the Gaussian integral passage used in the single-stream comparison, with all discrete masks retained inside the common draft. 8.24 Common drafts and Gaussian density factors The finite choice used in the preceding integral argument is precise. A draft records a gauge-slice variable, a store and a boundary choice, together with the fixed selector, scale and width. After fixing the gauge-slice integration variable, the paid conditions fix the latter three parameters. Only 212 ## Source page 215 A Wilson Yang–Mills construction J. Rodgers the store and boundary choice remain variable, and they range over a finite set at a fixed lattice step. For any ultrafilter one pair is therefore constant eventually: otherwise the complements of all its finitely many fibers would have an empty intersection in the ultrafilter. Replacing the gauge- slice coordinate by the fixed integration point produces one common draft. Viability transfers to it on that eventual set. No continuous gauge-slice variable is claimed to become constant along the original histories. In this common draft, the atmospheric physical fluctuation w is the same in the two histories. The restart map is zβ(w) = p β (w −tX). The protected-mean comparison and coupling comparison imply comparison of this map. More generally its corrected version along an allowed output-field family replaces w by the common smooth steered field before subtracting the protected mean. The exact identity tX+β−1/2zβ(w) = w ensures that the original real integrand is recovered; the analogous steered identity is used for its derivative integrand. The Gaussian density formula is ρC(z) = ( √ 2π)−D exp{−1 2 log det C −1 2zTC−1z}. The covariance comparison, its uniform positive floor, and the finite-dimensional inverse and deter- minant formulas compare every term in this expression. The floor also ensures that any subsequen- tial density limit at a fixed finite z is strictly positive: the logarithmic exponent has a finite limit and its exponential is positive. Thus division by the corresponding marginal density is legitimate in the local reweighting formula. The same argument applies to every principal covariance restriction on a fixed finite set of tested bond/color coordinates. Its positive floor is inherited from the full covariance, and its inverse and log determinant depend smoothly on that restricted matrix. The local correction factor is exactly its finite Jacobian factor times the density of the corrected local fluctuation divided by the density of the uncorrected local fluctuation. The Jacobian is a finite product of the smooth link steering factors. Each steering factor is evaluated near the identity in the common draft; the chosen local chart supplies its smoothness through the required derivative order. Consequently this entire correction factor compares through that order. This establishes the density and local- change-of-variable factors used in the ordinary and marked integral comparisons; comparison of their remaining vertex factors is treated separately. 8.25 The protected change of variables and its determinant The steering factors in the preceding ratio have the following construction. Fix a small positive radius d, independent of the later scale delay, on which the logarithm is a left inverse of the exponential through radius 2d. Shrink a matrix neighborhood of 1 so that log(exp a M) is analytic for ∥a∥≤2d, and so that for group elements M it exponentiates back to exp a M when ∥a∥≤d. These choices follow by continuity on a closed ball strictly inside the original chart. Let χd(a) be the smooth inner-share cutoff, one on ∥a∥≤d, zero for ∥a∥≥2d. Define, with this multiplication order, pM(a) = χd(a){log(exp a M)−a}, ΦM(a) = a+pM(a), jM(a) = det(1+DapM(a)). (PV1) Outside the cutoffsupport the first expression is identically zero in a neighborhood, so the map is smooth there regardless of the exterior values of the logarithm. On the support, smoothness follows from the chosen analytic chart. At M = 1, pM and its derivative vanish, Φ1 = id, and j1 = 1. 213 ## Source page 216 A Wilson Yang–Mills construction J. Rodgers Compactness of {∥a∥≤2d}, joint continuity, and the exterior zero identity supply one neigh- borhood of M = 1 on which, uniformly for every a, ∥pM(a)∥< d/2, ∥DapM(a)∥< 1/2, jM(a) > 0. (PV2) For the determinant assertion, use its continuity and its value one on the compact ball; outside the ball it is exactly one. The derivative bound makes pM Lipschitz with constant 1/2. Thus ΦM is injective, since equality of two images implies ∥a −b∥≤1 2∥a −b∥, and surjective, since for each target b the contraction a 7→b−pM(a) has a fixed point. Its derivative is invertible by the Neumann series. Consequently it is a global smooth change of real coordinates with positive Jacobian jM. For an original coarse background V0 and a nearby background V , the correction at a fine link is Me = (replica V0)e(replica V )−1 e , u′ e = ΦMe(ue). (PV3) The order matters: for ∥ue∥≤d, exp u′ e (replica V )e = exp ue (replica V0)e. (PV4) On inactive links the fine slice input remains unchanged. On active links the identity is needed only where their protecting cutoffs are nonzero. Those cutoffs imply ∥u′ e∥≤8a tfield. Choose the scale delay so that 8a tfield ≤d/2. Then (PV2) implies ∥ue∥≤∥u′ e∥+ ∥pMe(ue)∥≤d, so (PV4) applies there. This proves equality of the old and new reconstructed configurations on every protected read. A cutoffproduct vanishing at the read makes the corresponding term zero. A nonzero product forces equality on its whole actual footprint. This argument proves the equality for old activities, atom words, row predictions, sector profiles and their complements individually, including the hard decisions inside them. It does not differentiate a hard decision. Writing the physical fluctuation as u = μV0 + gz, g = β−1/2 > 0, the corresponding Gaussian coordinate change is z′ = g−1{ΦM(μV0 + gz) −μV }. (PV5) It obeys μV + gz′ = ΦM(u) exactly, and becomes the identity at V = V0. Restrict to the finite set I of bond/color coordinates required by the ground. Each link block of (PV5) has derivative g−1DΦMeg = DΦMe; the scalar dilation factors cancel. The coordinate permutation between link/color arrays and Euclidean vectors contributes no absolute determinant. Therefore det Dz′ = Y e∈Ilinks jMe(ue) > 0. (PV6) This is a product over tested links, with one color-space determinant per link, rather than one determinant per color component. Let C0, CV be the principal covariance matrices on I, with Gaussian densities ρ0, ρV . Finite- dimensional change of variables gives for an integrable local function f Z f(w)ρV (w) dw = Z f(z′(z)) Y e jMe(ue) ! ρV (z′(z)) ρ0(z) | {z } LV (z) ρ0(z) dz. (PV7) The denominator is strictly positive at every finite z. Both integrals can equivalently be written against the original full Gaussian, since the functions depend only on the indicated marginal and the unused coordinates integrate to one. This proves the local likelihood formula, including its direction and normalization. 214 ## Source page 217 A Wilson Yang–Mills construction J. Rodgers Along a smooth coarse field family based at V0, its finitely many matrices Me lie in the common neighborhood from (PV2). Differentiation of (PV1) on the compact cutoffball gives uniform finite- order bounds on the displacement and determinant; exterior derivatives vanish or equal those of the constant one. Composition with the coarse family gives a displacement price of the same small order as that family’s motion and a determinant price of order zero. Equations (PV5)–(PV7) then supply the Gaussian score derivatives. The covariance floor and the finite inverse/determinant formulas control the score coefficients. The integrable Gaussian envelopes established above justify differentiation under the integral. Thus a marked entry can remain evaluated on the fixed original fine input: the derivatives fall on its protecting cutoff, on the smooth ordinary vertices, and on LV . The protected equality (PV4) identifies this formula with the original integral on a neighborhood of the base point. This is the mechanism used in the marked primitive comparisons and their product estimates. 8.26 Finite-window covariance comparison For clarity, covariance comparison here is obtained from its actual construction. The form matrix consists of the selected physical row operators evaluated on coordinate impulses. Selection is determined by whether the row base is outside the store’s fixed six-radius interior. In an unselected row both histories have zero form entries. In a selected row the physical row comparison gives entrywise comparison through the required derivatives, with the fixed impulse held the same. The plate is the form Gram matrix plus the fixed dormant diagonal. Finite matrix multiplication and addition therefore compare its entries. For each covariance column, its window is the fixed finite set prescribed by the reference frame. The principal plate restriction inherits the uniform positive floor. Inversion on this set of positive definite finite matrices is smooth, with derivatives controlled by the same inverse bound; hence the two restricted inverse matrices compare through the required order. Entries outside the window are identically zero in both histories. Placing the inverse column back in the full coordinates and symmetrizing gives the actual reference covariance, so it too compares. This retains the column- window construction and does not replace it by the full plate inverse. The protected mean is obtained from these same window columns and its local bill vector. Their comparison gives the mean comparison used by restart. Together with the coupling-window comparison it supplies the affine physical-variable comparison. The scalar density proof above consequently rests on finite inverse matrices with a common lower eigenvalue bound. Its remaining physical-row and bill-vector inputs are precisely the stationary and local nonlinear comparisons, rather than an independent covariance convergence assumption. 8.27 Extracting the common activity comparisons At each fixed level and physical period, the activity data lie in one common normed presentation space. Its dictionary, supports and threshold are fixed, and its norm controls anchored regular derivative mass and forced absolute mass. Its completion is used only for limits. The actual retained finite histories remain elements of the original space. We spell out how a Cauchy sequence in this norm gives the evaluated comparisons needed above. For either the regular-occurrence map or the forced-coefficient map, the activity norm implies a nonnegative absolute budget of total at most C∥a∥, uniformly over configurations. At a fixed period one may take C to be the number of sites times the corresponding anchored budget constant. For regular terms sum the anchored masses over their unique anchors; for forced terms choose a site from each nonempty forced inventory contained in its support. The local mass dominates the 215 ## Source page 218 A Wilson Yang–Mills construction J. Rodgers unweighted bound. This yields the stated global budget. Choose its canonical least envelope q(a, i) = sup U |F(a, i, U)|. The existing absolute budget makes this supremum finite, nonnegative and summable with total at most C∥a∥. Linearity of F gives q(a, i) ≤q(a −b, i) + q(b, i). For a Cauchy sequence an, these envelopes have uniformly small tails. Indeed choose N with C∥an −aN∥≤ε/2 for n ≥N, and a finite set controlling the tails of the finitely many a0, . . . , aN by ε/2. On its complement the displayed subadditivity bounds every later tail by ε; the earlier tails are already controlled. If an, bn are Cauchy and ∥an −bn∥→0, then |F(an, i, U) −F(bn, i, U)| ≤q(an −bn, i) ≤C∥an −bn∥ uniformly in U for each label. The two envelope families also have the uniform tail property just proved. Thus regular occurrences with their actual fixed healthy masks, and forced entries with their actual inventories, have exactly the evaluated comparison and tail bounds used in the local construction. Masked regular evaluation is linear because the threshold and each labelled support are fixed in this presentation. For any two escaping subsequences of a single Cauchy activity sequence, their norm difference tends to zero by the Cauchy criterion. The coupling sequence has the same property and converges in R. The retained history’s actual activities are eventually these normed presentation elements at every fixed level. Hence all incoming regular, incoming forced and outgoing forced comparisons required by the two-history transfer argument follow from the common activity Cauchy sequence. Completion convergence suffices: the completion embedding is isometric, so convergence there makes the original presentation sequence Cauchy. The scalar and activity convergence themselves are obtained by the retained-history construc- tion as follows. At each level its normalized scalar coordinate is bounded in a fixed finite interval, by the positive width and the uniform normalized-coupling bound. Compactness of the countable product of these intervals gives one diagonal escaping subsequence on which all scalar coordinates converge. The fixed-scalar construction then gives convergent activity data and the associated in- termediate quantities at all levels along that subsequence. To promote this to the full sequence, repeat the same extraction on any escaping subsequence. The comparison of two such banked limits with the same limiting terminal coupling identifies their scalar limits. The subsequence criterion for convergence therefore gives full scalar convergence, after which the fixed-scalar construction gives the full activity convergence. The section on uniqueness of banked history limits proves the comparison identifying these limits. Compactness is used only to obtain the candidate limits. For the winning trajectories the bottom coupling at level zero is the same prescribed center ̄β, so its convergence hypothesis holds. The finite-horizon alignment identifies their paths with the original stationary paths and their normed activity presentations with the actual retained activities. This provides the Cauchy sequences used throughout the fixed-level insertion proof, on the original common choices. 8.28 Finite-depth approximation of the activity trajectory At a fixed finite horizon, regard the normalized scalar coordinates as an externally specified admis- sible sequence. The activity update maps the next-level activity to the current one and keeps its norm at most 1/8. On this ball its paid comparison is ∥Φi(λ, x) −Φi(λ′, x′)∥≤c0 max{|λ −λ′|, ∥x −x′∥}, c0 = 8/128 = 1/16. 216 ## Source page 219 A Wilson Yang–Mills construction J. Rodgers The accompanying scalar correction satisfies the corresponding bound with coefficient 4si. These are the finite-history estimates used in the banked-limit comparison. For any intermediate depth b, construct a comparison activity trajectory by setting its activity at depth b to zero and applying the same updates backwards below b; set unused later levels to zero. Finite decreasing induction constructs this trajectory. The paid ball-invariance estimate keeps every value admissible. Compare it with any full trajectory having the same scalar script. At depth b the distance is at most one (in fact at most 1/4); each backward step multiplies that distance by at most c0. Thus at m ≤b the discrepancy is at most cb−m 0 . Now compare two sequences of histories whose scalar scripts have the same limit at every fixed level. For the trajectories truncated at the common fixed depth b, the top activity is identically zero in both histories. Descending induction, using the fixed-input one-step comparison at each of the finitely many levels, gives a common limit of their truncated activities. Given ε > 0, choose b −m so cb−m 0 < ε/3. For sufficiently late histories both horizons exceed b and their truncated activities differ by less than ε/3. The two truncation errors prove that the original level-m activities differ by less than ε. Applying this argument to every pair of escaping subsequences proves that the activities have a common limit in their complete activity space. The scalar correction limit follows by the one-step comparison using the already convergent next-level activity. Finite initial segments discarded to ensure all required horizons are present do not change these limits. This supplies the fixed-scalar convergence used in the diagonal extraction and banked uniqueness argument. Its fixed-input one- step comparison is the local Gaussian/cluster/reset comparison developed above; the ball-invariance and contraction constants are the paid one-step estimates on that same update. The physical scalar identity also survives this passage. At finite horizon the correction equals the difference of adjacent physical couplings minus the committed path coefficient. The latter has its stationary limit. Taking the three scalar limits therefore gives the displayed banked flow identity, with no infinite telescoping sum. 8.29 From labelwise comparison to reset budgets At a fixed physical period there are finitely many site supports. The budget weight is linear in the nonnegative scalar bill, so there is a finite constant P∗> 0 with p.mass(D, t) ≤P∗t (t ≥0) for every support D. This constant may depend on the fixed period; no uniform volume assertion is used in this convergence argument. Suppose a labelled difference has a nonnegative uniform-tail majorant fn,i, and each fixed label eventually admits every positive tolerance. Given a desired total weighted budget ε, put t = ε/P∗. Choose a finite label set E with every majorant tail at most t/2, and eventually bound each label in E by t/[2(|E| + 1)]. Use these small bounds on E and the original majorant elsewhere. The resulting nonnegative array has total at most t, hence weighted budget at most ε. This applies both to the paid regular difference bounds and to the all-configuration forced difference bounds. It proves the quantitative difference-budget statements used to recover the activity norm. The same construction controls the projected coupling increment. Each source separately has a common finite paid total, obtained by multiplying its uniform-tail bound by P∗. The diagonal projection estimate gives t2 0|∆n| ≤C, t0 = p.threshold > 0. For the difference of two sources, choose its paid deviation budget below t2 0ε/2. Linearity of projec- tion and its diagonal drift estimate then give |∆n −∆′ n| ≤ε/2 eventually. Thus the two increment 217 ## Source page 220 A Wilson Yang–Mills construction J. Rodgers sequences are uniformly bounded and their difference tends to zero. This proves the increment comparison used in the exact coupling update, rather than assuming a separate scalar continuity principle. The reset dictionaries themselves are on common supports. The canonical stationary path presentations retain their fixed slot and atom supports, so the low homes and high footprints do not change with their numerical coefficients. The reset listing is therefore identical for both path families. Its regular and forced comparisons can be transferred through the final dictionary consumption on this same listing. The final vacuum and coupling scalars are irrelevant to that activity comparison. These observations isolate the remaining task in reset convergence: compare the actual residual, freshness, loan and attached-log coefficient functions with uniform tail budgets. 8.30 Comparison of the reset coefficient families The regular reset family is the disjoint union of the source rebates, low freshness terms and high freshness terms. For a rebate, the difference identity is the rebate of the source difference. Its paid bound is multiplied by the fixed reset-loss constant, so both the uniform tail bound and each label’s arbitrarily small comparison bound pass through. The fault difference has the same linear identity and the same all-field reset-loss bound, yielding the forced comparison. For a fixed low label, the actual slots and support are independent of the stationary numerical coefficients. Only its mark m varies. Choose once a reference stationary coefficient array for which this label’s mark m∗̸= 0 (all path coefficients equal to one suffice for this single-label algebraic test, and the direct mark is positive). Its freshness is then the fixed reference freshness multiplied by m/m∗. The coefficientwise path comparison makes these scalar ratios compare, so the fixed-label paid freshness difference tends to zero. Its uniform majorant is the original weighted low-mark budget times its fixed low-cost polynomial and the common threshold factor. Bounded-length low labels form a finite set, since each pair of path labels has total length bounded. The spare positive exponential width therefore makes these majorants have uniformly small tails. High freshness is minus the sum of the two bounded compared scalar parameters times the actual cubic or quartic atom. The atom coefficient comparison gives fixed-label convergence; the weighted atom budget with spare width gives uniform tails. Finite signed-seat replication and the fixed orientation multiplier preserve all these bounds. The projected source increment is a bounded compared scalar by the previous argument, so it may be used as one of those two parameters. For the low loans, the same reference-mark factorization gives exactly loanR(α, ∆) = loanR∗ (m/m∗)α, (m/m∗)∆ . This follows from the literal loan definition: its live pair term is proportional to the mark, its proxy is independent of the numerical path coefficients, and its home and kept tests are common. In the direct case the mark is fixed. Subtracting the two reference loans is the reference loan at the differences of their two scalar arguments. Its all-field shape bound tends to zero as those arguments tend to zero. Thus the loan differences converge uniformly over configurations, even on the boundary of a failed health test. The common absolute majorant for these loans is el/ρ|m|  2t(Bα + B∆) + 1 2B∆pairBill t2 . The width hypothesis leaves a positive exponential margin after el/ρ is spent. The weighted low-pair budget and the finiteness of bounded-length labels therefore give uniformly small tails. Combining faults and loans proves the attached-log comparison. Combining rebates and the two freshness 218 ## Source page 221 A Wilson Yang–Mills construction J. Rodgers families proves the regular reset comparison, all on the same canonical listing and with its original countable enumeration. The final consumption first turns every attached logarithm li into eli −1. If |li| ≤vi ≤B, the mean value estimate gives |eli −1| ≤eBvi. A common total absolute budget supplies such a B, so the increment family has uniformly small summable tails. At each fixed label exponential continuity, applied to the bounded compared values under arbitrary varying configurations, gives convergence of the increment difference, and the varying-configuration selection argument makes this uniform in configurations. The resulting forced family is indexed by finite subsets of the union of the old hard labels and these exponential-increment labels. Its coefficient is their finite product when the canonical sup- port/eligibility test holds, and zero otherwise. The same test is used on both sides. Finite products of bounded compared values compare, and multiplication by this common indicator preserves the comparison even if the indicator varies with the history index. The finite-subset product budget preserves uniform tails. Canonical injective reindexing places these coefficients in the drawer, with zero outside its image; finite sums over the preimage establish preservation of both totals and tails. The regular family is likewise placed by its canonical support embedding. Hence the completed output arrays have the quantitative weighted difference budgets used in the activity-norm closure. This closes reset convergence without differentiating a hard indicator. 8.31 Closing the one-step convergence in the activity norm Suppose two incoming activity presentations are Cauchy with norm difference tending to zero, and their couplings have a common limit. The canonical-envelope argument above gives the ordinary regular-occurrence and forced comparisons. The common stationary path and kernel limits then supply the microscopic comparisons for the two physical steps. Their actual output source dictio- naries are expressed on one common listing by the local construction. The path reset is performed with those same dictionaries and the same committed coefficients. The coupling after this step is exactly βout = βin + αpath + ∆source. The projection comparison gives a vanishing difference of the two source increments ∆source, and the stationary path coefficients have the same limit. Thus the two output couplings differ by a quantity tending to zero. The regular and forced reset comparisons give the analogous convergence of the output activity presentations. To see the last implication in the actual norm, its budget characterization has two positive normalizing constants, breg and bforce. For a target ε > 0, choose the regular difference budget below (ε/2)breg and the forced difference budget below (ε/2)bforce, simultaneously at every anchor. The regular and forced comparison tail lemmas supply these bounds eventually. The common support and format data permit their use for the actual difference presentation. The norm characterization then bounds its norm by ε/2 < ε. Consequently the output norm difference tends to zero. This step requires the quantitative tail comparisons after reset, not just pointwise convergence of individual output coefficients. The incoming completion data used by the finite-history update satisfy exactly these hypothe- ses. Their common completion limit makes their original presentation sequences Cauchy and their difference tend to zero by the isometric embedding. The continuous physical-coupling map sends their common normalized-scalar limit to a common physical limit. The chosen physical step iden- tifies its actual output activities with the update’s original presentation element; no new output 219 ## Source page 222 A Wilson Yang–Mills construction J. Rodgers representative is selected for the comparison. The displayed one-step coupling identity also iden- tifies the scalar correction as output coupling minus input coupling minus the committed path coefficient. This proves the fixed-input update convergence invoked in the finite-depth argument, once the regular and forced reset tail comparisons are established. 8.32 Uniqueness of banked history limits The banked contraction compares histories at the same retained normalization. Its tail error tends to zero after the common finite-depth approximation is fixed. The double-tail Cauchy argument proves convergence of the whole relevant inner family, not merely convergence of an extracted subsequence. This is the convergence input used for the eventual full fixed-volume mesh limit. Here is the comparison that makes the diagonal extraction unique. Consider two banked limits with normalized scalar coordinates fi, f′ i, completed activity data Xi, X′ i, and local scalar correction limits gi, g′ i. Their bottom coupling limits at level zero agree. Write di = max{|fi −f′ i|, ∥Xi −X′ i∥}, γ0 = 1/128. The one-step comparison, applied to the two limits by approximation with actual finite trajectories, gives ∥Xi −X′ i∥≤8γ0di+1, |gi −g′ i| ≤4sidi+1. The scalar flow identity and equality at the base give, in physical coupling coordinates, Ωi −Ω′ i = − X j 0, the scalar difference is at most 4γ0D. Hence di ≤8γ0D for every i, and taking the supremum gives D ≤8γ0D = D/16. Thus D = 0, proving equality of every scalar and activity limit. The one-step inequalities arise by taking limits in the paid finite-trajectory comparison; the nearby finite trajectories are constructed by (FCA7) and (FCB5), with positive comparison weights and cumulative bounds (FCB1)–(FCB3); the fixed-scalar comparison identifies their activity limits. This is the precise contraction used to upgrade the subsequence construction above. 8.33 Extracting stationary path coefficients and their higher jets The path book retains stationary coefficients, their higher jets and the true supports of the corre- sponding rows. Absolute convergence at a smaller analytic width permits termwise differentiation. Insertion transfer then consumes those same coefficient comparisons, so that changing the path presentation does not amount to changing the represented local insertion. 220 ## Source page 223 A Wilson Yang–Mills construction J. Rodgers What the path extraction does and does not select The already established contraction fixes the banked activity limits. The path coefficients and their higher jets are additional presenta- tion records and may require compact extraction. Their selected limits are used to represent those same activities and insertions. The extraction does not change the Wilson family or turn its already unique banked limit into a choice between different physical values. Conversely, uniqueness of the banked reader alone does not assert convergence of every unreduced path presentation. There is a separate extraction behind the stationary paths just used. It concerns numerical path coefficients, not completed activity presentations. Combine all repeated occurrences of the same transported-curvature slot in each row by adding their coefficients. Do the same for each color component of each ordered pattern of two, three or four slots in the loop, cubic and quartic atoms. Denote this countable coefficient array by aN,m(κ), where N is the upper horizon and m the fixed level. The length l(κ) is the slot length or the sum of the pattern’s slot lengths. There are finitely many keys of bounded length: a bounded-length walk has a bounded number of steps from a finite alphabet, and there are only finitely many plaquette and color types. The path bounds give, for each row or color component separately, X κ |aN,m(κ)|eσml(κ) ≤Cm eventually in N. (SH1) Here Cm is the row allowance, the loop allowance for two-slot atoms, or the common higher-atom allowance for three- and four-slot atoms, as appropriate. An individual coefficient is bounded by Cm, because the exponential weight is at least one. Enlarge these bounds to include the finitely many exceptional N’s. The product of the resulting compact intervals, over the countable set of pairs (m, κ), is compact and metrizable. Every escaping horizon sequence consequently has one subsequence on which all these coefficients converge simultaneously. Write its limit am(κ). For each finite set of keys, pass the inequality (SH1) to the limit. Taking the supremum over finite sets gives (SH1) for am at the original width σm. More is true at every strictly smaller width τ < σm: X κ |aN,m(κ) −am(κ)|eτl(κ) −→0. (SH2) Indeed the part with length at most M is a finite sum tending to zero, whereas the remaining part is at most 2Cme−(σm−τ)M. (SH3) Choose M first and then the horizon. This order proves (SH2) and permits pairing the coefficients with every function bounded by Deτl. It also preserves the zero and first loop moments, since each displacement coordinate is bounded by path length and a polynomial in length is bounded by an exponential of any positive width. The row symbol at zero is such a pairing with a bounded indicator of the plaquette type. Its limit preserves the finite metric identity. Color invariance passes by the finite sum over color coordinates. Reconstructing the rows and atoms from this common coefficient array therefore retains all the original path hypotheses and allowances. The convergence includes derivatives of the unmasked physical functions at every order. To justify this assertion, complexify matrix entries around a physical configuration and use inverse links as independent conjugate-entry coordinates. The extension of a slot of length lis a holomorphic polynomial bounded on a sufficiently small common ball by Ceτl. A product of q slots has the same bound with Cq and the sum of their lengths. Equations (SH2)–(SH3) give uniform convergence of the row and atom series on that ball. Cauchy’s integral formula on a smaller ball gives convergence of each derivative, with its fixed radius loss and factorial. Restricting to real smooth parameter 221 ## Source page 224 A Wilson Yang–Mills construction J. Rodgers curves then gives convergence of all real Fréchet jets. This argument applies to the actual group words; no finite cutoffof the limiting path is substituted for them. For completeness, the stationary-graph derivative argument used to pass through the nonlinear minimization is as follows. Let PN(x, y) be the finite profile, let P be its limit, and let y = uN(x), y = u(x) be their analytic stationary branches through zero. Put FN = ∂yPN, F = ∂yP, and HN = DyFN(0, 0), H = DyF(0, 0). (SH4) The preceding all-order convergence implies convergence of every jet of FN, in particular HN → H. The stationary gap makes these vertical Hessians invertible. Inversion is continuous near the invertible H, so H−1 N →H−1. This is the inverse needed here, acting from vertical covectors to vertical vectors. Differentiate FN(x, uN(x)) = 0 repeatedly. With gN(x) = (x, uN(x)), the exact diagonal chain rule is a finite sum over partitions π of the derivative slots: 0 = X π D|π|FN(0)[ D|B|gN(0)[v|B|] : B ∈π ]. (SH5) The single-block term contains HNDkuN(0)[vk] and the known horizontal derivative Dk id(0)[vk]. Every other block has size less than k. Thus (SH5), solved by H−1 N , proves convergence of the k-th derivative by induction from the lower derivatives. In an ordered-partition convention identical single-block contributions are grouped with their positive multiplicity; dividing by that multiplicity gives the same recursion. Polarization of symmetric real multilinear forms supplies mixed deriva- tives. The case k = 0 is uN(0) = u(0) = 0. Composing the convergent graph jets with the profile jets proves convergence of all stationary value jets. There is also a precise cancellation at the first unmatched profile order. On a stationary graph g = (id, u), DP(0)[(0, w)] = 0, D2P(0)[Dg(0)v, (0, w)] = 0. (SH6) The first equation is stationarity at zero; differentiate stationarity once to obtain the second, and use Hessian symmetry for the reversed order. Suppose P, Q agree through order k + 2. Their stationary graph jets agree through order k + 1, by the recursion above. In the chain rule at order k + 3, any term involving a graph derivative of order at least k + 2 has either one block or two blocks with the other block of size one. These are exactly the terms killed by (SH6). All remaining graph derivatives match. Moreover the difference P −Q has zero jets through order k + 2, so its composition at order k + 3 has only the all-singleton partition. Consequently Dk+3(P ◦g −Q ◦h)(0)[vk+3] = Dk+3(P −Q)(0)[(Dg(0)v)k+3]. (SH7) This is the stationary value identity used for the cubic and quartic response comparisons (k = 0, 1); higher unknown graph derivatives have not been discarded without compensation. In the lattice profile, the linear map in (SH7) is the derivative of the actual fine logarithmic inclusion of the slice graph. Its derivative is the fine stretching map applied to (v, Du(0)v). A different tree-gauge representative changes it by an exact lattice gradient. Gauge invariance, to- gether with vanishing of all lower jets of the profile difference, makes its first nonzero derivative insensitive to such a gradient: differentiate the gauge identity and observe that every correction in- volving a nonlinear gauge derivative multiplies a lower, zero profile jet. Hence the stationary matrix response and its tree-extracted version give the same first unmatched value. Passing an eventual one-sided bound through the convergent branch derivatives proves the corresponding bound for the limiting coarse profile. Interchanging the two profiles gives the reverse bound. This is the response 222 ## Source page 225 A Wilson Yang–Mills construction J. Rodgers comparison used together with the decay and base matching estimates in the stationary-history construction. The one-loop profile also has a literal decomposition before passage to the limit: " loop action + 1 2 log det(∂2 yPN) −log Y e offtree J(ye) # ◦gN. (SH8) The determinant is nonzero at zero by the same positive vertical Hessian. Determinant is polynomial in its entries, logarithm is analytic near this positive value, and the Haar logarithms are analytic in the common chart. The jet convergence just proved therefore applies to (SH8), including its second jet. A finite inequality comparing this second jet to the adjacent draft (the coupling coefficient times the quadratic action plus its loop action) passes to the limit by continuity of absolute value. Field-independent scalar offsets are not identified by this argument. Finally the extra physical covariance and response rows have their own quantitative, full- sequence convergence. In their weighted row and column norms the finite suppliers give, for i < N ≤M, ∥UN,i −UM,i∥row, ∥UN,i −UM,i∥col ≤B clockk(i + 1)r  aN ai+1 c , c > 0. (SH9) For the covariance row the distance weight has exponent σ/Li at unit output scale; for the response row it has exponent σ at output scale Li. Since aN →0, (SH9) gives an entrywise Cauchy limit. Fix a finite set in a weighted row or column, pass its absolute sum to the limit M →∞, and then take the supremum over finite sets. This retains (SH9) with U∞,i in place of UM,i, in both norms and at precisely these scales. The limits thus supply the physical rows on the same common witnesses, with quantitative tails; they are not an additional independently chosen stationary subsequence. There is a further full-sequence assertion for the actual metric rows, which we separate from book extraction. At every fixed torus, the unamended diagonal matrix has the limit given by flat quadratic convergence. Its inverse matrices have entry bounds from one common positive strip allowance. A finite-dimensional compactness argument shows that the limiting diagonal matrix is invertible: extract any convergent subsequence of the bounded inverses, pass the identity DND−1 N = 1 to the limit, and obtain a right inverse of D. A square matrix with a right inverse is invertible. Continuity of inversion then proves convergence of the entire sequence of inverses, independently of the extracted subsequence. For the actual amended rows the prescribed construction, not merely agreement on curls, gives at that torus M−1 N = D−1 N + u(d2D−1 N )T(d2D−1 N ), (SH10) where u is the fixed amendment coefficient at this level. All operations are continuous finite matrix operations. Thus M−1 N converges. Its inverse MN converges by the same bounded-inverse argument, using the metric’s original strip allowance. This explains why the amendment must be retained from the same construction; agreement of potential Hessians alone would not imply (SH10). To recover cover coefficients from these all-period limits, take two possible coefficient limits R, R′ of a uniformly exponentially bounded row. Their folded kernels agree on every scheduled torus. Fix a cover displacement z and choose a torus whose centered box contains z. The folded coefficient at its residue is R(z) plus the other congruent coefficients. Their sum is bounded by a constant times the exponential tail outside that box, uniformly in the history index. As the scheduled periods grow, this tail tends to zero, hence R(z) = R′(z). Compact coefficient extraction therefore has only one possible limit; the full row converges coefficientwise. Splitting into a finite 223 ## Source page 226 A Wilson Yang–Mills construction J. Rodgers displacement set and its exponentially bounded complement proves convergence in every strictly smaller strip norm, exactly as in (SH2)–(SH3). The argument applies to the diagonal, its dual and the amended metric. The physical covariance and response rows have in addition the rate (SH9). The extraction (SH1)–(SH3) for nonlinear path books alone remains subsequential. The auxil- iary response comparison is used through exactly the orders it supplies: base matching through the second jet, then cubic and quartic comparisons (SH7), and the loop second-jet comparison (SH8). In particular any two extractions share the second profile jet, since it is the same quadratic matrix limit just proved. The recursive first-unmatched-jet argument compares the next two jets with their stated bounds. It does not assert equality of arbitrary all-order path books or field-independent scalar offsets. Whenever the subsequent primitive-continuity argument uses common path coef- ficients, its two sequences are taken from the already selected common book refinement. Full quadratic-row convergence and retained nonlinear-book convergence thus have different, explicit scopes. 8.34 From retained coefficients to physical row comparisons At a fixed step the retained-history comparison data consist of the limits of the upper and lower path books, hence of their quadratic/cubic/quartic atom coefficients, along the single common horizon refinement selected in (SH1)–(SH3). They also include the three cover-kernel limits and the coupling and activity limits. In this paragraph, two escaping subsequences mean two subsequences of that retained horizon refinement. Both therefore have the same book limits; this is not an assertion that arbitrary subsequential path books extracted before the refinement are equal. The coupling comparison is passed to the same tail ultrafilter, while the retained activity comparisons supply the regular-occurrence and forced-coefficient evaluations at arbitrary varying fine configurations. This constructs the comparison data for the local formulas. The full fixed-volume limit of the Wilson smeared integrals is a separate conclusion proved in “Extending the full fixed-volume mesh limit”; it is not obtained by silently declaring path-book compactness to be full convergence. The cutoffmakes each physical linear expression finite. There are only finitely many canonical path labels of length at most k: their plane type and integer displacement lie in a finite coordinate box. In shortening with length multiplier d > 0, every retained label has length at most k. Thus the cutoffshortening is a finite sum of its compared scalar coefficients times the corresponding transported curvature slots. The direct plotted branch is the fixed curvature function and needs no coefficient limit. Likewise a coarse routed kernel at cutoffk uses only cover displacements of coordinate length at most k. For a fine-to-coarse route from a lifted residue to Lz, route length is |Lz −lift(r)|1. Since |lift(r)|1 ≤4L, the retained displacements lie in the box of radius k + 4L. Its endpoint and cutoffpredicates are the same in both histories. Each nonzero summand is its compared kernel coefficient times adjoint transport along the same finite route. Smooth finite word multiplication, inverse and adjoint action therefore compare these summands and their derivatives. Summing proves comparison of all three physical kernels. Coarse vision is a finite contraction of these kernels with the compared plotted coarse fields. The linear field on a logarithmic fine direction is either the fixed discrete curl, a finite cutoffpath sum of such curls with the compared path coefficients, or the fixed coarse noise linear map. An oriented row additionally applies the common signed displacement rule and carrier adjoint transport. A reversed link uses the negative adjoint-transformed displacement; its base shift is the one already fixed by the oriented link convention. These operations are smooth finite sums and products and hence preserve comparison. The nonlinear energy rows are then their original finite scalar expressions: the direct clipped 224 ## Source page 227 A Wilson Yang–Mills construction J. Rodgers fine part plus the two coarse-response half-squares and cross terms; the wire curvature deficit and effective noise square; the negative coarse curvature deficit together with the two departure terms; and the tight-response half-square. The clipped or kept fine tests act on the same fine configuration in the two histories. Their varying numerical values can therefore be retained as bounded common factors, without taking a continuity limit through the test. The inherited regular source uses the regular-occurrence comparison directly; the atom and path-tail sources use their retained scalar coefficients and the same actual masked words. These prove the physical row and inherited-source comparisons feeding the packet and mean calculations above. The underlying retained stationary and activity convergence is the history input, whose extraction must use the common choices described in the construction. 8.35 Mean, local source and chain expressions The bill vector comparison can be made explicit. A selected row’s strain is its free residual vision plus the physical row operator applied to the saved log field and the hot part of the varying fluctuation. Selection and hot-coordinate sets are fixed by the draft. The saved log is the same all- argument local function on both sides; its uniform derivative bounds give its comparison. On hot coordinates use the compared fluctuation; on dormant coordinates its lens contribution is identically zero. Physical row comparison therefore compares the strain. For the fine free residual, compare the actual base-stroke map and subtract the compared coarse vision. For a noise residual, the base-stroke part is common; its derivative bounds hold in the clean hull ensured by viability and selection. Adding and subtracting the same compared vision then gives the residual comparison. Thus the clean-hull condition is used exactly on selected noise rows, rather than postulating clean fields everywhere. The bill vector is the transpose form applied to the selected zero-fluctuation strains, with their original scaling. Finite matrix-vector multiplication compares it. The protected mean is minus the transpose window-column matrix applied to this bill; hence it compares as asserted above. Its scaled stationary defect is the plate times this mean plus the bill, multiplied by −√β, and therefore compares as well. The positive common coupling limit is guaranteed by the lower coupling-window endpoint; taking square roots and reciprocal scales is consequently valid. The chain terms are now finite expressions in compared matrices and vectors. Their successive factors are the appropriate negative plate–covariance defect while the index is within the prescribed itinerary, and the plate factor at its terminal position. Matrix products compare these factors, with their order unchanged. The quadratic fluctuation contraction and the linear stationary-defect contraction compare by finite coordinate sums; the local cone cutoffs multiplying them are smooth functions of squared fluctuation norm. The blank-reference chain uses the empty store and identity output field, whose viability is automatic. Its plate has the same common lower bound, so the same inverse and determinant comparisons apply. The condensation source is one half the difference of the blank and actual local determinant expressions. The Haar source is smooth for all fluctuation arguments after its cutoff. Inside the cutoff support its argument lies strictly within the positive Haar chart radius, so the chart logarithm is smooth. Outside that support the source is locally zero. The strict chart margin covers the cutoff boundary and proves the all-argument claim. This justifies its use in products without extending an uncut logarithm beyond its domain. Finally the packet logarithm is the original finite combination −β(chalky + crispy) + X visited condensation + X plaza HaarSource −|pens| log(chartMass(β)). 225 ## Source page 228 A Wilson Yang–Mills construction J. Rodgers Here chalky is the posted root energy minus its selected quadratic costs; crispy is the selected quadratic cost at the protected mean. Each cost is a finite sum of squared strains over the actual parent rows. The root-energy comparison is the corresponding nonlinear energy comparison minus that same cost. The chart mass is positive and continuous at the positive coupling limit, so its logarithm compares. All posted, visited, plaza and pen sets are fixed finite sets in the draft. This proves the packet-log comparison from the strain, root-energy and determinant comparisons, with the original normalization retained. 8.36 The four vertex comparisons After fixing the draft, the ordinary integrand has the same four vertex types as its absolute activity bound. The pointwise comparison treats their predicates explicitly. For a connected block packet, connectedness and the containment/empty-flag test are fixed discrete predicates. If a predicate fails the corresponding factor is identically zero in both histo- ries. Otherwise the packet is the exponential of its compared energy expression, multiplied by the common contract factor, tickets and profiles, and the compared smooth local cutoffon its actual enlarged plaza. The fine-configuration ticket and profile factors may vary with the history index but are the same on both sides and uniformly bounded. Their output-field dependence uses the already chosen smooth wire profile. Thus the finite product comparison applies without requiring a limit of the ticket indicator. For an old forced vertex, compare its actual forced coefficient evaluated on the same fine con- figuration, as supplied by the retained forced-activity comparison, and multiply by the compared cutoffon its true footprint. For a regular exponential vertex, compare its logarithmic expression first and then apply the smooth exponential and subtract one. Its branches are inherited regular terms, the short local energy source, the undercurrent condensation term, the Haar term and the chain-row terms. The tests for being inside the posted region, for voyage membership and for hot exterior coordinates are fixed by the draft; their zero branches agree exactly. The nonzero branches use the protected mean, covariance/window, local logarithm and row comparisons already needed for these expressions, with the actual signs and factors of β retained. For a warning vertex, its mandatory/hot-region predicate is also fixed. The factor is a compared cutofftimes the actual fine profile or contract charge minus its constant reference value, or an exterior cone minus one. Fine profiles and finite label charges are bounded common factors; the exterior cone is a smooth function of the compared fluctuation direction. Its mandatory branch is identically the stated constant in both histories. Thus all warning comparisons retain their original hard predicates. Finite products over the molecule give the integrand comparison, and multiplication by the compared density gives the transformed integrand comparison. The derivative version uses the same four factors in their local smooth extended formulas, the same fixed fine configuration, and the compared steered fluctuation and corrected Gaussian coordinates. Its full finite product is identified with the original differentiated integrand by the exact local formula on a neighborhood of the healthy base. These statements compare the true vertices rather than replacing hard indicators by smooth approximations. The underlying energy, row, cutoffand Haar comparisons are the finite-coordinate ingredients of these four cases. 8.37 The marked primitive comparison The marked premium contains a test that its distinguished badge occurs in the molecule. The badge is fixed by the common mark key, so this test agrees in the two histories. If it fails, both premiums 226 ## Source page 229 A Wilson Yang–Mills construction J. Rodgers vanish. If it holds, the integrand replaces precisely that vertex by the paid marked ingredient and retains the product over the molecule with that badge erased. At fixed physical fluctuation and common draft, the reconstructed fine configuration is the same on both sides. The ingredient comparison is therefore the comparison of the two mark entries evaluated on this common, possibly history-dependent, fine configuration. Multiply it by the common finite enforcement cutoff, the compared ordinary product over the erased-badge molecule, and the compared Gaussian density. This proves the pointwise comparison of the marked amplitude. The exact reconstruction identity between the paid ingredient and this expression uses viability; it is applied on the eventual viable set. For the derivative integrand the marked entry remains evaluated on the fixed reconstructed fine input. Output-field derivatives act on its smooth steered cutoffand on the remaining ordinary vertices, as prescribed by the local reweighting formula. The local chart identity identifies the actual marked ingredient with this expression near the healthy base. The compared steering map gives comparison of the cutoff; the four ordinary vertex comparisons give comparison of the remaining product; the local Gaussian ratio and Jacobian give comparison of the reweighting factor. Their product is exactly the derivative integrand, not a derivative of a newly substituted marked function. This explains why convergence of mark entries under varying-configuration evaluation is sufficient here. The same Gaussian-polynomial majorant bounds these marked derivatives and permits the integral passage already proved. Finite boundary and forest averages then compare the actual marked grove. Its all-field bound also gives convergence of its uncut amplitude under arbitrary varying output configurations, using the fixed-draft argument separately at each gauge-slice point. This supplies both the regular marked factor in the one-mark cluster stream and the forced marked comparison used in the later reset. 8.38 Uniform derivative comparison of a single cluster stream A one-mark connected stream with r occurrences is its fixed Ursell coefficient divided by r!, multi- plied by r X ν=1 lν Y μ̸=ν uμ, where uμ is an ordinary primitive entry and lν its marked replacement. Its actual support is the occurrence union when the mark speaks to that label, and its zero alternative otherwise. The speaking condition depends on the fixed mark key and label inventory, so corresponding histories have exactly the same support. A silent stream is identically zero. The stream’s derivative bill is summable uniformly in the history index. To see the essential bound, let aμ, bμ bound ordinary and marked entries through derivative order j. The ordered product rule bounds the term with mark ν by (1 + r)jbν Y μ̸=ν aμ. Replace aμ, bμ by 2jaμ, 2jbμ. Since 1 + r ≤2r, this absorbs that derivative factor. The ordinary bounds may be taken as the smaller of their uniform primitive envelope and their original purchased budget; taking the smaller preserves summable tails. A positive reserve d, selected by the one-mark budget argument, makes the inflated array fμ = 2jaμ + d 2jbμ fit the connected-cluster reserve. 227 ## Source page 230 A Wilson Yang–Mills construction J. Rodgers Expanding its product with nonnegative entries gives d X ν  Y μ̸=ν 2jaμ  2jbν ≤ Y μ fμ. Thus a valid stream bill is d−1 times the ordinary absolute connected-cluster weight for f. The rooted cluster estimate makes this family uniformly summable with small tails. Multiplication by the fixed Ursell factor and r!−1 is retained exactly. For a fixed speaking stream, a healthy union makes every occurrence support healthy. The primitive comparison along any allowed configuration family then compares all its ordinary factors and its single marked factor through derivative order j. Finite product and sum differentiation compares the stream itself. For a silent stream the assertion is immediate. Passing through every ultrafilter above the tail filter turns this into ordinary convergence of the derivatives of any fixed- order allowed test family. Here is why such test-family convergence gives a uniform paid derivative bound, including on the closed healthy set. If an order-k derivative bound failed uniformly over the allowed maps from Rk and healthy base configurations, select a violating map at each bad history index. The assumed convergence for that sequence of maps contradicts the violations. Take the finite intersection for 0 ≤k ≤j. This yields a single late cutofffor all these orders. To extend the test to an arbitrary finite-dimensional parameter space and directions v1, . . . , vk, define Lz = 1 k + 1 k X ν=1 zν bvν, bvν = ( vν/∥vν∥, vν ̸= 0, 0, vν = 0. With the coordinate maximum norm, ∥L∥≤1. The vectors aν = (k + 1)∥vν∥eν satisfy Laν = vν. Pulling an allowed map back by z 7→Lz + p preserves its allowed derivative bounds. A test bound b/(k + 1)k then bounds its derivative on the aν by b Q ν ∥vν∥, proving the full operator bound b. Smoothness for this subtraction is furnished by the separate primitive derivative bills at the healthy base; the mask itself is never differentiated. Finally convert these arbitrary-map derivative bounds to the anchored paid direction convention. At a fixed period 1 + dist(x, y) is bounded by one finite Y . The conversion costs at most Y p.power j, independent of the history index. Divide the desired tolerance by this constant before applying the uniform comparison. The identity configuration makes the healthy test set nonempty. This proves the regular comparison for a stream at its actual fixed support. Real and imaginary streams are treated separately, multiplied by their common normalization and recombined, then embedded by zero into the common enumeration. This supplies the microscopic regular estimate used above. 8.39 Passing microscopic comparisons through projection The microscopic regular comparison means that both arrays are regular on one fixed family of supports, have uniformly summable paid bounds with uniformly small tails, and, at each fixed label, their difference eventually has paid derivative bound below every positive tolerance. The forced comparison means both arrays have absolute bounds with uniformly small summable tails and, for each fixed label and every sequence of configurations Un, the difference of their values at Un tends to zero. The latter implies eventual uniform smallness over configurations: otherwise choose a violating configuration on each bad index to contradict that sequence test. Neither notion assumes continuity of the masks. The tail hypothesis gives a useful dominated-series principle. If two arrays have uniformly summable absolute tails and their fixed-label differences tend to zero, split the sum into a finite set 228 ## Source page 231 A Wilson Yang–Mills construction J. Rodgers and its complement. The finite difference sum tends to zero, and the two tails have an arbitrarily small uniform total. Hence the difference of the full series tends to zero. The same proof permits evaluation at arbitrary varying configurations. Forced comparisons are therefore preserved by summing a series, by collecting labels into finitely many barcodes, by a fixed injective reindexing with zero outside its image, and by finite sums and scalar multiples. They are also preserved by products: each fixed label is uniformly bounded, and fg −f′g′ = (f −f′)g + f′(g −g′); a global bound on one absolute-budget family times the other’s tail budget controls the product tails. Let Π denote the fixed weighted local section projection, with projection price Cread. The core of the collected raw insertion is its seed plus the sum of Π applied to its bulk terms. For regular germs this projection is linear, including subtraction. Its paid bound gives |Π(f −g)|1 ≤Cread paidBound(f −g). At each label the right side tends to zero, and the projected rows have uniformly summable tails by that same price. Thus the projected core difference tends to zero whenever the seed difference does. The residual and rebate operations used to remove this core are likewise linear on regular germs at the common threshold and support. Their fixed paid conversion price bounds the residual in the enlarged support and bounds the rebate everywhere. Apply these operations to the difference of the two bulk terms. Arbitrarily small paid difference bounds therefore give arbitrarily small residual paid bounds and arbitrarily small all-field rebate differences. Their uniform tail budgets are multiplied only by the fixed conversion price. Placing residuals in the common canonical support enumeration, with zeros at unused indices, preserves the regular comparison. The new forced profile is the old forced profile plus the threshold hook of the collected rebates. The hook is a finite sum over support classes, multiplied by the actual compatible products of forced activities. These products agree in their discrete support and flag predicates because the two representations have the same forced address data. Their numerical factors have the forced comparison by the assumed comparison of the original forced activities, finite-product stability, and summable product budgets. Multiplying by the compared rebates and summing therefore preserves the forced comparison. Finally the common keeper barcode embedding collects these profiles and extends them by zero outside its image; this preserves both the convergence and tail bounds. Thus projection and collection pass the microscopic regular and forced comparisons to the output formatted insertion. 8.40 The common input decomposition for a shot For fixed input capture x and frame index i, a shot is the merged sum of three kinds of contribu- tions: the primary frame insertion, one contribution for each regular remainder label, and one for each forced barcode label. The input is the same book in both histories. The common incoming orientation fixes its threshold, forced addresses and barcode profiles. Stowing the forced input through the keeper therefore gives one common function of the fine configuration for every fixed forced label, independent of the history index and of which of the two histories is used. Each of these contributions has a summable common tariff. For the regular inputs choose their paid regular bounds vj, and use tariffψ p.mass(Dj, vj). For forced inputs choose all-configuration bounds cj, and use ψ p.mass(Kj, cj)/κ1, with the same positive incoming forced scale κ1. The input capture’s wealth bound makes both arrays summable. The finite primary contribution uses the larger of its heading tariffand its heading-plus-core tariff. The paid shot estimates bound not only each remainder but also its output core coefficient by this array. Thus it is a valid majorant for the countable comparison argument. 229 ## Source page 232 A Wilson Yang–Mills construction J. Rodgers For a regular label the actual input ingredient is the real or imaginary part of its fixed masked regular function; the preparation construction has exactly that ingredient, its actual logarithmic key, and weight at most vj. For a forced label the ingredient is the real or imaginary part of the common stowed function and has its actual gas key. If that function is zero, the output contribution is identically zero in both histories. Otherwise its input bounds and eligibility allow the same preparation construction, with weight at most cj. These are genuine fixed ingredients across the two histories, although their auxiliary preparation data may vary. The nonconstant primary frame term likewise uses the fixed masked section insertion at the common incoming threshold, split into real and imaginary parts. For the constant frame index, preparation is unnecessary: the output is the same scaled unit insertion in both histories. Its forced profile is zero, and stowing that zero profile through any keeper leaves it zero; this proves equality even if the two keepers have different presentations. We now establish the microscopic comparison of these preparations. Fix the scale, its block ratio L, the degree d, and the incoming dictionary. Both histories have that same dictionary and budget at this scale. Their couplings approach each other, their regular and forced activity coefficients have the weighted comparison already proved, and their stationary kernels and path responses have the common limits. Thus their local Gaussian covariances, means, saved fields and vertex factors have precisely the comparisons established earlier in this chapter. A preparation of a fixed real ingredient retains its wing type and a common bound on its input weight. Converting the preparation to the common local chart preserves its ingredient, wing and weight. Consequently the two converted real ingredients agree exactly, as do the two imaginary ingredients. This is stronger than convergence of a possibly changing representative. The support address of a local integrated term depends on the wing and its fixed combinatorial fragrance address. It therefore agrees in the two histories and throughout the comparison sequence. Denote this support by Eα. The insertion format ensures 0 ∈Eα. Apply the mixed marked-integral comparison to each such address. Its hypotheses are now the identical ingredient just identified, the compared Gaussian data and vertices, and their common summable derivative envelopes. The change to a fixed Gaussian integration variable, the two dominated-convergence arguments, and the finite-order derivative estimate give: for every address α and every ε > 0, the difference of its two real bulk terms eventually has paid bound ε, on Eα, at the common output threshold and derivative order. The same statement holds for the imaginary bulk terms. Their separate envelope arrays have the uniform tails of the marked cluster majorant. Multiplying by the fixed block factor Ld and recombining real and imaginary parts preserves these conclusions. Extending the address enumeration by zero preserves them as well. We have therefore proved the bulk comparison, including both regularity on the common support and the summable tails needed for projection. The forced part also retains its actual address data. Its construction first restores the missing ordinary logarithms, then restores the marked slots, and finally applies the coupling reset to the common shelves. For the ordinary logarithms, the finite-set/tail comparison of the reset families proves convergence and common tails. The marked slots have the mixed-integral comparison just obtained; inserting the prescribed finite subsets and extending by zero preserves it. Products of the restored ordinary exponentials with the marked factors preserve convergence and their summable envelopes. Finally the reset comparison applies to the same path family, threshold, shelves and rebate operations on both sides. The forced output profiles consequently have uniformly summable bounds and satisfy Fn,α(Un) −F ′ n,α(Un) −→0 for every sequence of configurations Un. (HL1) This is the strong all-configuration comparison used here; no continuity of the masks is involved. 230 ## Source page 233 A Wilson Yang–Mills construction J. Rodgers For clarity, (HL1) implies the corresponding uniform comparison. Otherwise there are ε > 0, a subsequence nj, and configurations Unj where the difference exceeds ε. Complete these choices to a sequence by arbitrary configurations at the other indices. It contradicts (HL1). Thus the supremum of the difference tends to zero. Combined with the uniform tails, the finite-set/tail argument also proves convergence of each summed forced profile. The resulting real and imaginary forced comparisons recombine as Ld(Fn + iGn) −Ld(F ′ n + iG′ n) = Ld(Fn −F ′ n) + i(Gn −G′ n) , with the same fixed L and d. This proves the required forced comparison for the raw complex preparation. Its primary seed has the already compared coefficient. The linear projection estimate, residual and rebate bounds, and common keeper extension from the preceding section now give convergence of the projected core and arbitrarily small bound on the collected remainder. The countable ingredient summation and finite-frame assembly therefore use proved local comparisons, all on the original common input book. 8.41 Summing local insertion comparisons The local comparison feeding the transfer metric preserves the same coefficient supports in the two histories. It consists of convergence of their finite core coefficients to each other and, for every ε > 0, an eventual remainder bound ε for their difference in the common output orientation. The difference is formed by appending the negative of the second labelled remainder to the first, so it remains a legitimate formatted insertion on that common support. Here is the summation principle needed for countably many local comparisons. Suppose the i-th pair of insertions each has a common summable bound fi ≥0, and its difference has a remainder bound tending to zero with the history index. The coarse bound on that difference is 2fi. Given r > 0, choose a finite set E with P i/∈E 2fi ≤r/2. On the finitely many indices in E, eventually choose the common bound v = r/[2(|E|+1)]. Use v on E and 2fi outside. This is a valid per-index bound and its total is at most r. The paid merge bound therefore gives an eventual remainder bound r for the full merged difference. Absolute convergence and the same label enumeration identify this merged difference with the difference of the two merged insertions. The finite core sums converge by the same finite-set/tail split. Thus countably many local comparisons assemble into the full comparison. The marks of a fixed coefficient input are obtained by a finite sum over the frame indices and the prescribed motion chips. Their shot inputs are the same captured coordinate-axis data in both histories. The local comparison for each such shot passes through its fixed motion and through this finite sum by the formatted-insertion operations. Hence the resulting site marks have converging core difference and arbitrarily small common remainder bound. Finally each actual coefficient readout equals that site’s core, and its recorded capture equals capture of that site mark under the common output keeper. Each site mark has a finite paid remainder bound. The identity collecting a difference under the keeper identifies the difference of their recorded captures with capture of the formatted difference mark. Its arbitrarily small remainder bound bounds the capture norm by the same small amount. This proves convergence to zero both of the coefficient-isomorphism outputs and of their recorded capture outputs. These are precisely the fixed-input comparisons used in the transfer argument; the preceding microscopic comparison supplies their local analytic input. 231 ## Source page 234 A Wilson Yang–Mills construction J. Rodgers 8.42 The transfer comparison metric Let |x|1 = P ν |xν| be the coefficient size in the fixed finite frame. A Gram datum P contains a complex-linear capture map gP into the insertion-data seminormed space, with ∥gP x∥≤a|x|1. The comparison relation P ∼ε Q means ε ≥0 and ∥gP x −gQx∥≤εa|x|1 for every x. It need not separate distinct Gram data. It is symmetric, satisfies the triangle inequality in ε, and every pair is comparable with ε = 2. For coefficient maps, the corresponding comparison is |Sx −Tx|1 ≤ε|x|1. Both uniform comparisons follow from bounds on the finitely many coordinate axes: expand x in that basis and sum absolute values. An adjacent transfer consists of a flow P 7→Φn(P) on Gram data and a coefficient isomorphism Sn(P). Its established paid inequalities are |x|1 ≤b|Sn(P)x|1, |(Sn(P) −Sn(Q))x|1 ≤cε|x|1, Φn(P) ∼ε/2 Φn(Q) whenever P ∼ε Q. At fixed input P, both the flow outputs and the maps Sn(P) are Cauchy in their respective comparisons. These statements concern the actual integrated transfer maps; outside the eventual admitted schedule they may be completed by the zero capture map and the identity coefficient map. This harmless completion supplies total sequences while preserving all eventual statements. To recover flow comparison from coefficient-normalized quantities, suppose gXV = T, gZW = U, where V, W are coefficient isomorphisms, their difference has comparison cost c0, the maps T, U differ by at most c0|x|1, and |x|1 ≤b|V x|1. Put k = V −1i. Then gXi −gZi = (Tk −Uk) + gZ(Wk −i). The two terms are bounded by c0b|i|1 and ac0b|i|1. Thus X ∼lZ whenever (1 + a)c0b ≤la. This proves the normalized comparison without an inverse estimate on an untested ambient space. Pointwise convergence of the normalized maps along every pair of escaping history subsequences implies the uniform Cauchy comparisons. First apply it to each of the finitely many coefficient axes, obtaining a common bound and hence a uniform map bound. Next use the preceding inverse- normalization estimate for the flow. Finally, if an eventual pairwise tail bound failed, one could choose bad indices u(n), v(n) ≥n. Both escape to infinity and contradict the assumed subsequence convergence. This turns the subsequence statement into an actual double-tail Cauchy estimate. If inputs Pn themselves are Cauchy, the flow outputs are Cauchy too: insert one fixed late input PN between Φn(Pn) and Φm(Pm). The two outer errors are halved by contraction and the middle error is small by fixed-input convergence. Similarly insert PN in the coefficient maps; their two outer errors cost c times the input error. Taking the input tolerance to be ε/[4(c + 1)] and the middle tolerance ε/2 proves the uniform coefficient-map comparison. These establish the precise transfer properties used in the finite-depth approximation and coefficient Cauchy argument below. 8.43 Fixed-level convergence of transported insertions We explain the compactness mechanism for insertion coefficients before taking the continuum limit. At a fixed physical period, let Wi,n be the integrated Gram data at level i in the n-th retained history. Adjacent levels satisfy, eventually in n, Wi,n = Ti,n.flow(Wi+1,n). 232 ## Source page 235 A Wilson Yang–Mills construction J. Rodgers The paid transfer bounds make each flow contract the comparison error by one half and carry the two-part Cauchy property of its input to its output. All admitted Gram data at one level have mutual comparison error at most two. Iterating d levels and replacing the last input by fixed admissible data therefore approximates Wi,n, eventually, with error at most 2 2−d. For fixed d the approximating sequence has the two-part Cauchy property by the finite transfer bounds. Choose d so the approximation error is below a quarter of the desired tolerance, and then use this Cauchy property. The triangle inequality proves that Wi,n itself has that property at every fixed level. Only finitely many eventual identities are used for any such approximation. Here is the inverse-normalization argument for coefficient transport. In the finite-dimensional coefficient norm, suppose xn = Snyn eventually, xn is Cauchy, the maps Sn are Cauchy uniformly on bounded inputs, and |v|coeff≤b |Snv|coeff with fixed b > 0. First xn is bounded, so yn is eventually bounded by the same inequality. Next Sn(yn −ym) = xn −xm + (Sm −Sn)ym. The first term tends uniformly to zero for large n, m; the second does so by boundedness of ym and the uniform map comparison. The inverse bound proves yn Cauchy. Equivalence of the finite- dimensional coefficient norm and the ambient norm converts this to the usual Cauchy statement. Apply this successively to the actual adjacent transfer relations for the coefficients normalized by the reference inverse. Their lowest retained coefficient is eventually the fixed terminal coefficient y, by the based transfer condition, and is therefore Cauchy. The Gram comparison just proved supplies the Cauchy comparison of each transfer map; its paid inverse bound supplies the uniform b. Induction upwards gives Cauchy coefficient sequences at every fixed level. This uses inverse bounds on the actual graph input at that level, without assuming a bounded product through arbitrarily many future levels. The corresponding captured insertion data are Cauchy as well. If gn is the integrated capture map at that level and xn its Cauchy coefficient, write gnxn −gmxm = gn(xn −xm) + (gn −gm)xm. The uniform map bound controls the first term, and the second part of the Gram comparison controls the second on the bounded coefficient sequence. This constructs the Cauchy sequence of captured data used for the fixed-level density limit. In the retained-history construction the terminal horizons tend to infinity. Thus every fixed level eventually lies below the horizon, all coefficients and captured data above refer to the original common history choices, and the path at that level is the prescribed stationary path. The history conditions provide the corresponding uniform lawful, scalar, regular, forced and path budgets. At a fixed level the Cauchy coefficient sequence is bounded; multiplying its bound by the fixed normalization scalar gives the uniform insertion payment bound. The exponential insertion and each forced profile are measurable: they are sums of masked measurable regular terms, followed by continuous pullback and either complex conjugation or identity according to the chosen motion. These are the measurability inputs for the fixed-level integral passage. 8.44 Uniform convergence of fixed-level insertion integrands At a fixed level the comparison required for integration is uniform over all configurations; no continuity across changing masks is assumed. More precisely, a sequence of functions has the needed property if it is eventually uniformly bounded and is uniformly Cauchy over its entire 233 ## Source page 236 A Wilson Yang–Mills construction J. Rodgers argument set. This property is preserved by eventual equality, fixed pullback, finite sums and products, and continuous functions on the finite-dimensional bounded target ranges. For the last assertion the common bounded range lies in a compact ball, on which the continuous function is uniformly continuous. For captured insertion data x, let Lx(U) be its summed masked regular remainder and let Fx,B(U) be its forced profile with barcode B. The capture norm and paid local mass bounds give absolute convergence and |Lx(U)| ≤∥x∥, |Fx,B(U)| ≤k∥x∥, uniformly in U. Masks have absolute value at most one; for the regular terms apply the healthy paid bound when the mask is one and use zero otherwise. For the forced profile restrict the summable forced budget to the specified barcode. Both maps are linear in x: absolute convergence permits termwise subtraction, including subtraction under the fixed masks. Thus the same inequalities hold for differences. A Cauchy sequence of captured data consequently gives uniformly Cauchy regular remainders and every fixed forced profile, with eventual uniform bounds. The finite core insertion is also controlled uniformly in the configuration. For each coefficient- axis vector, its placed core function is continuous on the compact finite lattice configuration space and hence bounded; its healthy mask cannot increase that bound. Expanding a general coefficient in these finitely many axes gives a uniform Lipschitz bound in its coefficient norm. The constant coefficient is handled in the same way. Therefore a Cauchy sequence of core coefficients gives uniformly Cauchy core insertion functions, including their masks. The format and capture identities identify the actual exponential insertion entry with the sum of this core function and the captured regular remainder, eventually in the history index. They identify the actual forced profile with the captured barcode profile as well. The preceding uniform bounds therefore prove the fixed-level convergence assertions for both actual entries and all profiles. A fixed translation, signed motion, complex conjugation when required, and fixed normalization scalar preserve these assertions. This proves the translated insertion convergence at any fixed anchor from the origin construction, without choosing a new normalization there. For later use, integration of such a sequence against a probability measure is Cauchy whenever its terms are eventually measurable: the uniform bound gives integrability, and the norm of the difference of two integrals is at most the uniform norm of their difference. Applied to the finite polynomial restore expression, finite-dimensional continuity and finite sums/products preserve the same convergence. The common positive lower bound on its unmarked denominator then permits the normalized quotient limit. This connects the captured-data convergence to the fixed-level normalized expectations. 8.45 Exact unmarked density and component regrouping Here is the unmarked version of the molecule expansion, including its coverage condition. Fix one forest throughout, one probability Haar measure in its gauge coordinates, and the common output configuration. A packed address consists of a store, a finite warning-label set, and the remaining labelled lists. The map sending this data to its molecule uses the prescribed block layout of that store. Reading the roster, warning ownership and lists from the molecule is a left inverse. Conversely, every molecule satisfying that layout is recovered from the read data. Thus this map is injective and misses only addresses whose activity is zero. For each store and boundary choice, the local expansion has summable terms. Its absolute majorant is the scalar tribute times a product of nonnegative vertex tariffs. The tarifffamily is summable; hence its finite-subset products are summable, since their total is bounded by the exponential of the tariffsum. Restricting to the injectively encoded addresses preserves summability. 234 ## Source page 237 A Wilson Yang–Mills construction J. Rodgers Integrating the expansion and then summing the finite boundary choices gives the store’s exact cut entry. The scalar OX = levyX deductedX is unchanged by replacing the store and assigning a boundary choice: the neutrality identity cancels their apparent effect on the scalar correction, with the forest held fixed. If there are b boundary choices, the tribute for each is exactly OX/b. Their sum is therefore OX times the averaged molecule receipt. Summing over stores, and using the full discrete sector coverage, yields the absolutely convergent identity push(R; V ) = OX X p receiptX(p). This is a density relative to probability Haar in the output variable; no unit partition normalization has been introduced. To factor this sum into connected activities, partition each molecule into the connected com- ponents of its actual enlarged supports. These components have pairwise disjoint grounds and their union reconstructs the molecule. A compatible family contributes only if its roster flags cover exactly the output flag set, and each component satisfies its own layout, eligibility and flag-read conditions. These conditions are equivalent to the union molecule’s layout and approval. For exam- ple, intersecting the union flag set with one component’s ground leaves precisely that component’s flags: flags from another component cannot enter, by disjointness and the fact that a component’s flags lie in its ground. Define a component activity by planting all output flags in its local data and then applying its own eligibility test. On its ground, this planted data agrees with the data from the union roster. The same holds for the selected regular keys: a key belonging to another component cannot meet this ground. Locality of the integrated activity therefore identifies the planted component activity with the union-data activity. Independence on disjoint grounds factors the latter product, including the independent finite boundary averages. Hence receiptX  [ p∈A p  = 1{compatibility and exact flag coverage} Y p∈A aX(p). If the union approval fails, either flag coverage fails, making the displayed contribution zero, or at least one component fails its eligibility condition and its factor vanishes. Thus this equality includes all zero cases. The connected-component map is injective because union is its left inverse. Every family outside its image has zero compatible contribution: a nonzero compatible family consists exactly of the components of its union. Reindexing the absolutely convergent sum proves push(R; V ) = OX X A finite 1{compatibility and exact flag coverage} Y p∈A aX(p). Finally OX > 0: its Gaussian chart mass and kernel normalization factors are positive, and its other factors are exponentials or positive powers of those masses. The extensive scalar stays outside the connected gas. The argument has not asserted locality of that scalar or performed the subsequent coupling reset. The frozen data in this identity are the store and boundary choice after fixing the gauge slice; the threshold, width and selector belong to the fixed paid chart. The comparison of these expressions uses the common drafts, principal covariance matrices and Gaussian densities constructed earlier in 235 ## Source page 238 A Wilson Yang–Mills construction J. Rodgers this chapter, together with the bill and protected-mean comparison and the four vertex cases. For a marked product it uses the same comparison with its finite marker coefficient retained. The next subsection states how these addresswise comparisons pass through the common source, its products and the reset; the signed line estimate (LCN11) gives the corresponding parameter control. No varying discrete approval predicate is differentiated. 8.46 Comparison of the common source and its marked products Fix a period, a derivative order j, and a budget B of order at most j, with threshold equal to the common temperature. All addresses and all finite trial tori in this paragraph are fixed before either history is varied. A regular comparison means common sparse summable envelopes in the B-seminorms and convergence, for each address, in those seminorms. A forced comparison means common sparse envelopes in the uniform norm and addresswise uniform convergence. These are stronger than pointwise comparison of the total action; the distinction permits the subsequent products and reset operations. For the connected logarithmic stream, the local Gaussian comparison already proved gives convergence of every derivative through order j, along every sequence of healthy configurations and admissible test maps. This implies uniform convergence in the healthy-configuration derivative seminorm: otherwise choose a violating configuration and test map for each member of a bad subsequence, contradicting that sequential statement. On a fixed torus choose Y ≥1 + dist(x, y) for every pair of sites. Conversion from admissible derivatives to anchored B-derivatives costs at most T = Y B.power j. Apply the uniform derivative result with tolerance ε/T. The common stream envelopes, multiplied by T, remain sparse and give the required regular comparison. The forced connected stream is handled by the same bad-sequence argument with order zero; its separate sparse envelopes were obtained from the forced integral estimate. In particular, the ultrafilter statements used to express local Gaussian limits here imply ordinary sequential convergence, since every free ultrafilter on every bad subsequence would give the same contradiction. The path contribution is the sum of four already defined terms: the low path term with coef- ficient −vn, the cubic and quartic terms with coefficient −βn, and the loop quadratic term with coefficient −1. Their coefficient arrays converge in the common weighted path norms. The scalar sequences vn, βn in the two histories converge to the same respective limits and are uniformly bounded. For any such scalar pair an, a′ n and regular pair Fn, F ′ n, use anFn −a′ nF ′ n = an(Fn −F ′ n) + (an −a′ n)F ′ n. The uniform scalar bound preserves sparse envelopes, while the two terms on the right tend to zero at each address. Thus all four path terms compare. Finite frame averaging preserves this conclusion: transport by a frame costs the fixed factor turnFee(B), and multiplication by the averaging weight costs Rim(c). For a target tolerance ε, use the original comparison with tolerance ε/(Rim(c)turnFee(B)). The finite frame index multiplies each sparse envelope only finitely many times. Consequently the full regular source, the negative connected logarithmic stream plus these path terms and their prescribed frame average, has a regular comparison. The hard source has a forced comparison, including its zero branch and its canonical label embedding. The enumeration used to place these terms in the common dictionary is the same for both histories. The finite trial correction requires one additional observation. At the fixed period both selection panels are eventually contained in a single finite label set, and their cutoffparameters eventually 236 ## Source page 239 A Wilson Yang–Mills construction J. Rodgers exceed every label in that set. Their panels therefore agree eventually. Removing the selected terms from the regular source preserves its sparse envelopes and addresswise convergence. Likewise the selected site contribution is a finite sum over this common set with the same fixed selection weights. Its comparison follows term by term. There is no passage through a discontinuous moving selection rule. A trial on the fixed small torus is pulled back through its prescribed padded shape map. Its support lies within the fixed radius z, and a healthy neighborhood of that radius maps into the all-healthy trial torus. The derivative pullback costs at most M = 2B.order(1 + z)B.power B.order. Using tolerance ε/M on the trial torus proves comparison after pullback. The ornament is a finite sum over the trial types and sites, with fixed ray weights, so it has a regular comparison. The corresponding failed-trial contributions have a forced comparison by the same finite-sum argument. The site correction is exactly the selected regular contribution minus its value at the identity configuration minus the failed-trial contribution. The identity is healthy; hence evaluating there preserves the regular comparison. Subtraction proves the uniform site comparison, and the finite set of sites gives the forced comparison required for the correction. These facts assemble the common source: its regular part is the unselected regular source together with the trial ornament; its hard part contains the original hard source and the negative frame average of the site correction. Every sum uses a common disjoint label set. This proves comparison of the complete source, rather than merely of its total evaluated function. For clarity, the marked-product step can also be written without the source notation. A marked family un,i(t, V ) carries sparse bounds for |un,i(t, V )| + |∂tun,i(t, V )| and convergence of both quantities along arbitrary configuration sequences, at the prescribed pa- rameter values. An unmarked compared logarithm ln,i(V ) becomes a constant marked family. Its exponential factor is eln,i −1; on a common bounded interval the mean-value theorem transfers the logarithmic comparison to this factor and preserves the sparse envelopes with the fixed exponential multiplier. Extend both families by zero through the same label embeddings. A resulting forced atom is a finite product of the embedded marked factors and these unmarked exponential factors, multiplied by its eligibility indicator. The two histories have exactly the same eligibility predicate on the common configuration. Product differentiation, ∂t Y i∈A ui = X i∈A (∂tui) Y k∈A\{i} uk, therefore proves comparison for both the atom and its mark derivative. No continuity of the eligibility indicator is needed: it multiplies both differences by the same number in {0, 1}. The sparse product bound established earlier controls the sum over these finite subsets. Widening the source is another common zero extension, so the same argument applies after widening. Finally evaluate ∂t at t = 0. Its absolute value is bounded by the marked envelope, and its comparison is the derivative part just proved. This supplies the forced marked reset comparison used in the two-history output limit. 237 ## Source page 240 A Wilson Yang–Mills construction J. Rodgers 8.47 Analytic control of omitted insertion terms One estimate needed in the continuum comparison is obtained by scaling all restored insertion errors with a single complex parameter. The elementary analytic fact is this: if f is holomorphic on |z| < S, S > 2, and |f(z)| ≤B there, then |f(1) −f(0)| ≤2B S . (IC1) For every z on the segment from zero to one, its closed radius-S/2 disk lies strictly inside the radius-S disk. Cauchy’s derivative estimate gives |f′(z)| ≤B/(S/2). Integrate along that segment to obtain (IC1). Here the normalized insertion moment at level j is Stampj,t(P) = Z (P(U))t d priorj(U) ZN , where t is an insertion-polynomial coefficient and ZN > 0 is the original Wilson partition function. The denominator is the same at every intermediate level, by the exact transport identity. Suppose P is served by the insertion family at level j, and its restoration formula is P = restore(P0, P1). Define P(z) = restore(zP0, zP1), f(z) = Stampj,t(P(z)). Then P(1) = P, while P(0) is the restoration with the two error inputs removed. The retained insertion estimates give holomorphy on the disk of radius S = brisk(Bj) > 2. They also give integrability for every z in that disk. The scalar complex parameter uses the available support- exponent margin of two; no geometric support is changed. Transport P(z) to a fixed earlier level k < j. The retained transport theorem keeps its served insertion bound there. Integration is unchanged by transport, including after taking the finite polynomial coefficient t; coefficient extraction is a continuous linear map and commutes with this integrable finite-dimensional integral. The root small-field estimate at k therefore gives the uniform disk bound |f(z)| ≤exp | Sitek | treasuryk +n(x + y + 2)  toll(wk, t) =: Bk,t. Here n is the number of insertion slots, x, y are their two root norm allowances, and the toll uses their positive weights. All volume dependence is at the fixed reporting level k; there is no exponential cost involving the increasingly fine level j. Applying (IC1) gives Stampj,t(P) −Stampj,t(restore(0, 0)) ≤ Bk,t brisk(Bj)/2. (IC2) This is the uniform omission estimate. It follows from the exact insertion transport and the common analytic disk, and does not require convergence of the intermediate representations of the mixed pin terms. 9 Physical labels and their detection Physical labels are first finite Taylor-polynomial coefficient objects. Radial gauge fixes their po- tential representative before equations are imposed; the curl and radial reconstruction maps below 238 ## Source page 241 A Wilson Yang–Mills construction J. Rodgers are exact inverses on the Bianchi subspace. Three degrees must be distinguished: polynomial posi- tion degree, engineering weight and field-count degree. Later scaling formulas use the engineering weight, whereas Wick products also keep track of field count. 9.1 Radial Taylor variables and the free equation space The jet variables in this chapter are Taylor coefficients, with the factorial already divided out. At order j ≥0, a radial potential coefficient is a family Ai of symmetric (j + 1)-linear maps from R4 to the real color space. Its associated homogeneous potential is ai(x) = Ai(x, . . . , x), and the constraint is X i xiai(x) = 0 for every x. (RJ1) This finite-dimensional real space is the radial Taylor space at order j. Its polynomial-position degree is j + 1, its engineering weight is j + 2, and its field-count degree is one. Polynomial labels inherit their weight and field count by addition over their factors; constants have both degrees zero. Complex labels are obtained by complexification. No field equation has yet been imposed on these radial variables. Define the curvature coefficient by the homogeneous polynomial Qil(x) = ∂ial(x) −∂lai(x). (RJ2) On symmetric tensors this means freezing one input at ei, subtracting the reversed expression, and multiplying by j + 1. The curvature is antisymmetric and satisfies ∂aQbc + ∂bQca + ∂cQab = 0, (RJ3) since the second derivatives commute. Conversely let Q be a homogeneous degree-j antisymmetric two-form satisfying (RJ3). Set ai(x) = 1 j + 2 X k xkQki(x). (RJ4) Antisymmetry proves (RJ1). Differentiating (RJ4), subtracting the reversed derivatives, and using (RJ3) gives ∂ial −∂lai = 2Qil + P k xk∂kQil j + 2 = Qil, where the last equality is Euler’s identity for degree j. Conversely, for a radial potential, differen- tiation of P k xkak = 0 gives P k xk∂iak = −ai. Hence P k xkQki = (j + 1)ai + ai = (j + 2)ai. Thus curl and (RJ4) are inverse linear maps. Equality of homogeneous diagonal polynomials implies equality of their symmetric tensors by polarization, so these are identities of the coefficient spaces themselves. The free solution space consists of those Q which additionally satisfy X a ∂aQab = 0 for every b. (RJ5) The free gauge-jet map sends this subspace to radial potentials by (RJ4). It is injective because curl is its left inverse. In particular, saying that a radial coefficient is in its range is exactly saying that its curvature satisfies the free equation (RJ5); Bianchi is already built in by (RJ2). Taking normalized Taylor coefficients of a smooth Maxwell solution on an open set preserves (RJ3) and (RJ5): differentiate these equations at the point, commute the derivatives, and divide by the 239 ## Source page 242 A Wilson Yang–Mills construction J. Rodgers Taylor factorial. Conversely the polynomial attached to an element of the solution space obeys these equations everywhere. This justifies the solution-jet substitutions used in the equation-ideal argument below. Color conjugation acts on these coefficients by the adjoint action. A rotation g acts by ai(x) 7−→ X k gikak(g−1x). The radial constraint, curl, Bianchi and (RJ5) are preserved, directly by orthogonality and the chain rule. Thus the reconstruction (RJ4) intertwines both actions. The polynomial ideal is the kernel of restriction from radial coefficients to this free solution range, taken at each finite degree horizon. A choice of projection onto that range is a finite-dimensional linear choice; the polynomial telescoping argument below shows explicitly how its kernel is generated by the linear equation coordinates. This explains both the projection Pj used there and the radial reconstruction used in the Gaussian covariance comparison. 9.2 Polynomial labels, coherent frames and local sections The polynomial variables in this construction are coordinates of real radial jets, rather than coor- dinates of arbitrary connections. Write Vj for the real space of radial homogeneous potentials of polynomial degree j + 1, described in (RJ1)–(RJ4), and choose a real basis of each Vj. If zj,a are its coordinate functions, the algebra of labels before imposing gauge invariance is P = C[zj,a : j ≥0, 1 ≤a ≤dimR Vj]. (PF1) Each polynomial uses finitely many variables. Its evaluation on a real jet J = (Jj)j≥0 substitutes the real coordinates of Jj, regarded as complex numbers. Such evaluations separate polynomials: reduce to the finitely many variables present, and successively use the fact that a nonzero one- variable polynomial cannot vanish at every real number. Give zj,a the bidegree (j + 2, 1). Thus the bidegree of a monomial zm is  X j,a (j + 2)mj,a, X j,a mj,a  = (r, f). (PF2) The first entry is engineering degree and the second is number of fields. The horizon-d label space consists of the adjoint-invariant polynomials whose monomials have r ≤d. It is finite dimensional: only variables of weight at most d occur and every exponent is at most d. Adjoint transformations and orthogonal transformations mix coordinates within a fixed Vj; therefore they preserve both entries of (PF2). Taking a homogeneous component preserves adjoint invariance and the horizon. Conjugation of coefficients preserves these spaces as well, because evaluation takes place on real jets. There is a basis bi, with b0 = 1, consisting of real polynomials of individual bidegrees (ri, fi). Indeed decompose into the finitely many bidegrees, and then into real and imaginary parts. The resulting real homogeneous elements span the complex space; choose an independent spanning subfamily containing 1. At successive horizons extend the previous independent family inside the same spanning set. This gives coherent frames: previous basis elements remain present as elements, although their nonzero index numbers need not remain fixed. The only elements with field number zero are constants. Hence fi > 0 for i ̸= 0. 240 ## Source page 243 A Wilson Yang–Mills construction J. Rodgers Coefficients c = (ci) mean the polynomial P i cibi; use P i |ci| as a coefficient norm. There are two different scaling operations: Jj 7→hj+2Jj, Jj 7→tJj. (PF3) On bi these multiply the reading by hri and tfi, respectively. In particular coefficient scaling by field number is the diagonal map ci 7→tfici, invertible for t ̸= 0, and fixes the unit. The coefficient action of s ∈O(4) is obtained by applying the tensor action to P i cibi and taking its coordinates. This action is an action on the polynomial label space. Its compatibility with the physical quotient requires the separate rotation argument below; it does not follow merely from this definition. Here is the actual construction of a local lattice representative of each label. At horizon d put AJ i (x) = X j 0; continuity at zero leaves only N(k) ij (x, 0). Thus the top curl vanishes. Radial reconstruction (RJ4) makes the top potential zero. Descending induction eliminates all components. No interchange of the two iterated integrals in (PF6) is required. 241 ## Source page 244 A Wilson Yang–Mills construction J. Rodgers Only finitely many slots are needed. Among kernels of finite sampling families choose one of minimal dimension. Appending any further slot cannot lower that dimension, hence cannot lower the kernel. A vector in this kernel is therefore killed by every slot and, by the preceding argument, is zero. Write X = Q j 0 be the healthy-field threshold at level N, let Sd(c; o, U) be the section just constructed, and let Td,N,p be the terminal coefficient isomorphism of the selected history. Put Id,N,p(v; o, U) = a−d N Sd θ−fi N (T −1 d,N,pv)i  i; o, U  . (PF12) Thus calibration is applied first, threshold scaling second, and engineering normalization last. The exponent in the threshold scaling is the field number fi, not the engineering degree ri. A general horizon label contains several bidegrees, so neither diagonal scaling may be replaced by one scalar on that entire space. Although the coefficient vector in (PF12) depends on N, p, its stencil is the single stencil of Sd. Linear combination and multiplication by constants do not enlarge it. Consequently (PF12) is continuous, gauge invariant and supported on that translated stencil for every cutoff, including cutoffs at which it wraps around the torus. The finite-Taylor normalization and the calibrated symmetry statements are used only after the reference level fits below N; this distinction is needed when constructing a family defined at every cutoff. At such a cutoffthe terminal map commutes with conjugation and signed orthogonal actions. Its inverse does too, by applying the terminal map to the two candidate values. Threshold scaling commutes with those actions because they preserve field number. Together with the corresponding identities for Sd, these facts prove the reality and signed covariance of (PF12). If e0 denotes the coefficient vector of the constant polynomial, the physical unit at level m is ad me0. The based-history identity is Td,N,p(ad Ne0) = ad kde0. (PF13) Substituting its inverse in (PF12), using f0 = 0 and Sd(e0) = 1, proves that the reference physical unit gives the constant cylinder one. Translating the placement translates precisely the same coefficient vector; no new history or new calibration is chosen at the new site. 243 ## Source page 246 A Wilson Yang–Mills construction J. Rodgers 9.3 Calibrated coefficient changes and exact solutions Calibration is a finite-dimensional coordinate change on the specified coefficient space. Equation cancellation is proved on the entire relevant cylinder, with masks and gauge averaging retained, before passing to a quotient. A detector separates the physical classes only after this equation image has been shown to be silent against every allowed spectator reading. These are algebraic and analytic steps with separate hypotheses. The inverse of the local slot sampler in (PF8) is an inverse on truncated radial jets. The coefficient inverse required for a physical insertion is a different map. We construct it in the norms and units in which it is used. Fix a degree horizon d, its coherent frame, and a reporting level k ≥kd, where kd is the reference level for that frame. All assertions in this paragraph are uniform in N ≥k and the period p, with k fixed. Let Td,N,p be the calibrated terminal coefficient isomorphism of the insertion history. If v is its reference coefficient, transport T −1 d,N,pv along that same history to level k and take its primary coefficient. Denote the result by Cd,k,N,pv, Ad,k,N,pv = a−d k Cd,k,N,pv. (PC1) At the reference level Cd,kd,N,pv = v, by the normalization of the history. At an adjacent step its coefficient estimates give, in the frame norm | · |1, |Civ|1 ≤Ri+1Ld i |Ci+1v|1, |Ci+1v|1 ≤Ri+1|Civ|1, (PC2) where Ri+1 ≥1 is the actual column allowance and Li ≥1 the block side. These are bounds on the primary coefficients of the original transported entries. They do not replace the primary coefficient by its class in the physical quotient. Define the finite number Od,k = Y i 0, (Dξz)i = ξfizi, (PC6) where θk is the original barrier threshold, not a restarted threshold. Define the linear readout Td,k,N,p by requiring that the polynomial obtained from (PC5) have coefficient vector DξTd,k,N,pv. This determines T because every diagonal entry of Dξ is nonzero. It also explains the exact calibration identity: the retained polynomial, including its Gaussian contraction, is the polynomial with those coefficients. There is no assertion that the whole untruncated exponential insertion is polynomial. The analytic comparison estimate for this construction is measured before the rescaling Dξ: |Ad,k,N,pv −Td,k,N,pv|1 ≤1 2|Ad,k,N,pv|1. (PC7) We derive this relative estimate before solving any calibrated columns. Let n bound the ordinary polynomial degree, let z = θk/g ≥1, g = β−1/2 N,p,k, and multiply every variable of the unweighted insertion polynomial by z; call that polynomial q. Its coefficient sum obeys coin(q) ≤B|Av|1. The actual nonlinear transported-variable substitution has the form Xα 7−→fα + θ−1 k SθkAα, (PC7a) where fα is linear, St multiplies every variable by t, and every monomial of Aα has ordinary degree at least two. Their coefficient sums are at most B. For 0 < t ≤1, coin(t−1StAα) ≤tB, coin(fα + t−1StAα) ≤2B. (PC7b) The first assertion follows term by term from tm−1 ≤t for m ≥2; the second is the triangle inequality. Telescoping each monomial of degree at most n, replacing one variable at a time, bounds the nonlinear substitution error by coin(q)n(tB)(2B)n. For a covariance with entries at most ε ≤1, Wick’s finite pairing expansion bounds coin(WQq −q) by coin(q)ε[2(n+2)]n. After the linear substitution this costs a further factor Bn. These two errors give A(t, B, ε, n) = n(tB)(2B)n + ε[2(n + 2)]nBn. (PC7c) Engineering truncation can only decrease coefficient sum. For the actual holonomy Taylor reading, the calibration before Wick contraction has error at most ε|Av|1 in polynomial coefficients. This is the Taylor remainder of the transported slots in (PF8)–(PF11), with their ordinary nonlinear part separated as in (PC7a). On the finite-dimensional image of the polynomial exposure map, fix a constant c ≥1 bounding its inverse from polynomial coefficient sum to frame coefficient sum. The total frame error is therefore |Av −Tv|1 ≤c{ε + BA(θk, B, ε, n)}|Av|1. (PC7d) The covariance here is the rescaled covariance z−2Q, since scaling variables by z and Wick contrac- tion commute with that change of covariance. The band estimate gHk ≤2θk implies |(z−2Q)αβ| ≤ε when |Qαβ| ≤B and 4B ≤Hk. 245 ## Source page 248 A Wilson Yang–Mills construction J. Rodgers Choose a = [4(n + 2)]−1, B = Ha k, ε = H−1 k . The original threshold obeys θk ≤H−1 k eventually. The terms on the right of (PC7d), including its outer B, have powers of Hk at most −1, −1 + a(n + 2) = −3 4, −1 + a(n + 1) < −3 4. (PC7e) Their fixed prefactors depend on d, n, c, not on N, p. Thus 2c{ε + BA} ≤1 eventually, proving (PC7) uniformly in those remaining indices. Also B + Bε[2(n + 2)]n ≤Hk eventually, so the Wick polynomial fits its stated monitor price. These estimates use the original threshold and the actual rescaled covariance; no solved-column bound has been used to prove the inverse. The unit is exact: its representative is the constant polynomial, contraction and truncation fix it, and both rescalings have weight zero on it. Thus Tvunit = e0. We retain (PC7) in its relative form; an absolute error on a few selected fields would not imply the inverse assertion that follows. The triangle inequality gives |Av|1 ≤2|Tv|1, |Tv|1 ≤3 2|Av|1 ≤2|Av|1. (PC8) Combining the first inequality with (PC4) proves |T −1z|1 ≤2Od,kad k|z|1. (PC9) To justify existence of this inverse before using it, Tv = 0 implies Av = 0 by (PC8), then v = 0 by (PC4). An injective endomorphism of a finite-dimensional space is surjective. This proves both existence and the bound (PC9). For a desired reference coefficient b and a position magnification ρ, let (Wρb)i = ρribi. The exact solved calibration is DξTu = Wρb, u = T −1Dξ−1Wρb. (PC10) The two diagonal operations use the distinct entries fi, ri of the bidegree. The solution is unique. Real b gives real u, because conjugating the equation gives the same equation and uniqueness applies. If b = e0, the solution is vunit by the exact unit identity. This constructs all the solved basis columns simultaneously for the one readout, rather than choosing a different readout for each column. The finite cutoffbounds on β give ξ−1 = θk q βN,p,k ≤2Hk, (PC11) where Hk is the actual monitor at level k. If b1, . . . , bs are the finitely many chosen real represen- tatives of a free quotient basis, (PC9)–(PC11) imply the uniform fixed-level bound ∥uj∥≤2Od,kad k s X j′=1 |D2HkWρbj′|1. (PC12) Indeed bound each component of Dξ−1 by the corresponding component of D2Hk, use ∥· ∥≤| · |1, and enlarge the single basis-column bound to their finite sum. This is the compact ball used in extracting the solved columns. It is allowed to grow with the reporting level. For the subsequent scale-dependent prices write Hk = eBk, choose ρ = Hb k, and put q = maxi fi. Formula (PC10) bounds |Tu|1 ≤(2Hk)qρd|b|1, Hk|Au|1 ≤2q+1|b|1Hbd+q+1 k . (PC13) For fixed b and d, the finite prefactor is eventually paid by one further factor Hk. The allowed polynomial coefficient price is therefore Hbd+q+2 k . This derives the degree-dependent price used for the calibrated polynomials. It also separates it from the fixed-level inverse bound (PC9) and from the inverse of the separating-test matrix used in (WS5). 246 ## Source page 249 A Wilson Yang–Mills construction J. Rodgers 9.4 The full free Maxwell polynomial hierarchy The free comparison hierarchy is specified on all polynomial labels, with no restriction to its two- point function. Let Sj be the real space of homogeneous curvature Taylor coefficients of order j satisfying antisymmetry, Bianchi and the Maxwell divergence equation in (RJ3),(RJ5). Choose a real basis and write Xj,α for its coordinate symbols. The free polynomial algebra is W = C[Xj,α], weight(Xj,α) = j + 2. (FM1) A real solution tower A = (Aj) evaluates these symbols in its real coordinates. Evaluation separates polynomials by the finite-variable argument of (PF1). The restriction map from radial labels substitutes the radial potential reconstructed from Aj into each original jet coordinate. Its kernel at horizon d is exactly the equation ideal: it consists of polynomials vanishing on every such solution tower. Its image Winv d is the invariant sector of weight at most d. To see surjectivity onto this sector, use the color-equivariant projection of radial curvature coefficients onto their Maxwell subspace and then radial reconstruction to lift every linear solution coordinate. Polynomial substitution preserves weight and equivariance; restriction of this lift is the identity. This is an algebraic operation on coefficients, not a claim about interacting equations. The basic covariance is fixed by G0(z) = 1 4π2|z|2 , Cab,kl(z) = δbk∂a∂lG0 + δal∂b∂kG0 −δbl∂a∂kG0 −δak∂b∂lG0, z ̸= 0. (FM2) Colors contract by their invariant Euclidean inner product. This is the curl at the first endpoint and the opposite curl at the second endpoint of the potential covariance; the opposite endpoint derivative accounts for its signs. Its two legs are closed and divergence free away from zero, by commutation of derivatives and harmonicity of G0. The contractions of higher solution coordinates use normalized Taylor coefficients, so the facto- rial convention must be retained. If p ∈S∗ j , first extend it linearly to the ambient two-form Taylor tensors. Apply it to the order-j normalized Taylor tensor of the first leg of (FM2), obtaining a two-form in the second indices and color. Pair its order-l normalized Taylor tensor with an exten- sion of q ∈S∗ l , and multiply by (−1)l. Denote the resulting scalar by Kp,q(z). Both extensions are immaterial: at each point away from zero the differentiated leg is in the corresponding solution sub- space, so their differences annihilate it. This defines the contractions on precisely the constrained variables in (FM1). For monomials at distinct points x1, . . . , xn, regard every power of every symbol as that many distinguishable occurrences. Sum over perfect matchings of all these occurrences, allowing a pair only if its two occurrences belong to different insertion points. A pair (i, p), (k, q) contributes Kp,q(xi −xk). The product over its pairs, summed over the permitted matchings, is the monomial moment. The empty matching contributes one and an odd total number contributes zero. Extend by complex multilinearity in the n polynomial inputs: Mn(x; P1, . . . , Pn) = X monomial choices Y i ci ! X perfect matchings no pair within one insertion Y pairs Kp,q(xi −xk). (FM3) The restriction on pairs is the Wick ordering of each composite insertion. There are finitely many terms for every tuple of polynomials. Pulling (FM3) back by the restriction map defines the free table on the original coherent label frames. 247 ## Source page 250 A Wilson Yang–Mills construction J. Rodgers The elementary contraction is symmetric under exchange of endpoints and legs. Reindexing matchings therefore proves permutation symmetry. It depends only on differences, proving transla- tion invariance. The chain rule applied to (FM2), orthogonality of the color action, and orthogonal covariance of the two curls give the full orthogonal tensor law. All entries are real on real coeffi- cients, giving the reality law. Inserting the constant polynomial one introduces no occurrences, so it removes that slot. These prove the algebraic hierarchy laws for every arity. Each derivative of a contraction of orders j, l is bounded by a constant times a power of the inverse separation; at order zero the power is 4 + j + l. This follows by differentiating the rational homogeneous function (FM2), with the finite normalization factors included in its constant. Taking products and finite sums in (FM3) proves analyticity on the separated configuration space and a power bound there. For the product bound add the exponents and multiply constants; for a sum take their sum and enlarge to a common exponent using r−1 ≥1 when 0 < r ≤1. These are the positional bounds needed for its smeared hierarchy. 9.5 Positivity and strictness of the reflected free pairing We derive the reflected one-particle positivity needed for polynomial detection. Fix an axis a, let θa reflect that coordinate, and let Ra act on all tensor indices of a probe by this reflection. Take finitely many real probes pi of possibly different jet orders di, at points xi with (xi)a > 0. Coincident points among these sources are allowed. Float all sources by tea, t ≥0, and put H(t) = X i,j K(pi, Rapj; xi + tea, θa(xj + tea)), where the last two arguments indicate the endpoints of the difference kernel. Let Ft be the sum of their curvature pulses. The derivative in the first endpoint and the opposite motion of the reflected endpoint give H′(t) = −2L(t), L(t) = X i,j −K(∂api, Rapj; xi + tea, θa(xj + tea)). (GP1) Here ∂api is the raised probe, with the normalized derivative convention of (FM2). The factor two is the sum of the two endpoint motions. On the reflecting plane the reflected pulse is Ft,bc 7→εbεcFt,bc, where εa = −1 and all other εb = 1. Substitute these signs into the bilinear Maxwell stress. Its normal-normal component is −1 2 P b,c ∥Ft,bc∥2. The mixed terms with exactly one normal index change sign; skew symmetry gives Ft,aa = 0, and combining the remaining terms yields exactly this negative half-sum. Integrate its divergence over the positive half-box, removing small balls about its finitely many upper sources. The plane term has the negative outward normal; the small-sphere terms are the translation case of the pulse-current identity (FS2). The reflected sources are in the negative half-space and cause no extra sphere term. The other box faces disappear as their size tends to infinity. Indeed each pulse of order d is O(|y|−4−d); its bilinear stress is O(|y|−8−di−dj), while the three-dimensional face measure is O(R3). Their fluxes are therefore O(R−5−di−dj). Summing the finitely many terms proves L(t) = lim R→∞ 1 2 Z [−R,R]3 X b,c ∥Ft,bc(ya = 0)∥2 d3y ≥0. (GP2) The same argument bounds each finite-box integral by L(t), by monotonicity of the nonnegative integrand. Since every source-to-reflected-source separation tends to infinity, the pole bounds give H(t) →0. Equations (GP1)–(GP2) make H decreasing, so H(t) ≥0. This proves the real positive 248 ## Source page 251 A Wilson Yang–Mills construction J. Rodgers semidefiniteness of every finite reflected one-particle matrix. Symmetry of that matrix follows from endpoint exchange and reflection covariance. Its complex extension is Hermitian positive semidefinite by splitting a coefficient vector into real and imaginary parts. The equality case is stronger when all source points equal one point x with xa > 0. Suppose H(0) = 0. Monotonicity and nonnegativity imply H(t) = 0 for all t ≥0, hence L(t) = 0 for t > 0. Equation (GP2), together with continuity, forces every component of Ft to vanish on the entire reflecting plane: a nonzero value would have a neighborhood with positive squared integral inside some finite box. Translation covariance then makes the unshifted summed pulse vanish on ya = −t, for every t > 0. Each scalar color component is real analytic on R4 \ {x}, which is connected. The analytic identity theorem extends this half-space vanishing to every y ̸= x. Apply now the radial version of (FS2) to this zero summed pulse and any polynomial Maxwell solution f. It yields X i (di + 2) pi(jdif(x)) = 0, (GP3) where jk is the normalized solution jet of order k. For any homogeneous solution tensor Ak, the field f(y) = asWave(Ak)(y −x) has normalized jet Ak at order k at x, and zero jets there at all other orders. Thus (GP3) gives P i:di=k pi(Ak) = 0, since k + 2 > 0. Summing over the finitely many jet orders proves that a zero reflected norm annihilates every solution tower. In particular, for any finite alphabet of independent solution-coordinate probes, its one-point reflected matrix is positive definite: evaluate the last identity on a tower with a single basis coordinate equal to one to force each coefficient to be zero. This establishes strictness on the constrained solution spaces themselves, including mixtures of jet orders. We pass from legs to polynomial words algebraically. On finitely many complex variables define ⟨P, Q⟩F = X ν Y i νi! ! PνQν. (GP4) Every weight is positive; hence ⟨P, P⟩F > 0 for P ̸= 0, including the scalar polynomial. Factor a finite reflected leg matrix as L∗L, which is ordinary finite-dimensional positive matrix factorization. Associate to a leg i the creation polynomial li(Z) = P k LkiZk. To represent a word, process its occurrences successively: append its creation polynomial and, for each possible contraction to a previous occurrence on the same side, add that covariance times the polynomial for the remaining word. This finite recursion is Wick’s pairing recurrence. Pairing the two resulting polynomials by (GP4) contracts their remaining occurrences across the reflection plane, since ⟨li, lj⟩F = (L∗L)ij. The factorial Q νi! counts the bijections between equal creation occurrences. Induction on word length therefore identifies the reflected Wick moment with this polynomial inner product, including all permitted contractions internal to either half. A finite linear combination of words has reflected norm equal to the nonnegative squared norm of its associated polynomial. Expanding polynomial insertions into monomials proves reflection positivity at arbitrary arity, and smearing preserves it by integration. At a single source point there are no contractions within either half because each insertion is Wick ordered. The recursion is then simply the substitution Zi 7→li(Z). The one-point leg matrix is positive definite by (GP3), so L is invertible; substituting L−1 gives the inverse polynomial substi- tution. Thus a nonzero polynomial remains nonzero, and (GP4) makes its reflected pairing strictly positive. This is the strict pairing used in the hosting argument and the detector construction. It proves that the free all-spectator null space has precisely the equation ideal already removed in forming the solution coordinates. 249 ## Source page 252 A Wilson Yang–Mills construction J. Rodgers Accordingly the free physical field space is canonically Ed/Nd ≃Winv d . (FM4) Under this identification the horizon inclusion is literal polynomial inclusion. A derivative in direction v is the polynomial derivation determined on a linear symbol by Dv symbolj(p) = symbolj+1(p ◦lowerv). (FM5) Here lowerv Aj+1 is the order-j Taylor coefficient of the derivative of the homogeneous polynomial represented by Aj+1; its coefficient is j + 1 times contraction with v. Thus (FM5) has the actual Taylor normalization. The Leibniz rule differentiates one occurrence at a time. Differentiating a matching product in (FM3) selects its incident covariance edge, while replacing an occurrence by (FM5) differentiates precisely that same edge. Summing proves the insertion-derivative identity. It raises the horizon by one and stays in the invariant sector. Conjugation and signed transformations obey (QD6). Hence the free differential system is the same type of all-spectator system as the interacting one. For s ∈O(4) the free field action is the polynomial substitution induced by the tensor action on the solution tower. It preserves weight, equations and color invariance, with inverse given by s−1; conjugating it through (FM4) gives the full field action. Infinitesimal skew rotations likewise act by the derivation obtained by differentiating that linear tensor substitution. Both descriptions are on the same field space and preserve the moment law (FM3). The free symmetric traceless stress is the actual weight-four invariant polynomial T free ab = X c,i F c aiF c bi −δab 4 X c,i,k F c ikF c ik. (FM6) Products here mean the Wick polynomial product used in (FM3). Adjoint invariance follows from preservation of the color inner product, symmetry by exchanging its two factors, and reality from real coefficients. Its trace is zero by summing the four diagonal entries. Its divergence is zero as a polynomial on solution jets. Indeed the derivative of the first term is P c,i F c ai∂aF c bi, since ∂aF c ai = 0. Bianchi and antisymmetry give X a,i F c ai∂aF c bi = 1 4∂b X a,i F c aiF c ai; (FM7) multiply the Bianchi equation by F c ai, sum, and interchange a, i in its second derivative term to obtain the factor two on the left and the half derivative on the right. This cancels the divergence of the subtracted term in (FM6). Evaluation on all real solution towers then proves the polynomial identity. Through (FM4) it gives a real, symmetric, conserved free field at horizon four, with the exact normalization used in the later free comparison. 9.6 The free low-degree stress and vector spaces The low-degree classification uses simplicity of the actual color Lie algebra, without assuming that the matrix group is connected. Differentiate its adjoint action along the actual exponential curves. This gives d dt Adexp(tx) y 0 = [x, y]. (FC1) The span of brackets is a nonzero ideal, hence is the whole simple nonabelian color algebra. An invariant linear functional annihilates each bracket by (FC1), so it is zero. An invariant real bilinear 250 ## Source page 253 A Wilson Yang–Mills construction J. Rodgers form satisfies B([x, y], z) = −B(y, [x, z]). Its alternating part is zero: three cyclic applications to B([x, y], z), with bracket antisymmetry, identify this value with its negative, and brackets span. For the symmetric part write B(x, y) = ⟨Ax, y⟩with A self-adjoint. Invariance says A[x, y] = [x, Ay]. Every eigenspace of A is therefore an ideal. A real self-adjoint operator has an eigenvalue; its nonzero eigenspace is the whole algebra, so A is scalar. Applying this argument to real and imaginary parts proves that every invariant complex-valued real bilinear form is a complex multiple of the fixed color inner product. A monomial of weight at most five has at most two fields, since each variable has weight at least two. With two fields its jet orders are either (0, 0) or (0, 1); the latter has weight five. Decompose a polynomial into its constant, linear and these quadratic pieces by ordinary field number and engineering weight. Invariance of each piece follows from the linearity of the color action. The linear piece vanishes by (FC1). Polarizing the (0, 0) piece, and separating the two jet orders in the (0, 1) piece, gives invariant bilinear forms. Resolve their spatial tensor coordinates; (FC1) then makes every color contraction a multiple of the color inner product. Consequently all weight-four labels are spanned by the unit and ⟨Fab, Fcd⟩; through weight five one adds ⟨Fab, ∂eFcd⟩. This is a spanning assertion about the actual invariant labels before further free equation relations. Put Qab,cd = P c F c abF c cd, with products interpreted as in (FM3), and define Na = X i Qai,ai, Ra = X i,k: i̸=a, k̸=a Qik,ik, Pab = X i Qai,bi. (FC2) The sums in Ra are over ordered pairs. Consider centered real symmetric tensor fields transforming under the signed coordinate group as rank-two tensors. Reflection in one axis multiplies Fab by the product of its two axis signs. Thus a diagonal component permits only products in which every axis occurs evenly; for two antisymmetric pairs this forces the same unoriented plane twice. The permutations fixing axis a have just two orbits: planes containing a and planes disjoint from a. They give Na, Ra. A constant could also occur, but centeredness removes it: the Wick quadratic one-point functions are zero and the unit expectation is one. For the off-diagonal component ab, a ̸= b, the parity character is the product of the signs of a, b. The two planes must therefore share exactly one other axis i, and be ai, bi, up to antisymmetry and exchange of the two factors. Permutations fixing a, b identify the possible remaining axes, giving precisely Pab. The characters of the other products differ in at least one reflection, so averaging against the required character kills them. Permuting axes then determines every tensor component from 00 and 01. Reality makes the coefficients real. The whole centered signed tensor space is therefore spanned by Aab = δabNa, Bab = δabRa, Cab = 1{a̸=b}Pab. (FC3) For a real polar vector of weight at most five, fix component zero and average under its stabilizing permutations and reflection character. Constants and FF terms have the wrong parity. In an Fab∂eFcd term, parity requires that the symmetric difference of the two plane-index sets be {0, e}. If e = 0, the planes coincide, giving the first two sums below. Otherwise they share a remaining index; the location of the derivative leg gives the last two sums. Thus its zeroth component is spanned by n = X i̸=0 ⟨F0i, ∂0F0i⟩, c = X i,j̸=0 i̸=j ⟨Fij, ∂0Fij⟩, a = X i,j̸=0 i̸=j ⟨F0i, ∂jFji⟩, h = X i,j̸=0 i̸=j ⟨Fji, ∂jF0i⟩. (FC4) 251 ## Source page 254 A Wilson Yang–Mills construction J. Rodgers The free divergence equation gives n + a = 0. Bianchi gives ⟨Fji, ∂jF0i⟩+ ⟨Fij, ∂iF0j⟩= ⟨Fij, ∂0Fij⟩. Summing over ordered distinct nonzero i, j gives 2h = c. Hence only n, h are needed. A signed vector is determined by its zeroth component, since exchanging axis zero with any axis transports that component. The polar vector space has real dimension at most two. These relations hold on all solution towers, so they are zero in the actual free quotient by (FM4). The lower rank bounds are obtained by explicit separated correlations, not by declaring the formal expressions independent. Let Hab,cd(z) = Cab,cd(z) be (FM2) and set Qab,ik;ce,fm(z) = Hab,ce(z)Hik,fm(z) + Hab,fm(z)Hik,ce(z). (FC5) The correlation of Qab,ik with Qce,fm at z, 0 is (dim g)Q(z), because the two cross matchings force the same color. At z = et, writing ν = (4π2)−1, the Hessian and third derivative of the Green function are ν−1∂j∂iG0(et) = −2δij + 8δtiδtj, ν−1∂r∂j∂iG0(et) = 8δijδtr −48δtiδtjδtr + 8(δriδtj + δrjδti). (FC6) These follow by differentiating |z|−2. Substituting them into the four signed Kronecker terms in (FM2), and applying the two-factor product rule to (FC5), gives every sample below. The common nonzero factor removed is lG = (dim g)ν2 > 0. Take the divergence component zero of (FC3), correlate it at e0 with Q01,01 and Q12,12 at zero, divide by lG, and take real parts. The two resulting linear functionals have columns A B C Q01,01 −256 0 128 Q12,12 0 −512 −128 . (FC7) For example the first A entry is P3 k=0 ∂0Q0k,0k;01,01(e0)/ν2 = −256; the corresponding B entry is the same sum over ordered i, k ̸= 0, which is zero. For C sum ∂rQrk,0k;01,01 over r ̸= 0, k. The other row uses 12, 12. These finite sums and (FC6) give respectively the three displayed columns. A third, undifferentiated sample correlates tensor component 01 at e0 with Q02,12 at zero; its columns are (0, 0, −16), directly from (FC5). It separates the last tensor direction. Thus A, B, C are independent, so the tensor space has dimension exactly three. The two divergence samples in (FC7) are independent; combined with the upper bound two from (FC4), they identify the entire polar vector space and prove surjectivity of divergence. Rank-nullity gives a one-dimensional conserved kernel. Solving (FC7) yields its generator 1 2A −1 4B + C = T free. (FC8) For its diagonal entry use P i,k Qik,ik = 2Na + Ra; its off-diagonal entry is Pab. This is exactly (FM6), already shown conserved. The last sample is nonzero on it, so the generator is nonzero. Every conserved centered real signed tensor in this weight-four space is its real scalar multiple. Any later nonzero scalar normalization, including either sign, preserves the rank and the conserved- line statements; the normalization of shell fluxes is a separate argument. There is a raw degree-five vector direction which must be kept until the equation quotient is taken. In polynomial notation its components are Eb = X i,a pairDeriv(b, i, a, a, i), (FC9) that is, the contraction of Fbi with the equation ∂aFai, in the normalized radial-jet coordinates used above. The two other raw vector directions are the polynomials called momentum1 and 252 ## Source page 255 A Wilson Yang–Mills construction J. Rodgers momentum2 in the preceding classification. Every real signed-equivariant raw vector has a unique expression uV1 + vV2 + wE. To see the independence after imposing the Maxwell equations, take a unit color vector and the two solution jets with respectively (F (0), F (1)) = (leg12, leg0;12), (leg01, leg0;01 −leg2;21). (FC10) Here the legs are radial potential jets, with the normalization made explicit below. Both satisfy the linear divergence equation and Bianchi identity. The first reads V1 = 0, V2 = 4/3 in component zero; the second reads V1 = 1. An equation polynomial reads zero on both. It follows first that v = 0, then that u = 0. Conversely E vanishes on every solution jet, directly by the divergence equation. A non-solution jet with nonzero divergence shows that E itself is a nonzero raw polynomial. Thus the raw vector space has dimension three, and the kernel of its map to the free vector quotient is exactly the one-dimensional space RE. This statement is independent of coefficient coordinates. For a coherent degree-five frame, apply its inverse coordinate map to each component of a real signed-equivariant coefficient vector. This is a real linear isomorphism to the raw polynomial vector space, because the coordinate map intertwines signed permutations and conjugation. Composing with the free quotient projection gives a surjection whose kernel is the inverse-coordinate image of RE. A section is obtained by applying the corrected polynomial lift to each quotient component. That lift commutes with both symmetries, and its quotient projection is the identity. These facts justify the vector kernel subtraction used later: it removes precisely the equation direction, and does not discard either of the two nonzero free vector classes. To make the raw coordinates completely explicit, write Pab;kcd = ⟨Fab, ∂kFcd⟩, (V1)b = X i Pbi;bbi, (V2)b = X i,j Pij;bij, Eb = X i,a Pbi;aai. (FC11) The pairings are the actual invariant polynomial labels, evaluated on radial potential jets; imposing the Maxwell equation is a later quotient operation. The basic legs in (FC10) are the homogeneous potentials (legab e)i(z) = 1 2(δibza −δiazb)e, (legk;ab e)i(z) = 1 3(δibza −δiazb)zke. (FC12) Taking their exterior derivatives defines the curvature legs, including the factor 1/3 in the first curvature derivative. This factor explains the 4/3 sample in (FC10). A constant curvature is auto- matically divergence free. For a linear curvature the Maxwell condition is exactly P i curl1(i, i, b) = 0 for every b; direct differentiation of (FC12) checks it for the two combinations in (FC10). Higher homogeneous jets can be set to zero. Here is also the raw upper-dimension argument, before using those solution samples. A weight- five label decomposes as a constant, an invariant linear term, an FF term and an F ∂F term. Simplicity kills the linear term as above. Total axis inversion fixes the two F’s but negates a polar vector, so both its constant and FF terms vanish. Resolve its remaining invariant bilinear color form into the color inner product. To recover its spatial coefficients it suffices to use the mixed evaluations m(A, B; v) = Re{v(A + B) −v(A) −v(B) + v(0)}, (FC13) where A occupies jet order zero and B order one. Take, in component zero, the three pairs (leg01 e, leg0;01 e), (leg01 e, leg2;21 e), (leg21 e, leg2;01 e). If all three readings vanish, every mixed coefficient vanishes. Indeed reflections require the sym- metric difference of its two plane-index sets to equal {0, k}, where k is the derivative index. For 253 ## Source page 256 A Wilson Yang–Mills construction J. Rodgers k = 0, equal planes containing zero and equal planes avoiding zero are the two possible types. For k ̸= 0, the planes share one other index, and the two placements of the derivative factor are the other types. Permutations fixing zero identify all coefficients of each type; antisymmetry fixes their signs. The three displayed tests kill three types. The cyclic identity leg0;12 + leg1;20 + leg2;01 = 0, paired with leg12 e, kills the fourth. The legs span each radial jet space by the expansion A = 1 2 X i,j legij(curl0(i, j)A), B = 1 2 X i,j,k legk;ij(curl1(k, i, j)B). Bilinearity therefore kills the entire mixed part. Reality recovers the complex coefficient from its real part. The three evaluations form an injective real map, proving the raw upper bound three. The two solution evaluations in (FC10) and the nonsolution pair (leg01 e, leg0;01 e), on which E0 = 1, prove independence of V1, V2, E. Thus (FC11) is a basis and every raw vector has unique coordinates. For later use, the divergence of the raw centered tensor with coefficients w = (w0, w1, w2), respectively diagonal mountain, diagonal total and off-diagonal shear, has these coordinates: Div(w) = 2(w0 −w2)V1 + (2w1 + 1 2w2)V2 + w2E. (FC14) The product rule gives 2V1 for the first tensor and 2V2 for the second. For the shear use Bianchi and interchange the ordered indices to obtain P a,i Pai;abi = 1 2V2,b; deleting the a = b terms subtracts 2V1,b, while the other product derivative is Eb. This proves (FC14). Its coordinate matrix is invertible, with inverse w2 = y2, w0 = y0/2 + y2, w1 = y1/2 −y2/4. It defines the actual scalar contraction on raw vectors: ρ(Div(w)) = −4(w0 + 4w1) = −2y0 −8y1. (FC15) If Q = 1 4 P i,j⟨Fij, Fij⟩, the tensor trace is 4(w0 + 4w1)Q; hence ρ reads minus that trace in Q-units. It kills E and the traceless conserved tensor w = (1, −1/4, 1). Since (FC10) proves that the raw equation kernel is exactly RE, ρ annihilates that entire kernel. This is the scalar sensor used later, with its sign and normalization fixed. The identification of the free quotient here uses the Maxwell polynomial evaluation and the reflected Gram norm proved in (FM3)–(FM4) and in the detector’s reflected polynomial calculation; it is not an additional assumption of nondegeneracy. 10 Free shell charges and separation of the Ward kernel The free shell calculation is the comparison model for physical Ward charges. Compact cutoffs ensure that all shell integrals avoid the insertions, and the fixed exterior degree horizon controls the tests used to separate defects. The argument does not assume that a low-degree test alone detects an arbitrary high-degree label; the displacement and Wick calculations below provide the additional separating probes. We give the polynomial calculations underlying the free charge separation used in the rotation and trace arguments. This also specifies the exterior tests needed when a charge acts on high-degree fields. 254 ## Source page 257 A Wilson Yang–Mills construction J. Rodgers 10.1 The compact stress shell identity Let χ be a smooth compactly supported function equal to one near a distinguished point x, whose support avoids all spectator points. For a current J, define its outward shell functional by Eχ(J) = − X a Z (∂aχ)(y)Ja(y) d4y. (FS1) The integrand is supported in a compact annulus away from all insertions. All exchanges of finite Wick sums and this integral are consequently justified by smoothness there. A one-leg source p at x produces the Maxwell pulse obtained by applying its jet functional to the covariance. For a smooth source-free Maxwell field f on a neighborhood of the support of χ, Green’s formula for the bilinear Maxwell stress current gives EχJv(pulsep,x, f) = −samplex(p, δvf). (FS2) Indeed the divergence vanishes offthe source by the Maxwell and Bianchi equations. Integrating the differentiated fundamental covariance against the cutofftransfers the derivatives to the smooth field. The fundamental-solution identity leaves precisely its jet at x. The boundary term at infinity is absent because the cutoffis compact. The variation of a two-form is δvfab = vc∂cfab +(∂avc)fcb + (∂bvc)fac. For the three vector fields used here this variation preserves the Maxwell equations. The stress is symmetric and traceless, so its contraction with the symmetric derivative of a constant, skew-linear, or radial vector field contributes zero to the divergence. Here is a regularized derivation of the fundamental-solution step in (FS2). With the covariance normalization c4 = 1/(4π2), put gt(y) = c4 t2 + |y|2 , ρt(y) = 8c4t2 (t2 + |y|2)3 , t > 0. (FS2a) Direct differentiation gives −∆gt = ρt. The radial integral gives R ρ1 = 1: the sphere area is 2π2 and R ∞ 0 r3(1 + r2)−3dr = 1/4. Also t4ρt(ty) = ρ1(y). Hence for every continuous compactly supported h, R ρt(y −x)h(y)dy →h(x), by the change of variable y = x + tz and the integrable majorant ρ1(z)∥h∥∞. For a plane source il, form the potential Ab = δib∂lgt−δlb∂igt, and take its exterior derivative. It satisfies the Bianchi identity by commutation of derivatives, while its divergence is δlb∂iρt −δib∂lρt. The divergence of its bilinear stress with the source-free field f is therefore the contraction of this expression with f; the terms from the symmetric derivative of v cancel by tracelessness and the other terms cancel by skew symmetry. Integration by parts against χ transfers the source derivatives to the smooth compact test made from χ, v, f. The preceding approximate-identity limit evaluates their difference at x, giving −δvfil(x). On the shell the regulated exterior derivative converges smoothly to the pulse, since that compact shell avoids x. Its flux thus converges to the left side of (FS2). This proves the plane-source case with the stated sign. Differentiate the source center in any direction while keeping the cutoffand vector field fixed. All differentiated shell integrands have a common bound on a fixed compact annulus, so differentiation passes through the integral. A center derivative raises the source leg by that direction; on the right it differentiates the sample of δvf. Induction gives (FS2) for all derivative legs; linearity gives every solution-jet functional. The wave property of δvf for the three motions ensures these samples respect the Maxwell jet relations. For a constant vector ei, (FS2) gives minus the derivative-leg sample. For v(y) = A(y −x), with A skew, it gives minus the infinitesimal tensor rotation. For v(y) = y −x, a homogeneous solution 255 ## Source page 258 A Wilson Yang–Mills construction J. Rodgers jet of derivative order j has Euler contribution j, and the two form indices contribute two more. Thus the one-leg radial charge is −(j + 2) times that jet. This specifies the engineering weight, including the two contributed by curvature rather than its derivatives. When both pulses are centered at x, use cutoffindependence and homogeneity instead of mul- tiplying distributions at their common pole. Two cutoffs equal to one near x have a difference supported away from x; source-free integration by parts makes their fluxes equal. Replace the cutoffby its dilation by a factor two about x. A jet-order-j pulse scales by 2−4−j; the other scales by 2−4−l; the volume and cutoffderivative together contribute 23, and a motion homogeneous of degree e = 0 or 1 contributes 2e. Thus the common flux Z equals 2e−5−j−lZ. That coefficient is strictly smaller than one, so Z = 0. When neither pulse is centered at x, their fields are smooth throughout the support of the cutoff, and the source-free divergence identity gives zero. When precisely one pulse is centered there and the other is at a spectator point, (FS2) gives minus the covariance with the transformed distinguished leg. Now insert the quadratic Wick stress in an arbitrary Wick monomial word. Double Wick extraction expresses its correlation as one half the sum over ordered choices of two occurrences, times the bilinear pulse current for those occurrences and the Wick pairing of all remaining occurrences. There are exactly the three cases just considered. The terms with both distinguished occurrences or with neither distinguished occurrence vanish. In each remaining term exactly one occurrence of the distinguished monomial is transformed. The two orders of the selected pair cancel the one half. Summing over its occurrences is the Leibniz rule for the derivation on that monomial. By multilinearity this proves, for arbitrary polynomial words, Eχ X b vb(y)⟨Tab(y)P(x) Y l Pl(xl)⟩ ! = − * (DvP)(x) Y l Pl(xl) + . (FS3) Here Dv is respectively translation, rotation, or the engineering-weight derivation. This proves the three shell laws, including their common minus sign, and includes every contraction term. In particular conservation of the free stress alone was not being used as a substitute for its charge normalization. 10.2 Low-degree vector charges Let Sj be the real space of homogeneous order-j Maxwell solution jets and Vj = S ∗ j . Their elements are the real one-leg test functionals. A variable at order j has engineering weight j + 2. For p ∈Vj, write Bc(p) for its linear polynomial in color c, and set [p, q] = X c Bc(p)Bc(q). This bracket denotes a color-contracted quadratic polynomial, not a Lie bracket. For a linear map D : Sj →Vl, its polynomial is C(D) = X ν [eν, Deν], (FS4) where (eν) is any basis and (eν) its dual. This is basis-independent, and C(p ⊗q) = [p, q]. If j ̸= l, assigning independent jets in these two orders in one nonzero color proves that C is injective. Consider a vector row Ti on the invariant polynomial filtration, killing the unit, increasing filtration by at most one, real under conjugation, and signed-covariant. Suppose it is silent, meaning X l ⟨P1(x1) · · · (TiPl)(xl) · · · Pn(xn)⟩= 0 (FS5) 256 ## Source page 259 A Wilson Yang–Mills construction J. Rodgers for every allowed exterior word and separated configuration. At input weight four the invariant space is spanned by the unit and [p, q] with p, q ∈V0. At output weight at most five the only further possibility is a color-contracted pair of orders zero and one. Total inversion is even on weight four and odd on such a pair. The vector law anticommutes with total inversion, and therefore removes the weight-four part of the output. Reality permits real coefficients in (FS4). Consequently Ti[p, p] = C(Di(p)), Di(p) : S0 →V1. (FS6) Let Kz(p, q) denote the real single-color covariance between two test legs at displacement z ̸= 0. To extract the constraint on (FS6), put two quadratic insertions at tu and zero, and a third at a fixed nonzero point, and multiply (FS5) by t5. The covariance of jet orders zero and one is homogeneous of degree −5. The triangle terms where the transformed order-one leg is on either near insertion have a finite limit; the term with that leg at the distant insertion tends to zero. The double contractions contribute 4 dim g. Choose the distant one-leg functional so that its covariance with a zero-order test is evaluation on a prescribed X ∈S0. Cancelling the common nonzero factor gives Ku(Di(p)X, q) q(X) = Ku(Di(q)X, p) p(X). (FS7) The availability of these evaluations follows from covariance separation of solution jets. First suppose a test functional of order j has zero covariance with every order-zero plane test at every nonzero displacement. Its pulse is then zero offits source, in every color component. Pair its radial stress shell with the homogeneous polynomial Maxwell solution having any prescribed order- j solution jet. The shell integral is zero because the pulse is zero there; (FS2) makes it −(j + 2) times the test evaluated on that arbitrary jet. The functional is zero. For partner order l + 1, vanishing against all such partners makes the radial derivative of its covariance with any order-l partner zero. That covariance is homogeneous of degree −(4 + j + l); Euler’s rule makes the radial derivative that nonzero scalar times the covariance. Thus it vanishes, and induction reduces to order zero. Finite-dimensional duality now makes these covariance evaluations span the required full dual. Thus no inverse of a covariance matrix at a single momentum is being assumed. This also proves that the solution-jet space is nonzero at every order: otherwise separation against that order would force the nonzero plane evaluation at order zero to vanish. If p(X) = 0, (FS7) first makes Ku(Di(p)X, q) = 0 for q(X) ̸= 0. Given q(X) = 0, add a test with nonzero value at X, apply that result to the sum, and subtract. Covariance separation gives Di(p)X = 0. Hence ker p ⊆ker Di(p). Choose X0 with p(X0) ̸= 0; subtracting p(X)X0/p(X0) from X proves Di(p) = p ⊗fi(p), Ti[p, p] = [p, fi(p)]. (FS8) Set fi(0) = 0. For two nonzero linear functionals there is an X on which both are nonzero: use a witness for one, a witness for the other, and their sum if needed. Substitution in (FS7) yields Kz(fi(p), q) = Kz(fi(q), p). (FS9) Pair the difference fi(p + q) −fi(p) −fi(q) with an arbitrary one-leg test, use (FS9), and cancel by linearity in its other slot. Separation makes the difference zero. The same argument with a scalar proves homogeneity. Thus fi : V0 →V1 is real linear. Signed covariance of Ti, injectivity of (FS4), and cancellation of a nonzero test functional give simultaneous signed covariance of the family fi. We classify this family explicitly. Write cab for evaluation of a zero jet in the ab-plane and siab = ∂icab for the corresponding first-jet functional. Under reflection of axis z, the latter has character equal to the product of the three signs belonging to i, a, b. Project onto a reflection character by the four commuting projections (1 ± Rz)/2. The character of f0(c01) leaves only 257 ## Source page 260 A Wilson Yang–Mills construction J. Rodgers s001, s212, s313. Average the permutation interchanging axes two and three. The Maxwell relation gives s212 + s313 = s001, so this fixed subspace is the line spanned by s001. Similarly the character of f0(c12) leaves only s012, s102, s201. Antisymmetrize the interchange of axes one and two; the Bianchi relation s102 −s201 = s012 leaves the line spanned by s012. Hence these two values have scalar coefficients a, b. Use (FS9) with p = c01, q = c12, and z = e0 + e2. The explicit differentiated covariance (FM2), evaluated below after (WE4), gives the same nonzero value −2c4 for s001, c12 and s012, c01. Therefore a = b. Permutations fixing zero give all remaining plane evaluations, which span V0; permutations moving zero give the other directions. We conclude fi = a∂i, Ti[p, p] = a[p, ∂ip]. (FS10) Since ∂i[p, p] = 2[p, ∂ip], the scalar of the translation derivation is a/2. Squares span the symmetric quadratic polynomials by polarization, and both rows kill the unit. This proves the entire degree- four assertion, with its factor two, rather than just a classification of invariant labels. 10.3 Fixed-displacement evaluation and the triangular Wick calculation Here are the three explicit evaluations used in (FS7)–(FS10). Put c4 = (4π2)−1, and let Hz = I −2zzT/|z|2 for z ̸= 0. Its action on two-forms is the solution action ρ(Hz). If Eab is the unit antisymmetric plane tensor, the differentiated covariance (FM2) gives Kz(p, cab) = κz p(ρ(Hz)Eab), κz = 4c4|z|−4 > 0. (WE1) Indeed substitute the second derivatives of c4|z|−2 into the two curls: their radial terms combine to the product of the two Householder matrix entries on the antisymmetric indices, with the displayed common radial factor. This is an identity on the six-dimensional real space of antisymmetric order- zero solution tensors. Given X ∈S0, set W = ρ(Hz)−1X. Antisymmetry gives the plane expansion W = 1 2 P a,b cab(W)Eab. Define the one fixed functional wz,X = κ−1 z 1 2 X a,b cab(W)cab. Bilinearity and (WE1) now give Kz(p, wz,X) = 1 2 X a,b cab(W)p(ρ(Hz)Eab) = p(X). (WE2) Thus the evaluation witness used at displacement −z in (FS7) exists at that displacement itself. No change of the distant point or sum over different displacements is necessary. Write Tx,y,z(A, B, C) for the three-point Wick moment. At each point its two letters are Wick ordered, so a contraction cannot join two letters at that same point. In a pairing of the six letters, each point must therefore have one leg going to each of the other two points. For letters a, v at the first point, two equal letters p, p at the second, and q, q at the third, there are two choices of which first-point letter goes to the second point. For either choice there are two choices of its partner among the equal p’s and two choices of the partner among the equal q’s. The third edge is then forced. Hence Tx,y,z([a, v], [p, p], [q, q]) = 4 dim g {Kx−y(a, p)Kx−z(v, q) + Kx−z(a, q)Kx−y(v, p)}Ky−z(p, q). (WE3) 258 ## Source page 261 A Wilson Yang–Mills construction J. Rodgers The color factor is exactly dim g: the three edges give Kronecker deltas requiring all three summed colors to coincide. All unequal-color summands are zero and each common color gives the same real scalar expression. Let x(t), y(t) →0, x(t)−y(t) = tu, u ̸= 0, with the third point z ̸= 0 fixed. The near covariance of two order-zero legs scales as t−4, while that of orders one and zero scales as t−5. Multiplication of (WE3) by t5 therefore gives 4 dim g {tKu(a, p)Kx(t)−z(v, q) + Kx(t)−z(a, q)Ku(v, p)}Ky(t)−z(p, q). (WE4) Every covariance involving the distant point is continuous in this limit. The first term tends to zero; the second tends to 4 dim g K−z(a, q)Ku(v, p)K−z(p, q). If the order-one letter is instead at the distant point, its two edges stay bounded and the only short edge is an order-zero covariance. Multiplication by t5 leaves an extra factor t, so that whole term tends to zero. Expand the panel in (FS6) in a basis as in (FS4), use (WE2) for the distant test, and apply these two limits to the three silent terms. The common nonzero factor 4 dim g cancels; the odd sign of the order-one covariance under u 7→−u gives exactly (FS7). Finally take p = c01, q = c12. Substitution in (FM2) gives, at every z ̸= 0, Kz(c01, c12) = 8c4z0z2(|z|2)−3. Differentiation of the first leg in direction zero gives Kz(s001, c12) = 8c4{z2(|z|2)−3 −6z2 0z2(|z|2)−4}. At z = e0 + e2, its squared norm is two and both displayed coordinates are one. The value is 8c4(1/8 −6/16) = −2c4 ̸= 0. The order-zero covariance is symmetric in its two plane tests; differentiating that symmetry gives the same value for Kz(s012, c01). Thus (FS9) reads −2c4a = −2c4b, proving a = b in (FS10). 10.4 Detection with a fixed exterior degree bound We first establish the collision extraction that will be used in the polynomial argument. Put Ra(x) = |x|2a. Say that F is cleared at order a when RaF, initially defined offzero, extends to a real analytic function on all of R4. A harmonic pole of degree r is analytic offzero, homogeneous of degree −r under positive dilations, harmonic there, and cleared at order r −1. The differentiated single-leg covariance of jet orders k, j is such a pole with r = 4 + k + j. A cleared homogeneous term of degree j means a function homogeneous of degree −j cleared at order max(j −2, 0). A product of poles of positive degrees a, b is a cleared homogeneous term of degree a + b, because the product of their analytic numerators clears it at order (a −1) + (b −1). The pole assertion for the differentiated kernels has a short direct proof. The Green function c4|x|−2 is homogeneous of degree −2; multiplying it by |x|2 leaves the constant c4. Away from zero its Laplacian vanishes, since the radial Laplacian in dimension four gives f′′(r) + 3f′(r)/r = 0 for f(r) = r−2. Suppose F is a pole of degree r ≥2 with numerator g = Rr−1F. For a fixed direction v, differentiation gives the numerator identity RrDvF = R1Dvg −(r −1)(DvR1)g. (HC0) Its right side is defined with the same regularity on all real space. Differentiating the dilation identity makes DvF homogeneous of degree −(r + 1), and commuting ordinary derivatives with the Laplacian shows it is harmonic offzero. Thus differentiation raises the pole degree by one and 259 ## Source page 262 A Wilson Yang–Mills construction J. Rodgers retains the required clearing order. Finite linear combinations of equal-degree poles retain all these properties. The two curls in (FM2) are a finite combination of second derivatives of the Green function, so have degree four. The two jet orders add k + j derivatives, with the factor (−1)j for derivatives at the second endpoint. Color pairings and arbitrary probe coordinates are fixed linear maps. This proves exactly the degree 4 + k + j and clearing hypothesis used above. It also gives a global bound |F(x)| ≤C|x|−r: bound the continuous function on the unit sphere and use positive dilation homogeneity. All uses here are offthe real collision point; no contact value or complex cone division is involved. For a harmonic homogeneous F of degree −r, Euler’s identity x · ∇F = −rF, together with the product rule for the four-dimensional Laplacian, gives ∆(Ra+1F) = 4(a + 1)(a + 2 −r)RaF. (HC1) For example ∆(|x|2F) = 8F +4x·∇F; the general formula follows either by the same differentiation or induction. If r ≥2 and the pole is cleared already at order r −2, it is zero. Indeed, starting with its analytic numerator at that order, (HC1) repeatedly produces an analytic numerator one order lower. At each step a ≤r −3, so the displayed coefficient is nonzero. At the last step F itself extends analytically. Its homogeneity then implies F(x) = trF(tx) →0 as t ↓0. For r = 2, clearing at order zero gives this conclusion immediately. Here is the exact finite expansion needed; it includes terms more singular than the pole being extracted. For integers r, M, write tr  F(tx) − X j r would diverge after scaling by tr; multiplying the convergent expression by tJ−r forces WJ = 0. Repeat until only degrees at most r remain. Degrees below r tend to zero; the limit is the degree-r term. Thus −K = Wr, or is zero if that degree is absent. In either case K is cleared at order r −2, and (HC1) makes it zero. This finite peeling, rather than a choice of a large scaling exponent, removes the more singular double contractions. To apply this statement, let P contain jets through order j, let a pair of test legs of orders k, l, with l < k, be inserted at x0 + z, and keep the other vertices separated. The contraction of an order-k probe p with the order-a variables of P is the differential operator Ba(p, z)P = X ν Kz(p, ea,ν) ∂a,νP, B(p, z)P = j X a=0 Ba(p, z)P. (HC3) The index ν includes the jet basis and color. Wick’s rule applies the two operators to every ordered pair of vertices. Set r = 4 + k + j and M = 8 + k + l + 2j + 1. There are four cases. 1. Both legs contract at x0. The resulting finite sum consists of second partial derivatives of P times products of two kernels. Their degrees are 8 + k + l + a + b < M, with a, b ≤j. They are the cleared terms W in (HC2), with harmonic part zero. 2. The order-k leg contracts at x0, and the order-l leg at another vertex. The distant contraction is analytic at z = 0. Multiplication by tr selects only Bj(p, z); the terms a < j have smaller pole degree and vanish. The distant contraction tends to its value at z = 0. 260 ## Source page 263 A Wilson Yang–Mills construction J. Rodgers 3. The order-l leg contracts at x0, and the other leg is distant. Its largest pole degree is 4+l+j < r, so the scaled limit is zero. 4. Both vertices are distant. Every coefficient is regular at z = 0, so the scaled limit is zero. These conclusions require only continuity of the distant Wick coefficients for the scaled limit; the coefficients are in fact analytic. Summing the finitely many placements proves (HC2). In case 2, summing the distant placements is Wick’s first-leg identity at x0. Summing colors in the invariant two-leg tester therefore gives harmonic part K(z) = ⟨(Cj(p, q; z)P)(x0) Y i Vi(xi)⟩, Cj(p, q; z)P = X c B(q, c) Bj(p ⊗c, z)P. (HC4) Here B(q, c) is the order-l, color-c jet variable. Orthogonality of the color action makes the sum invariant. The operator replaces a variable of weight j + 2 by one of weight l + 2, so it strictly lowers engineering weight whenever l < j. If P is a ghost at exterior cap d, and 4 + k + l ≤d, the inserted invariant pair is an allowed spectator. For every fixed nonzero direction z, all the inserted points stay separated for sufficiently small positive t. Permuting the inserted pair past P, ghost vanishing makes the left side before subtraction in (HC2) zero. Finite peeling and (HC1) now give K(z) = 0 for every z ̸= 0. Since the spectator word was arbitrary, Cj(p, q; z)P is itself a ghost at the same cap. This is the cluster extraction used below; it retains arbitrary spectator arity while keeping each spectator’s degree bounded. The passage to arbitrary input weight needs a uniform form of polynomial separation. An exterior bound d limits the weight of each spectator field, while allowing arbitrarily many spectators. Call a polynomial a ghost at bound d if all its separated correlations with all such spectator words vanish. We prove that one finite d, depending on the group, detects polynomials of every input weight. This is stronger than a Gram statement at a fixed input cap. First, separated single-leg covariances with fixed partner jet order span the dual of each homoge- neous solution-jet space. Their annihilator is zero by the differentiated covariance separation used in (FS7), and finite-dimensional duality proves the span assertion. Consequently if every single contraction of a polynomial with those tests is zero, all its partial derivatives in the corresponding jet variables are zero. Indeed a single contraction is the sum of those partial derivatives multiplied by the covariance coordinates; linear combinations of the spanning covectors isolate each derivative. Next we must separate colors using invariant tests. Put q = dim g. For every color index c = 0, . . . , q −1, choose a nonzero functional pc ∈Vc and a solution jet Xc with pc(Xc) = 1. Such jets exist at every order. Form the polynomial matrix Mcb = Bb(pc). (FS11) Its determinant is not the zero polynomial: assign the order-c jet to color c with value Xc, and all other color components at that order to zero. This evaluation makes M the identity. The argument takes place in the polynomial ring, which is an integral domain. It does not assert that M is invertible at every field configuration. Suppose a polynomial has no jet variables above order j +1. Extract a mixed cluster consisting of one highest-order contracted leg and one of the low-order color tests just chosen. Vanishing of these extracted correlations says MD = 0, where Db is the single contraction in color b with the highest-order variables. Multiply by the adjugate: det(M)Db = 0. Since the determinant is a nonzero polynomial, Db = 0. The preceding dual-span argument makes every highest-order partial derivative zero. Over characteristic zero a variable occurring with a positive power has a nonzero 261 ## Source page 264 A Wilson Yang–Mills construction J. Rodgers partial derivative, so the polynomial contains no variable at that order. This removes the highest jet order one layer at a time. We spell out the bounded-degree and induction aspects of the cluster extraction. Set j = q, L = j+1. The polynomial ring on jets through order j has finitely many variables and is Noetherian. Form its ideal generated by all positive polynomial-length homogeneous invariant polynomials. Choose finitely many such generators of this ideal, and let r bound their engineering weights. They generate every invariant polynomial as an algebra: for an invariant polynomial of length k > 0, express it in the ideal, take its length-k component, and average the coefficients over the compact group. Averaging is a projection onto invariant polynomials and satisfies R(hg) = R(h)g for invariant g. For a generator of positive length l, its coefficient after this projection is the averaged length-k −l part of the old coefficient; it has length strictly smaller than k. Induction on length expresses it as products of the same generators. Constants are empty products. Thus every invariant polynomial homogeneous of length k on this shelf lies in the span of products Q i Vi, with each Vi invariant of engineering weight at most r, homogeneous of length ki, and P i ki = k. This is the finite hosting bound; the number of factors is unrestricted. Here is the precise coalescence argument that uses it. Suppose a ghost P has all length compo- nents below k zero. Fix x ̸= y and a hosted product Q = Qn−1 i=0 Vi of total length k. Place P at x and Vi at y + tie0, for t > 0 sufficiently small. These points are distinct and avoid x. Expand all polynomials into monomials and apply Wick’s rule. A monomial of P of length l > k has too many letters to pair with the k exterior letters: it has no internal pair at x, so its moment is exactly zero. If l = k, every exterior letter must pair with a letter at x, leaving no exterior-to-exterior pair. All the resulting kernel arguments approach the nonzero vector x −y; their finite products therefore converge. Terms l < k are absent by hypothesis. The limit is exactly the two-point Wick pairing Bx,y(Pk, Q), where Pk is the length-k component. Ghost vanishing at every positive separated t makes this limit zero. Linearity extends it from products to their span. No estimate of kernels between coalescing exterior points is needed, since the pairing count removes those kernels before taking the limit. Induct on k. Let Pk be the first possibly nonzero length component and reflect its complex conjugate across the time hyperplane, giving Q = Pk ref. The shelf, invariance, engineering cap and length are preserved by this operation. The hosting result puts Q in the allowed span. At points −e0, e0 its reflected Wick pairing with Pk is strictly positive if Pk ̸= 0, by the polynomial Gram calculation above. Coalescence has made the same pairing zero, a contradiction. Thus Pk = 0. Every polynomial has only finitely many length components, so this induction proves that a ghost lying on this fixed shelf is zero. Define d = max{r, 4 + j + L}. (FS12) Induct on the engineering weight of the candidate ghost. If its highest jet order is J > j, cluster a pair of jet tests of orders L and l < q. Its weight is 4 + L + l ≤d, and l < L, l < J. Apply (HC2)–(HC4) with highest shelf order J: the finite cutoffis 8 + L + l + 2J + 1, and the extracted pole degree is 4 + L + J. The extracted mixed contraction is a ghost of strictly smaller engineering weight, hence is zero by induction. The color-matrix argument then removes the highest jet layer. Repeating leaves a polynomial on the shelf through j, where the hosting tests separate it. The weight-zero case is the constant, detected by the empty word. This proves ∃d ∀w ∀P of weight at most w, P a ghost at bound d =⇒P = 0. (FS13) The same finite bound is used before the finite-dimensional Ward readers are selected. In particular one takes their cap at least max(d, 4); lower-cap conclusions then follow by the filtered 262 ## Source page 265 A Wilson Yang–Mills construction J. Rodgers inclusions. Arbitrarily high output weight in (FS13) is permitted, but arbitrarily high individual spectator degree has not been silently introduced. 10.5 Elimination of higher-degree vector and angular defects Subtract from a silent vector row the scalar translation obtained in (FS10), and denote the residual row by Qi. It is silent, kills every degree-four field, and in particular kills the free stress. Insert that stress as an additional spectator in (FS5), apply its radial shell identity (FS3), and subtract the silent identity with the engineering-weight derivative applied to the distinguished field. The terms where the row acts on a spectator cancel in pairs. The stress term is zero. What remains says that [E, Qi]P is a ghost, E = X ν wνXν∂Xν, wν = jet order of Xν + 2. (FS14) All additional stress fields have degree four and the spectator cap is the fixed one in (FS13). Thus (FS13) gives [E, Qi]P = 0. Every polynomial is a finite sum of engineering-weight components Pk, and EPk = kPk. A map commuting with E preserves each such eigenspace: projecting its image onto weight l ̸= k gives (l −k) times that projection equal to zero. Total inversion acts on a weight-k monomial by (−1)k, because a derivative-order-j curvature variable has sign (−1)j = (−1)j+2. But the vector covariance law gives RQi = −QiR. On a preserved weight component these two rules give simultaneously RQiPk = (−1)kQiPk and RQiPk = −(−1)kQiPk. Hence QiPk = 0, and Qi = 0 on the whole cap. This proves the full one-dimensional free translation-charge kernel used after (RT17). For the internal angular row the degree-four quarter-turn argument in (RT18)–(RT19) already makes it zero on the free stress. Apply the angular version of (FS3) in the same silent identity. It makes the commutator with each angular generator a ghost, and (FS13) makes that commutator zero. On every finite filtered space, exponentiating a finite matrix gives the unique solution of its linear differential equation. Commutation with the generator therefore gives commutation with its quarter-turn. Signed covariance of the two alternating Lie-algebra slots and the plane-basis calculation following (RT19) then make every component zero at that cap. This supplies the higher-degree angular separation with the required bounded exterior tests. Finally the analytic segment argument following (RT19) converts the collinear scene readings and all their position derivatives into the full silent identity (FS5). Taking the cap in (FS12)– (FS13), the preceding arguments identify its kernel. Finite-dimensional decreasing-kernel selection gives a finite list with exactly that kernel. Its nonzero minors remain nonzero for the convergent interacting reading matrices. This proves the rank transfer used to normalize the actual translation charge and eliminate the actual internal angular defect. 10.6 Obtaining a detector on the physical coefficient space The detector used below is finite dimensional. At fixed degree let E be the coefficient space, I its equation ideal, and r the dimension of the free-field quotient E/I. The foundation identifies the kernel of the physical projection πi with an invertible image of I. Rank-nullity therefore gives dim ran πi = r, uniformly in the cutoffs. In the notation of the detector trials r = dim + 1. This dimension statement does not assert that every bare coefficient survives the equations. Choose r trained trial vectors ej,i, uniformly bounded by v, and let the detector matrix consist of their tested responses. The limiting response matrix has full column rank. In finite dimensions the least norm on its unit sphere is positive; compactness of that sphere makes the lower bound 263 ## Source page 266 A Wilson Yang–Mills construction J. Rodgers stable under sufficiently small entrywise perturbations. Denote the allowed perturbation by δ > 0 and the resulting left-inverse bound by L. Train with tolerance ε = δ/4. The actual tested trial vector differs from its limiting response by at most two errors of size ε: the finite training error and the short-distance probe error. There is a third error when ej,i is replaced by πiej,i. Indeed ej,i−πiej,i has exactly zero projection. The equation-cancellation bound makes its probe tend to zero, hence eventually smaller than ε, for all finitely many j simultaneously. Thus the projected trial response matrix is within 3ε < δ of the full-rank reference matrix. Its columns are independent. Since ran πi has dimension at most r, the projected trials span that range. The projection has a uniform operator bound A, so each spanning vector has norm at most Av. Apply the stable left inverse to the detector response of any vector in the range and expand in these r spanning vectors. Summing their norms gives ∥πix∥≤ArvL ∥Liπix∥. The choices are ordered: first choose the training index sufficiently large, then a sufficiently small positive probe scale, and only then pass sufficiently far out in the whole-mesh filter. Constants are independent of the remaining cutoffs and volume. The exact equation-cancellation input is obtained by applying the banked insertion estimate to the identified kernel; its cost is the vanishing flight of exponent (-1/32). The banked estimate is proved in (IC3)–(IC5); the separated reflected Wick pairing below constructs the full-rank reference matrix. The stability assertion has an explicit elementary proof. Write G : Cr →Cs for the reference response matrix, with sup norms, and choose K > 0 such that |u| ≤K|Gu|. Set δ = [2K(r + 1)]−1. If every column of H −G has norm at most δ, then ∥(H −G)u∥≤rδ∥u∥, ∥u∥≤K∥Hu∥+ Krδ∥u∥≤K∥Hu∥+ 1 2∥u∥. Hence |u| ≤2K|Hu|; one can take L = 2K. The injectivity of the reference matrix follows directly from separation of the free quotient by the chosen correlation functionals: a vector anni- hilated by all rows is zero. Expressing these functionals in a basis identifies their matrix with G. This explains both the rank lower bound and its stability on precisely the same varying projected trial columns used in the rank upper bound. 10.7 Bounding physical coefficients before taking limits Here is the normalization argument that upgrades the available insertion estimates to estimates for lattice differences. Write xi for the coefficient vector representing a difference insertion, and Xi = piixi for its physical projection. The presentation is idempotent: pi2 i = pii. The detector construction supplies linear maps Li, a constant Z ≥0, and, eventually in the mesh limit, ∥Xi∥≤Z∥LiXi∥. An insertion already bounded in the Wilson laws, together with a vanishing error in the tested insertion seminorms, gives |Lixi| ≤c eventually. This alone does not bound Xi, because the equation part xi −Xi need not have small coefficient norm. Instead put fi = (1 + ∥Xi∥)−1, yi = fiXi, zi = fi(xi −Xi). Then |yi| ≤1, 0 < fi ≤1, and zi belongs to the same admissible coefficient class: that class is closed under bounded scalar multiplication and subtraction of bounded coefficient families. 264 ## Source page 267 A Wilson Yang–Mills construction J. Rodgers Idempotence gives piizi = 0 exactly. The equation-cancellation estimate in the literal Wilson law therefore makes the tested observables associated with zi tend to zero. In particular, eventually Z|Lizi| ≤1 2. By linearity, Z∥Li(Xi −xi)∥≤1 2(1 + ∥Xi∥). The detector inequality and the triangle inequality now give ∥Xi∥≤Zc + 1 2(1 + ∥Xi∥), ∥Xi∥≤2Zc + 1. Thus the projected coefficients have a fixed bound before any limit is taken. Their primary insertions consequently have fixed cost. The unprojected remainder has zero physical projection, so the equation-cancellation estimate gives it a vanishing cost, hence also a fixed cost. Subtracting this remainder and the original vanishing approximation error proves the fixed-cost estimate for the difference insertion itself. This argument uses cancellation for the actual number of copies in the detector; it does not replace equation terms by a coefficient-norm estimate. For completeness, the algebraic passage from differences of correlations to these insertions is exact. In a Wilson integral of a product of slot observables, a lattice difference in the position of slot j acts only on the j-th factor. Finite subtraction commutes with integration, since the factors are continuous and integrable on the compact configuration space. Iterating this identity expresses every prescribed mixed lattice difference as a Wilson integral of the corresponding shifted and differenced slot observables. The preceding fixed-cost argument therefore yields the stated bounds for those differences once the detector and cancellation estimates have been supplied. 10.8 Equation cancellation in the physical detector The physical detector must annihilate the calibrated equation image exactly. The proof therefore corrects complete cylinders, including all lower jets read by the detector, rather than prescribing a single leading coefficient and ignoring its induced lower terms. Equivariance and reality are then restored by averaging operations that preserve this exact cancellation. We supply the cancellation input used in the detector argument. At fixed grade d, let Id be the free equation ideal, and let Rd be its marked lift into local lattice insertions. The marking identity says that an insertion evaluated on Rdy, y ∈Id, has zero Wilson expectation against any bounded measurable spectator depending outside a fixed finite neighborhood, once the lattice is large enough. Both the neighborhood radius and the minimum size can be chosen uniformly over y ∈Id: choose a finite basis of this ideal, take the maxima of its basis radii and size thresholds, and use linearity and finite summation. Multiplication by arbitrary complex coefficients does not enlarge the neighborhood. This local identity makes the replicated root seminorm exactly zero. Fix a coarse reporting level k. Eventually a fine-scale chamber contains the common marking neighborhood strictly inside it, because the ratio of fine to reporting spacing tends to zero. Tile the fine torus by the prescribed chambers and put one translated copy of the insertion in each chamber. Keep the distinguished copy in chamber zero, and multiply all other copies to form the spectator. Distinct chamber interiors keep that spectator outside the marking neighborhood. Its measurability and boundedness follow from the local insertion bounds on the compact finite configuration space. The marking identity then gives Eβ Y chambers a fa = 0. 265 ## Source page 268 A Wilson Yang–Mills construction J. Rodgers The distinguished copy is exactly the original local equation insertion, including its scale normaliza- tion. Taking the absolute-value root defining the root seminorm gives zero. The same conclusion holds on the whole mesh and for every volume, with the reporting level fixed before taking the fine tail. To transfer this exact cancellation through the trained coefficient charts, write Zi for the termi- nal chart at cutoffi and Pi for its calibrated invertible coefficient change. The physical projection has kernel equal to the image of Id under the composite marked chart. For a coefficient family xi in this kernel define yi = P −1 i Z−1 i xi. Then Ziyi belongs to the marked equation carrier. The calibrated inverse estimate bounds the normalized coefficient size of yi by twice that of xi. Hence a coefficient family with the permitted subexponential growth retains that growth after this inverse change. The trained-chart comparison is upper(ZiPiyi) −hallmarks(yi) = Oroot(flight−1/32(i)). (IC3) Its finite-dimensional derivation is explicit. Decompose yi using the fixed finite signed-orbit tally of frame basis vectors. For each tally component the trained local comparison has root bound flight−1/16(i). Its scalar coefficient is bounded by the coefficient norm times the fixed orbit- coordinate constant. Sum over the finite tally. The sum of these scalar bounds is subexponential, so it is eventually at most flight−1/32(i)/ flight−1/16(i). This proves (IC3). The algebraic recombina- tion is exact: the trained basis expansion is Piyi, whereas the original frame expansion recombines to yi. Signed translations and conjugations commute with the local insertion by its covariance. The hallmark term in (IC3) has zero root seminorm by the chamber argument. Since ZiPiyi = xi, we obtain upper(xi) = Oroot(flight−1/32(i)) when xi ∈ker πi. (IC4) This is the banked equation-cancellation estimate. In the detector proof apply it to xi −πixi, using π2 i = πi. The projected trial vectors and their unprojected versions then have the same limiting tested responses. Their common rank is the free quotient rank: the equation carrier is an invertible image of Id, and the free observable map has kernel precisely Id. Rank-nullity gives this dimension statement without assuming independence of all bare coefficients. The reference response matrix is injective on that free quotient. Each row is a free correlation functional: insert the variable quotient field in the first slot and the prescribed fixed basis fields in its spectator slots. Multilinearity makes it a linear functional. The separating family of these tests has zero common kernel, so its map into the finite row space is injective. Composing it with a quotient basis gives exactly the reference matrix used earlier. Thus (IC4) and the finite-dimensional stability calculation together apply to the same varying projected trial columns, completing the detector’s cancellation step. The local marking identity underlying (IC4) originates in Haar integration by parts. Vary one stored positive link e by Ue 7→exp(tA)Ue, with all other links fixed, and denote this derivative by ∇e,A. The change preserves product Haar measure. The Wilson action derivative is ∇e,AS(U) = ⟨A, Ee(U)⟩. Here Ee is the sum of the incident plaquette curvatures, with a backward incidence transported by the inverse preceding link. More explicitly a plaquette type with axes i < j contributes, at a link of axis i, the difference Fij(x) −AdU−1 x−ej,j Fij(x −ej), 266 ## Source page 269 A Wilson Yang–Mills construction J. Rodgers and at a link of axis j the negative of the analogous difference in direction i. Other plaquette types contribute zero. There is one contribution for each positive-axis plaquette type. Differentiating its four ordered factors gives this formula, including the inverse-link sign. The normalization follows from −Re tr(colorMat(A) mat(g)) = repDim ⟨A, curvature(g)⟩. This identity uses skew-adjointness of the color matrix and the orthogonal projection defining curvature. Let W and ∇e,AW be continuous, and let S be bounded measurable and unchanged by this link variation. Haar invariance gives Eβ ∇e,AW −β⟨A, Ee⟩W S  = 0. (IC5) To justify differentiation, translate the derivative formula from zero to any t by the one-parameter group law. On the compact configuration space the continuous derivative of the Wilson-weighted W is bounded; multiply its bound by the bounded envelope of S. This is an integrable uniform bound for difference quotients by the fundamental theorem of calculus. Differentiation under the Haar integral is therefore valid. The integral is constant by invariance, so its derivative is zero. Dividing by the positive partition function proves (IC5). The spectator need not be continuous or gauge invariant. With W = 1, β ̸= 0, it gives the exact equation expectation zero. The marking construction retains the derivative term when a multiplier uses the varied link. Its local equation identity means zero expectation against every bounded measurable spectator depending outside some lattice radius r, on every torus of side at least a fixed M, at every nonzero real β. The class of continuous local functions with this property is closed under finite sums, scalar multiplication and conjugation: enlarge the radii and side thresholds to common maxima; for conjugation use the conjugate spectator and reality of the Wilson law. These operations are exactly those used to assemble the equation section. One must also arrange the desired continuum label while enforcing these exact identities. The finite graded construction supplies a linear pair p 7−→(Qp, fp), where fp is a gauge-invariant local polynomial insertion with leading jet Qp. At every graded piece, Qp −p lies strictly above that piece in the finite lexicographic weight ordering. The construction retains the exact marking identity on the equation ideal. Triangularity makes Q invertible: a nonzero vector has a first nonzero graded piece, which Q leaves unchanged, so Qp = 0 is impossible. In finite dimensions injectivity implies surjectivity. The same first-piece argument shows that Q−1 preserves each descending weight filtration. Define a linear local realization by Y (p) = fQ−1p. It has leading jet p and retains the required power filtration. Real conjugation and signed coordinate symmetry are imposed on the paired construction, so Q, its inverse, and Y obey these symmetries, and the unit is fixed. There is one finite stencil realizing all its frame basis vectors: choose their finitely many polynomial presentations, enclose all their underlying stencils in a common centered box, and rewrite them as polynomials in the common box coordinates. Define the section on arbitrary coefficients by linear extension from this basis. Equality with Y on the basis proves equality everywhere. The box contains zero and is invariant under signed coordinate permutations. With R = Q, the resulting fixed section satisfies T(coords(Rp)) = fp. 267 ## Source page 270 A Wilson Yang–Mills construction J. Rodgers In particular the equation ideal is sent to the exact local marking class, as required above. The same section supplies its finite derivative-credit constants by compact polynomial bounds. The section and R are chosen once at grade d, before any block side, period, coupling, refinement or later derivative budget is specified. This explains why the local equation cancellation survives all subsequent scheduling choices. The graded linear lift just used is assembled as follows. For each weight pair m, let Em be its homogeneous label space and Jm = Em ∩Id. The ordinary polynomial section supplies a linear lift on Em with the correct leading form and power bound. On Jm, choose a finite basis and lift each basis vector by the positioned equation construction. Its output is a pair (q, f) with q −p of higher weight, the prescribed continuum form, the same power bound, gauge invariance, and the exact local marking identity. Extend these finitely many pairs linearly on Jm; all these properties are linear, including membership in the marking class. Choose an algebraic projection Lm : Em →Jm. Define the lift on Em by Fm(p) = F eq m (Lmp) + F ord m (p −Lmp). For p ∈Jm the second term is zero. For arbitrary p ∈Em both summands have the required grade and higher-weight difference, so their sum has leading form p. At the zero weight, J(0,0) = 0, since the equation ideal contains no nonzero unit, so this construction sends the unit to (1, 1). The whole lift is the finite sum F(p) = P m Fm(pm) over the homogeneous components of p. The equation ideal is preserved by taking these components, so its entire image has the marking property. This proves the linear assembly used in the triangular argument. Its equivariant version involves only finite averages. Let θs act on labels by a signed coordinate transformation, and let φs(q, f) = (θsq, f ◦pulls). Replace F by Fav = |S|−1 X s∈S φsFθs−1. Each summand has the same leading label and higher-weight remainder, because the signed action preserves grades and the equation ideal. The local marking class is preserved by this coordinate change. The averaging coefficients sum to one, so the leading label and the unit stay fixed. Rein- dexing by left multiplication in S proves signed equivariance. Finally let c(q, f) = (q, f), with the prescribed semilinear real structure on labels. Set FR(p) = 1 2 (Fav(p) + cFav(p)) . The composition of the two semilinear maps in the second term is complex linear. The real structure commutes with the signed action and preserves every grade, higher-weight space, form and marking identity. Thus this midpoint is both signed-equivariant and real, retains the unit, and still has the triangular leading-label map. These averages are performed before constructing the common finite stencil or choosing any cutofffamily. Gauge invariance of a positioned equation lift is obtained by averaging its constant-color or- bit, while retaining a finite polynomial realization. If f is a powered local polynomial, its orbit fg(U) = f(gaugeg U) spans a finite-dimensional space: each factor uses a fixed finite matrix repre- sentation and finite polynomial degree, and conjugation acts linearly on its finitely many coefficient tensors. The marking property is preserved by g: in its expectation change variables by the con- stant gauge transformation, use invariance of the Wilson law, and pull the spectator back by the inverse transformation. Its locality radius and bounded measurable envelope are unchanged. 268 ## Source page 271 A Wilson Yang–Mills construction J. Rodgers The paired vectors (1, fg) therefore lie in a finite-dimensional space. Their first coordinate and all configuration-evaluation coordinates are continuous functions of g. These functionals separate that space, so their span is the whole dual; hence every linear coordinate is continuous. Integrate the coordinates of (1, fg) against probability Haar measure to get a vector in their span. Its first coordinate is one and its second coordinate is exactly ̄f(U) = Z G f(gaugeg U) dg. Membership in this span expresses the pair as a finite linear combination of orbit pairs. The leading-form, higher-weight, power and marking properties are all preserved by those finite linear combinations, with the first coordinate keeping the leading label equal to the original p. Thus the average is still represented by an actual finite polynomial; no infinite-dimensional integral closure is needed. For these transported local polynomials, a general gauge transformation acts through its value at the root. Right invariance of Haar measure therefore proves full gauge invariance of ̄f. The polynomial leading form is color-invariant as well: compute its constant-color transformed form by equivariance, then use uniqueness of the leading-form polynomial to identify it with the original. This puts that form in the label space and proves the gauge-invariant version of the positioned lift used above. The generation of the equation ideal in this construction is ordinary polynomial algebra. On the degree-j jet space let Pj be a linear projection onto the range of the injective free gauge-jet map. Substituting Pj in every jet variable defines an algebra homomorphism P. A label polynomial belongs to the equation ideal precisely when Pp = 0: restriction to the free solution range is zero if and only if the polynomial after this projection is zero on every jet. Polynomial evaluation separates polynomials. For a linear jet functional l, split l= (l−lPj) + lPj. The first part vanishes on free solution jets, the second is its projected part, and both have the same bidegree as l. For a product of linear functionals, telescope the difference between that product and the product of their projected parts. Every resulting term contains one literal factor vanishing on free solution jets. Expand an arbitrary polynomial into its finitely many monomials. If Pp = 0, this telescoping writes p as a finite linear combination of such terms. When p is homogeneous, every term has its same bidegree, since the projection preserves each jet degree. It also follows that each homogeneous component of an equation polynomial is an equation polynomial. A polynomial of field-count degree zero is a constant; its restriction to free solution jets is that constant, so the equation ideal has no nonzero such element. To lift a term leql1 · · · lr, the source first represents the equation functional leq as a linear combination of linearized transported lattice forces. For each individual force, it constructs the other factors on stencils avoiding that force’s varied edge. Their product is therefore unchanged by that edge’s one-parameter variation, and (IC5) makes the resulting force product an exact local equation insertion. Different force summands may use different avoiding stencils; they are combined only after their individual identities have been obtained. The leading-jet bookkeeping keeps all linear terms through the retained degree. Each avoiding ordinary factor agrees with its desired linear functional modulo terms of field-count degree at least two. Their product agrees with the desired r-factor monomial modulo field-count degree at least r + 1, by the finite product expansion. Multiply by the force factor, truncate to the retained jet 269 ## Source page 272 A Wilson Yang–Mills construction J. Rodgers degree, and the error has field-count degree at least r + 2, strictly above the target count r + 1. The force’s retained linear functional may have higher jet weights than leq; these are also retained and lie strictly above its target weight. Multiplication by the other factors preserves this strict inequality. Adding the finitely many force terms therefore gives a polynomial with target leading term and only higher-weight remainder, while the actual lattice function remains exactly locally silent. Finite linear combination and the ideal generation just proved yield the positioned raw lift for every homogeneous equation label. This accounts for the triangular rather than exactly homogeneous form of its retained continuum polynomial. The force-span assertion is a finite-dimensional annihilator argument with a concrete kernel calculation. Linearize a force transported from a lattice source to the root along a path avoiding that source edge. At linear order its value is a finite sum of transverse differences of the sampled curvature jets: d−2 X l=0 X a 0 is the common separation reserve and A the insertion allowance. Summing gives |Mi(f; x) −Mi(f; y)| ≤4(n + 1)An+1R/r. (IC6) 271 ## Source page 274 A Wilson Yang–Mills construction J. Rodgers The bounds for reflected labels are uniform as well, since their finite-dimensional real-linear action maps a fixed coefficient ball to a bounded set. A detector average with probe scale b > 0 samples points displaced by at most 2b per direction in its fixed direction list. Equation (IC6) makes its difference from the unaveraged monitor arbitrarily small by taking b ↓0, uniformly on the subsequent whole-mesh tail and over every volume. The lattice placements have exactly the prescribed physical positions, because the refinement ratio satisfies aN count(N, k) = ak. There is no additional moving-position error at that step. For the trained columns, the unaveraged monitor tends to its free reference response in the order: training index first, then the whole mesh. Split its error into the finite-level training error and the free hierarchy’s evaluation at the approximate rather than target generic placement. The former tends to zero by the trained insertion estimates. The latter tends to zero by smoothness of the free hierarchy on the distinct-position configuration space. There are only finitely many rows and columns, so all their tails can be intersected. Take the training error below δ/4, then take b so that the averaging error is below δ/4, and finally use (IC4) so that projecting each trial changes its response by less than δ/4. This recovers the three errors used in the detector argument, with one common mesh tail and no extraction needed to establish rank. The trial coefficient bounds and projection norm bounds are already uniform over N, p, so the resulting detector inequality has exactly the uniformity required before passage to continuum limits. The training errors used above are bounds on complete corrected cylinders. Introduce commut- ing square-zero markers ε1, . . . , εn. Let F(ε) = X A⊂{1,...,n} Sfree(xA; qA)εA, D = log F, where the empty moment is one and εA = Q a∈A εa. The logarithm and exponential are finite polynomials in the nilpotent marker ideal. For actual lattice insertions Pa, form H(ε, U) = e−D(ε) Y a (1 + εaPa(U)). The training bound controls every nonempty coefficient of H in the replicated root seminorm by a common gapk →0, not just its singleton coefficients. The finite identity FH = Q a(1 + εaPa) gives, coefficient by coefficient, Y a∈A Pa = FA + X B⊊A FBHA\B. Thus the Wilson moment error from the free moment at the actual rescaled placement is bounded by the finite absolute coefficient price of F times gapk. This price stays bounded as the placements tend to the fixed distinct configuration: every subconfiguration stays distinct and every free submoment is continuous there. This proves the training-error convergence used after (IC6). In particular the singleton coefficient is Pa −F{a}, and for two distinct slots the pair coefficient is (Pa −F{a})(Pb −F{b}) −(F{a,b} −F{a}F{b}). The detector rows use at most these two slots. The pair correction is therefore its actual free cumulant at the rescaled placement; it cannot be replaced by a product of separately centered one-point terms. All trial coefficient limits can be taken on a common refinement. For each fixed grade and training index the finitely many lifted coefficients lie in a fixed compact coefficient ball, uniformly 272 ## Source page 275 A Wilson Yang–Mills construction J. Rodgers in the two cutoffs. The set of triples consisting of grade, training index and column index is countable. Apply successive compact subsequence extraction to these triples and use the diagonal refinement of the two cutofforders. This preserves all previous hierarchy and projection limits. Replacing the trial’s placeholder limiting columns by the limits so obtained changes none of its finite-cutoffestimates or placements. Passing the corrected-cylinder moment bound through the fixed-placement hierarchy limit proves convergence of every limiting trial response to the same free reference matrix. Its stable rank, together with the equation dimension bound, makes these limiting projected columns span the continuum physical quotient. The same trial data are retained when null labels are removed, so the cleaned hierarchy has exactly the free quotient dimension at every grade. The detector’s free columns can all be taken from the same reference frame, and one separated point pair suffices. The images of the frame basis span the free quotient, since the coefficient projection is onto. Its unit image is nonzero. Extend that unit to a basis by selecting further vectors from these finitely many spanning images, and enumerate the basis with the unit first. The original basis is real and homogeneous, and projection preserves conjugation and each bidegree. The selected quotient basis b0, . . . , bm therefore has all these properties; no orthonormality at a probe separation is required. For any x ̸= y, let Bx,y(p, q) be their free two-point Wick pairing. The free quotient identifies injectively with its polynomial realization on Maxwell solution jets. Reflect every tensor index of the conjugate polynomial p through the hyperplane perpendicular to x −y; call the resulting quotient field p♯. The reflected Wick positivity estimate gives Bx,y(p, p♯) > 0 (p ̸= 0). Hence the map p 7−→(Bx,y(p, b0), . . . , Bx,y(p, bm)) is injective. If all its entries were zero, bilinearity and the basis expansion of p♯would make its positive pairing zero. The same positivity shows that the free all-spectator null space has no residual beyond the equation ideal: any nonzero solution polynomial is detected by this particular reflected partner at one pair. Choose x = 0, y = (1, 1, 1, 1). This pair differs in every coordinate and is therefore a generic detector placement. There is one row per basis partner and exactly one spectator in every row. Its response matrix is the injective two-point map just displayed. This specifies the finite free detector used in (IC6) and the subsequent rank argument. The reflected positivity used here holds jointly for different jet orders, not just separately on each order. Fix a nonzero separation z, write r(z) = ∥z∥−2, and let Rz be reflection perpendicular to z. A real solution-tensor probe P of homogeneous order j has the Kelvin response reply(P)(y) = sjr(y)2R⊗2 y P(inv(y)), sj = 4(j + 1)(j + 2)(−2)j normal . Tensor indices and arguments transform together. Translate this polynomial probe in its inverted argument by −inv(z) and decompose it into homogeneous degrees 0, . . . , j. Multiply its order-k component by sj/sk; denote these lower-order probes by Tjk(z)P. The diagonal component is P. The sum of their responses is consequently r(y)2R⊗2 y sjP(inv(y) −inv(z)). Along y = z+tv, the inversion difference is tV (z, v, t), where V is smooth near zero and V (z, v, 0) = r(z)Rzv. Homogeneity shows that this response has a factor tj. Thus all its normalized Taylor jets 273 ## Source page 276 A Wilson Yang–Mills construction J. Rodgers at z below order j vanish, and its order-j jet is sjr(z)j+2RzP. Polarization turns the equality along every line into equality of the symmetric derivative tensors. Pairing with a reflected probe introduces the derivative sign (−1)j. Hence the transformed reflected Gram form, on a tower of orders 0, . . . , n −1, is n−1 X j=0 cj(z)⟨Pj, Qj⟩tensor, cj(z) = (−1)jsjr(z)j+2 > 0. The change of probes is triangular with identity diagonal. Alternatively its displayed positive Gram identity immediately proves injectivity, and finite dimension gives invertibility. The original simultaneous reflected form is therefore positive definite on the entire tower. Choose a real orthogonal basis e1, . . . , eM for this positive form and let di > 0 be its squared norms. Every finite Wick polynomial is a polynomial in such a finite tower. The cross-insertion Wick rule gives ⟨eα, eβ⟩refl= 1α=β M Y i=1 αi! dαi i . Indeed remove one factor on the left. Only factors of the same index on the right can pair with it, giving βidi; induction proves the formula, starting with the empty monomial. These are Wick products at each insertion, so within-insertion contractions are absent. Therefore for a polynomial P = P α pαeα, ⟨P, P⟩refl= X α |pα|2 Y i αi!dαi i > 0 (P ̸= 0). This proves reflected positivity, including arbitrary mixtures of engineering weights and field-count degrees and the constant monomial. For completeness the Kelvin response follows from differentiation of the scalar Green function with the source’s normalized Taylor tensors. For a symmetric order-j tensor P, define DP f(x) = X ν1,...,νj ∂ν1 · · · ∂νjf(x) j! P(eν1, . . . , eνj). A solution tensor S obeys the trace constraints corresponding to harmonicity. With G(x) = normal r(x), DSimG(x) = (−2)j normal r(x)j+1Sim(x, . . . , x). (IC7) Induct on j. The order-zero identity is the definition of G. The normalized derivative recursion is DP f = (j + 1)−1 P a ∂aDP(ea,·)f. Use ∂ark = −2krk+1xa. The derivatives falling on the tensor polynomial sum to its frozen trace and vanish. The remaining sum resolves P a xaea = x; division by j + 1 cancels that factor from the derivative of rj+1, leaving the next factor −2. This proves (IC7). Put Vi(x) = P a xaSai(x, . . . , x). The Bianchi identity and Euler’s homogeneous identity give ∂iVm −∂mVi = (j + 2)Sim(x, . . . , x). Indeed differentiating xa contributes twice Sim by antisymmetry; the remaining difference is P a xa∂aSim = jSim. Applying (IC7) and the vanishing divergence of S gives X a ∂aDSaiG = −2(j + 1)(−2)j normal rj+2Vi. 274 ## Source page 277 A Wilson Yang–Mills construction J. Rodgers The contraction of the two-curvature kernel with S is twice the curl of this last vector, with the order dictated by the opposite-side derivative. Differentiating once more yields sjrj+2 [Sim −2r(x)(xiVm −xmVi)] . The bracket equals reflection of both antisymmetric form indices through Rx; its term quadratic in the two radial contractions vanishes by antisymmetry. Since S is homogeneous of degree j, replacing its argument x by inv(x) = r(x)x converts the prefactor rj+2 into r2. This is precisely the Kelvin response used above, with sj = 4(j + 1)(j + 2)(−2)j normal. The sign (−1)k in an order-k probe at the second point comes from differentiating x −y there; it is included once. All these identities hold offthe origin, where the Green function is smooth. 10.10 Selecting the physical detector trials The trials used in the detector argument can be selected on the physical history already constructed. Here is the selection, with its quantifiers. Fix a finite detector panel, its real reference representa- tives, and its positive comparison width b. At arbitrarily late comparison indices k, the nonlinear construction supplies its metering bounds, the restarted representations and their admissible paths, the one- and two-insertion comparisons, and the marked estimates. For every sufficiently large refinement index N ≥k, these data are available for every volume parameter p. The representation and the readout may depend on N, p; the comparison index and its bounds do not. Only insertion arities at most two are needed, since each panel row has at most one additional insertion. The finite label transport is chosen once for all rows at these k, N, p. It is a linear map on the finite label space with the previously proved filtration and reality properties. Its inverse transports the reference representatives to a real tuple xN,p, with its zeroth member equal to the distinguished unit. The coefficient bound for this inverse gives sup N,p,ν ∥xN,p,ν∥≤Dk < ∞. (IC8) To make this statement include the finitely many refinements preceding the availability threshold, put every member of the tuple equal to the unit there, and enlarge Dk to include its norm. No estimate at those exceptional refinements will be used. The comparisons at the middle of the admissible scale interval propagate to its starting width along the actual representations at every intermediate scale. On this interval the reference coupling lies between one half and one times its value at k. Thus the same metering bounds apply through- out. The conditioned insertion estimate, followed by the path propagation estimate, bounds each requested correlation by its assigned bill at the starting width. Its error is the common shipping budget ε(k), which tends to zero. This is a statement about the transported physical insertions: it requires neither an assertion of a stronger density comparison nor a newly chosen scale history. There are finitely many panel rows and coefficient requests, so all their eventual thresholds can be met together. For completeness, the passage from arbitrarily late indices to a sequence makes all the geometric limits simultaneous. Write w(k) for the starting width and g(k) > 0 for the testing scale. Along admissible indices, w(k) −→∞, g(k) −→0, ε(k) −→0. The first assertion follows from the depth property of the scale calendar: if an admissible width stayed below a fixed j, its plateau identity would contradict the eventual strict increase of the 275 ## Source page 278 A Wilson Yang–Mills construction J. Rodgers calendar beyond the depth at j. Given integers i, j and η > 0, intersect the arbitrarily late supply set with the eventual conditions k > i, w(k) > j, g(k) < η, the banking bound, the reference-grade threshold, and the finitely many geometric fit conditions. This intersection is nonempty. Recursively select kn+1 from it with kn+1 > kn, w(kn+1) > w(kn), g(kn+1) < g(kn). The resulting indices and widths are cofinal, the positive testing scales decrease to zero, and the error budgets tend to zero. At each selected index choose the bounded real lifts in (IC8). The trial marks are the prescribed lattice approximations to the panel’s fixed physical configu- rations. Their fit bounds place them within the fixed quarter box, give their required separation and stencil admissibility, and imply convergence to those configurations. For each selected trial index, all sufficiently fine N, and every p, apply the physical conditioned estimate to these same lifts and marks. The stencil realization identity identifies the resulting correlations with the trial matrix entries. Consequently each entry has the required corrected comparison bound with error ε(kn), uniformly in p. This supplies the actual trial sequence used in the compact extraction and positive free-Gram argument, including the bounded lifts, common readout, unit and conjugation conditions, and the uniform volume quantifier. 11 The common continuum hierarchy Two bound families enter compact extraction. One is available at each fixed physical period, with constants allowed to depend on that period. The other holds uniformly over sufficiently large periods, eventually in the mesh at each such period. The second is supplied separately; it is not a consequence of the first. The full inner Cauchy limit and the compact selection are identified before the outer period refinement is taken. 11.1 From convergent transported coefficients to fixed-level integrals Fix the intermediate level m, period p, degree d, and an insertion coefficient y. Along the chosen increasing finest cutoffs, the transport construction gives a Cauchy sequence of primary coefficient vectors and a Cauchy sequence of captured interaction coefficients on one common dictionary at level m. The latter statement includes equality between each captured coefficient’s realization and the interaction part of that same transported insertion. Thus coefficient convergence is applied to the actual transported insertion, rather than to an independently chosen representative. The coefficient-to-observable estimates convert these two Cauchy statements into uniform boundedness and uniform Cauchy convergence over all configurations, both for the primary insertion and for every forced local profile. Spatial pullback, conjugation when required by the motion, and multiplication by its fixed scalar preserve these properties. Boundedness of a Cauchy coefficient sequence also bounds its paid coefficient norm, since that norm is a fixed scalar times a continuous finite-dimensional norm. Measurability follows from the local continuous terms, the measurable masking functions and the convergent countable sums in the insertion formulas. Continuity of the masks is not used. To see how this enters the normalized integral, remove the scalar vacuum factor from the restored density and denote the result by Tn(U). Its mode coefficient factors as the cold density times the selected coefficient of a product: the polynomial exponential of the primary insertion, and the field 276 ## Source page 279 A Wilson Yang–Mills construction J. Rodgers polynomial built from the background activities plus the forced profiles with the prescribed flags. The fixed-level flux bounds give uniform Cauchy convergence for the cold densities and background field polynomials. The preceding insertion bounds do the same for primary and forced profiles. Finite sums, products and continuous maps on common compact bounded ranges preserve uniform Cauchy convergence, so every mode of Tn has that property. Integration against the probability Haar prior therefore gives Cauchy numerator integrals. The stripped partition functions are Cauchy as well and have a common strictly positive lower bound, as proved earlier. Completeness gives their limits; the lower bound makes the denominator limit nonzero. Taking the quotient proves that the normalized mode integrals are Cauchy. Finally, the scalar vacuum factor cancels exactly in numerator and denominator: this quotient equals the common-normalization integral of the restored insertion. This is the link between the fixed-level coefficient construction and the earlier fine-to-intermediate comparison. The transport recurrence and captured-coefficient estimates are proved in the fixed-level convergence sections below. 11.2 Identification of the public hierarchy with the reconstructed one The final public projection introduces no second quantum field system. Its Hilbert space is the reconstruction’s completed state space, its common domain is the reconstructed word domain, its vacuum is the reconstructed vacuum, and its field operator is the same operator on that domain. The label space is the union of the graded field spaces already obtained from the continuum table. The phase space and probability measure are those of the character decomposition. For a character χ, the public separated correlation is the table’s evaluation in that character; the unselected cor- relation is the table’s unselected evaluation. The OPE reader supplies the channels, channel fields and coefficients. These identifications keep the lattice limits, OPE terms and reconstructed fields attached to the same table. The equality of vacuum moments can be checked without any limiting interchange: recursively evaluating a list of smeared fields in the public definition gives the identical reconstructed word vector, starting with the same vacuum. Taking its inner product with the vacuum proves equality of the moments. Similarly translations are the original strongly continuous translation operators and the energy-momentum measure is the original projection-valued measure. A Poincaré transforma- tion is represented by its Lorentz matrix and translation vector; its test action is the original inverse affine pullback. Thus the public covariance and spectral assertions follow by literal substitution once their reconstructed counterparts have been proved. Each graded field injects into the label space by the table’s pin map. Every label has a graded representative by the field-covering property; every graded field has an insertion-token representa- tive by the table’s quotient representative map. These are separate surjectivity assertions. Non- triviality follows from the degree-zero unit: its image is nonzero by injectivity and the unit’s nonvanishing; faithfulness then supplies a Schwartz test with nonzero field operator. The boundary label-quotient argument in the reconstruction chapter proves faithfulness. For separated configurations, a whole word can be placed in one of the decomposition’s resolved configurations. The character evaluation of that resolved word is continuous in the character by the decomposition’s weak continuity property. Its integral equals the unselected word expectation by the barycenter identity. Reading that word through the table gives both character continuity and the public mixture formula. The compression-valued moments and character readings in the reconstruction chapter provide the resolved-word placement and barycenter identities used here. The analytic continuation used publicly is the reconstruction’s holomorphic wave evaluated on the pinned graded labels. At strictly ordered Euclidean times, assembling the spatial coordinates and times reconstructs exactly the original four-dimensional configuration, so the heat-root iden- 277 ## Source page 280 A Wilson Yang–Mills construction J. Rodgers tity gives its selected Euclidean value. For the boundary limit, materialize the product of Schwartz tests as one joint test. The public complex shift ti + irhi is exactly the reconstruction’s shift by the ordered height vector rh. Its boundary theorem therefore gives the required vacuum mo- ment. The substantive holomorphy, heat-root and boundary-existence arguments remain part of the reconstruction proof, rather than consequences of these notational identifications. 11.3 Compatibility of the chosen family and its refinements The construction chooses its Wilson family before a volume subsequence or an observable is re- quested. The explicit order of its pre-length floors, post-length numerical bills and countable width demands is (CAL12)–(CAL18). Their consumption gives the flat delay condition at entrance index zero on the run’s actual horizon, as proved there. The same fixed family is used below. At the continuum extraction stage there are two successive refinements of a requested order f. The first chooses a subsequence g along which the running reference parameters have limits at each fixed level. The second extraction gives a subsequence w, one continuum table, its final structure F and scaling fields. Use the composite refinement g ◦w. Every convergence assertion from the first extraction survives the second, because a strictly increasing natural sequence tends to infinity. Composition of the subsequence maps gives (f ◦g)◦w = f ◦(g ◦w), so all conclusions have precisely the same final volume order. On that final table choose the reconstruction for every character, one equivariant OPE reader, and the trace test. The reader’s coefficients and local fields are retained as one prescription at every truncation degree; the contact devices are then chosen with that same reader basis. This is the quantifier order used by the public theorem and its detailed extension. The preceding construction estimates and the subsequent reconstruction, operator-product and trace arguments establish these services on the retained table. The compatibility argument here proves that using these supplied results together preserves their actual family, table and subsequence, rather than treating unrelated existence witnesses as interchangeable. 11.4 Extending the full fixed-volume mesh limit At fixed intermediate level the configuration space is fixed, and the source’s Confluence condition means eventual uniform boundedness and uniform Cauchy convergence in that configuration vari- able. Finite sums and products preserve it. A continuous map preserves it when the range lies in a common compact set: uniform continuity on that compact set controls the Cauchy modulus. In particular continuous polynomial coordinate maps preserve it for the finite insertion algebra. For eventually measurable functions on the fixed Haar probability space, uniform boundedness gives integrability and | Z fN − Z fM| ≤ Z |fN −fM| ≤sup U |fN(U) −fM(U)|. Thus the integrals are Cauchy. The finite polynomial insertion numerator is built from the scalar-normalized density, transported primary insertions and local error insertions by exactly these operations. Its Confluence follows from the corresponding fixed-level field and density statements. The scalar-normalized partition integrals are likewise Cauchy. Their common positive lower bound is kept before division. At fixed level j and volume p, let E be the number of sites, v the positive small-field threshold, H = 2 bβj, and let K be the uniform upper bound for the action and activity costs on the selected small-field neighborhood. Write Dmat for the fixed finite real matrix-coordinate dimension entering this estimate, denoted dimR Mat(G) 278 ## Source page 281 A Wilson Yang–Mills construction J. Rodgers in the supplied notation. Its precise value is not used in the quotient argument; the lower bound is positive for every such fixed finite dimension. The denominator estimate gives exp[−K −#Edges Dmat log(16/v)] for the scalar-stripped partition function, independently of the finer cutoffN ≥j. Its equality to the fixed-level normalized denominator follows from the selected trajectory’s partition identity. The numerator and denominator limits therefore form a valid convergent quotient. Write cvac for the represented scalar vacuum term. The factor e−cvac cancels exactly in this quotient, recovering the represented expectation. This proves the fixed-level ratio convergence once the uniform density and insertion convergence and the small-field denominator estimate have been established by the nonlinear construction. The Taylor bound used here follows from bounds on two further lattice differences. On a com- pact collision-free K, choose a compact thickening still disjoint from collisions. All first differences of a fixed difference word are bounded there. A coordinate-by-coordinate lattice path joining two nearby mesh points stays in this thickening and has at most a fixed dimension factor times their physical distance divided by aN steps. Telescoping the first differences along the path gives a local Lipschitz bound for the word. Apply this to each first difference, using its own first differences, to obtain a common Lipschitz constant A for the discrete gradient. Subtract the affine function with that gradient frozen at the starting point. Each unit difference of the residual is aN times the change in the gradient, hence bounded by aN A times the total endpoint distance. A second path telescope gives a dimension factor times A times the squared endpoint distance. This is the uniform quadratic Taylor estimate above. FixedBounds supplies every fixed finite difference order, so the two extra derivatives are available without changing the physical volume. One can obtain the next-difference limit without selecting another mesh subsequence. Fix x, a coordinate a and ε > 0. Choose a positive physical displacement q small enough to stay inside the compact neighborhood and to make Bq small compared with ε, where B is its Taylor constant. Put kN = aN⌈q/aN⌉, so kN →q and eventually q ≤kN ≤2q. The long difference quotient of the known convergent word between its address near x and that address shifted by ⌈q/aN⌉ea is Cauchy, because both endpoints have cluster limits and kN →q > 0. The quadratic Taylor estimate places its distance from the unit next-difference value at x below BkN ≤2Bq. Choose q first to make this error ≤ε/4; then take N,M large enough that the two long difference quotients differ by < ε/2. The next unit-difference sequence is Cauchy at x. Its eventual oscillation bound follows from the preceding local Lipschitz estimate and the O(aN) error of mesh addressing. The dense-set extension lemma, now applied to this derivative sequence at every x, gives its locally uniform limit along the original mesh sequence. Repeating proves the existence of all next limits used in the derivative-identification argument. The discrete-to-classical derivative identification uses a quadratic Taylor bound. Suppose fl is the locally uniform limit of the lattice difference word l, and ga,l is a locally uniform limit of its next coordinate difference. The fixed bounds supply, on a compact neighborhood, a uniform estimate |FN,l(Y ) −FN,l(X) − X a (Ya −Xa)FN,a::l(X)| ≤B|Y −X|2 for physical lattice points sufficiently close together. Approximate real points x,y by their mesh addresses. Pass to the limit in the inequality to obtain |fl(y) −fl(x) − X a (ya −xa)ga,l(x)| ≤B|y −x|2. 279 ## Source page 282 A Wilson Yang–Mills construction J. Rodgers After division by |y-x|, the right side tends to zero. Therefore fl is Fréchet differentiable, with derivative h 7→P a haga,l(x). The continuous limits ga,l identify its coordinate derivatives. Starting with the undifferenced limit and repeating for every finite word proves smoothness to all orders, since each derivative matrix is a finite linear combination of the continuous next limits. The constants and mesh thresholds may depend on the order and compact set. The coefficient-history convergence and uniform derivative estimates provide these next limits and Taylor bounds at every fixed finite order. The result is independent of the particular fixed forward, backward or centered stencil. For nonnegative integers b,f with e = b + f > 0, the normalized difference F(x + fea) −F(x −bea) (b + f)aN is the arithmetic average of the e unit forward differences based at x−bea +jea, j = 0, . . . , e−1. Unit differences commute with these finite translation averages. Iterating expresses any finite mixed stencil word as a finite average of the corresponding unit-difference word. Every averaged argument has physical displacement O(aN) and hence the same limiting physical point. The cluster limit is therefore the same derivative, with no extra stencil-length factor. This proves the claimed normalization for all fixed reach stencils. For the actual interpolant, take the surrounding right-closed mesh cube. If θa is the coordinate fraction measured from its right endpoint, assign a vertex choosing the left endpoint in coordinates I the weight Q a∈I θa Q a/∈I(1 −θa). These weights are nonnegative and sum to one. Every vertex is within O(naN) of the physical point. Cluster convergence at those vertices, continuity of the limit and compactness imply uniformly small vertex errors on each compact collision-free set. The error of their weighted sum is bounded by the maximum vertex error. Thus the multilinear interpolants converge locally uniformly, also at cell ties. This assertion does not require or claim that a finite- mesh interpolant is smooth at its cell boundaries; lattice differences are taken before interpolation. The dense-position Cauchy argument uses a second uniform approximation. Let pN denote the fine-cutoffmoment for a fixed separated configuration. At a fixed intermediate level j, replace it by its transported moment Vj,N. The fixed-level confluence estimates make Vj,N Cauchy in N. The comparison theorem bounds |pN −Vj,N| by a fixed finite cost divided by a scale function g(j) →∞, uniformly for all sufficiently large N. To use that theorem simultaneously for a finite word, choose positive weights xi = ζi/(|wi| + 1), where wi bounds the charged coefficient cost and ζi is its allowed margin. Then xi times the charged cost is at most ζi. The finite collection of transported observables and localized errors has one eventual weighted bound by summing its individual bounds. This supplies the common comparison cost. Choose j making the uniform error below ε/4, then N,M large enough that |Vj,N −Vj,M| < ε/4. The triangle inequality proves the Cauchy property of pN. The convergence-of-histories chapter establishes the confluence and comparison estimates used in this argument. The equicontinuity lemma used in the fixed-volume construction does not require that the approximating functions themselves be continuous. Let U be open in a locally compact metric space and let TN : U →C have eventual uniform oscillation bounds on each compact K ⊂U: for every ε > 0 some δ > 0 makes |TN(x) −TN(y)| < ε for all sufficiently large N and x, y ∈K with distance less than δ. Assume TN(x) is Cauchy on a dense subset D ∩U. For an arbitrary x choose a compact neighborhood K ⊂U and a dense point d within the oscillation radius. Then |TN(x) −TM(x)| ≤|TN(x) −TN(d)| +|TN(d) −TM(d)| + |TM(d) −TM(x)|. 280 ## Source page 283 A Wilson Yang–Mills construction J. Rodgers Taking all three terms below ε/4 proves the Cauchy property at x. Completeness of C gives a pointwise limit f. Passing to the limit in the oscillation estimate proves continuity of f. If convergence failed uniformly on a compact K, choose Nj →∞and xj ∈K whose errors stay above a fixed ε. A convergent subsequence xj →x, the oscillation bound for TNj, pointwise convergence at x and continuity of f contradict that error. Thus TN →f locally uniformly on U, with no mesh refinement. In the source U is the periodic collision-free position set at a fixed physical volume. The nonlinear bounds supply its oscillation estimate. The dense set is the schedule-compatible set of ancestor positions. At those positions the common-history comparison gives the Cauchy property for the full mesh sequence; the two-level approximation and fixed-level confluence arguments give this Cauchy property. The preceding lemma then extends this full-sequence convergence to all of U. The Taylor and next-difference arguments above identify all derivative limits and prove smoothness. To pass a known cluster limit of lattice values to tests, express the lattice sum as the integral of the piecewise constant function on the physical lattice cells. Each cell has volume a4n N , exactly the mesh prefactor in the sum. The compact test support stays inside a suitable physical period box and away from collisions. Cluster convergence gives an eventual uniform bound on all lattice values whose physical positions lie in this compact support: otherwise choose an unbounded sequence of such values, extract a convergent sequence of physical positions in the compact set, and contradict the cluster limit. The test is uniformly bounded, so these piecewise constant integrands have one integrable bound on the fixed volume box. The sampled physical positions converge to each point, and continuity of the test and the cluster limit gives pointwise convergence. Dominated convergence therefore yields the integral of the test against the limiting correlation. The same argument applies to every already identified derivative cluster limit. This proves the smearing passage once those cluster limits have been established. The scalar coupling refinement has a direct compactness proof. For each level m the represented action supplies the interval bound bβm/2 ≤βN,p,m ≤2 bβm whenever m ≤N. Extend βN,p,m to N < m by the value bβm, solely for this compactness argument. Regard the extended array at N as a point of the countable product Q p,m[ bβm/2, 2 bβm]. This product is compact metrizable. Extract one mesh subsequence along which every coordinate (p,m) converges to ap,m. Then regard (ap(k),m)m along the requested volume sequence as points of Q m[ bβm/2, 2 bβm], and extract one volume subsequence along which they converge to Bm for every m. For fixed m, the auxiliary extension agrees eventually with the original coupling because the mesh index tends to infinity. Thus the chosen refinement has the stated iterated limit for every level. Any later mesh and volume refinements preserve it. These limiting coefficients retain the running slope. Write tm = log(a−1 m ), bβm = ̄β + stm, s = 2b0, b0 = 11CA 48π2 , where CA is the adjoint Casimir fixed in the conventions. For every ε > 0 and all sufficiently large m, actual-history running gives |βN,p,m −bβm| ≤εtm (N ≥m), uniformly in p. Taking the ordered limit at fixed m preserves this closed inequality. Hence (Bm − bβm)/tm →0 and Bm/tm →2b0. Let m(μ) be the first level with tm ≥log μ. The schedule mesh condition gives tm+1/tm →1. Since tm(μ)−1 < log μ ≤tm(μ), 281 ## Source page 284 A Wilson Yang–Mills construction J. Rodgers squeezing gives tm(μ)/ log μ →1 as μ →∞. The positive auxiliary history coupling g(μ) = B−1/2 m(μ) therefore satisfies g(μ) p 2b0 log μ −→1, g(μ) −→0. This concerns the coupling read from the represented history. Its identification with a separately defined position-space diagnostic requires the corresponding comparison theorem and is not made by this scalar compactness argument. 11.5 Common compact extraction and the field quotient The physical field space is a quotient by the equation image, followed by identification through the complete spectator detector. Uniformly bounded projections permit its finite-dimensional co- ordinates to be retained on the common extraction. Kernel dimension stability uses closeness of idempotents: if u ∈ker P, u = Qu and ∥P −Q∥< 1, then ∥u∥= ∥(Q −P)u∥≤∥Q −P∥∥u∥, hence u = 0. Applying the same argument in the opposite direction gives equality of the two kernel dimensions. Inclusions and derivatives descend only after their compatibility with this equation image has been proved. We describe explicitly the compactness step that puts the preceding estimates on one continuum table. It has two different inputs. At every fixed physical period p, each fixed finite list of insertions has eventual centered root bounds, with constants allowed to depend on p. Separately, on sufficiently large periods there are centered bounds uniform in those periods, eventually in the mesh at each period. The nonlinear insertion estimates give both inputs. The second is not inferred from the first. Here is the bound for an ordinary correlation derived from these centered bounds. Fix n insertions and a compact separated set of physical positions, with separation margin r > 0. Choose an intermediate spacing am < r, and then a finer reporting level k ≥m so that 6nak < am. The stencil of each insertion fits its reporting chamber at every sufficiently fine mesh, uniformly in the period: each fixed integer stencil offset times the fine spacing tends to zero, whereas ak > 0 is fixed. Include the finitely many signed images and reflected insertions in the common centered bound A. Choose the origin of the m-tiling so that all n insertion points have distance at least three reporting widths from its chamber boundaries. The inequality 6nak < am leaves an allowed origin in each coordinate. Separation and am < r place the insertion points in distinct chambers. Pull each chamber back to the reference chamber, with its prescribed signed motion and optional conjugation. The insertion locality and stencil fit give genuine chamber-supported functions, and each has rooted bound at most A. The reflection-positive chessboard inequality therefore gives |⟨I1(x1) · · · In(xn)⟩| ≤An. (CV1) This argument applies to arbitrary separated tuples; it does not require every coordinate projection of the tuple to be separated. For equicontinuity first work on the open set of tuples whose coordinate projections are pairwise distinct, using periodic coordinate differences at a fixed volume. Choose a compact thickening within that open set and a margin 4r for all its projected gaps. The centered translation-difference estimate, with a reporting spacing below r, bounds a displacement h of one slot in one coordinate by |⟨I(x + he) · · · ⟩−⟨I(x) · · · ⟩| ≤(An/r)|h|, |h| ≤r. For two nearby tuples change their 4n coordinates one at a time. Each intermediate tuple is obtained by choosing coordinates from one endpoint or the other, so its distance from either endpoint is at most their distance. It stays in the chosen thickening. Summing the 4n one-coordinate bounds 282 ## Source page 285 A Wilson Yang–Mills construction J. Rodgers gives a local Lipschitz constant 4nAn/r. The same construction is available at each fixed period and, separately, uniformly in all sufficiently large periods. Only countably many functions need be selected. At each degree choose a finite basis of the label space and take finite words in these basis vectors, all arities, and a countable compact exhaustion of their generic position sets. Multilinearity recovers arbitrary labels from these basis words after selection. Boundedness (CV1) and the equicontinuity estimate give, by the countable local Arzelà– Ascoli argument, a common mesh subsequence with a locally uniform limit Mp for every fixed p. Apply the large-period bounds to these limits: passing a boundedness or oscillation inequality through the mesh limit preserves it. A second countable Arzelà–Ascoli selection along the requested period sequence gives a table S, locally uniformly on generic configurations. All degrees and words use these same two selections. For clarity, the countable compact selection itself introduces no hidden uniformity assumption. If the i-th coordinate lies eventually in a compact set Ki, choose one of its values in Ki and replace its exceptional earlier values by that value. The resulting sequence belongs to the countable compact product Q i Ki, which is compact metrizable in the cases used here. A convergent subsequence in that product converges in every coordinate. The replacements are eventually absent in each fixed coordinate, so the original sequence has the same coordinate limits. This also retains additional countable numerical data Yi(N, p). For every active pair (i, p), first use its own compact fixed-period bound and select their mesh limits zi(p) simultaneously. For each i, a separate compact set Ki bounds Yi(N, p) at all sufficiently late periods and sufficiently fine meshes at each such period. Closedness of Ki puts zi(p) in Ki eventually in p. A common period subsequence then gives zi(p) →yi for every i. Fixed-period limits are kept at every active p, including periods absent from the selected period sequence. Apply this construction to the finite-dimensional coordinate projections and the coupling data used above. Here is the construction and uniform bound for those finite-cutoffprojections. At degree d, let KN,p be the image of the free equation ideal under the exact calibrated label map already constructed. This is a subspace of the single fixed coefficient space Ed, although it varies with N, p. Choose once a linear coordinate isomorphism W : Cdim Ed →Ed, with Euclidean norm on its source, and let πK be orthogonal projection onto W −1K. Put FK = I −WπKW −1. (CVP1) It kills K, and x −FKx ∈K for every x. Moreover ∥FK∥≤∥I∥+ ∥W∥∥W −1∥, (CVP2) independently of K, since ∥πK∥≤1. Thus no conditioning estimate on a moving basis of KN,p is required. When K is invariant under the signed label action Vs and coefficient conjugation, average FK to QK = 1 |S| X s∈S VsFKVs−1, PKx = 1 2{QKx + QK ̄x}. (CVP3) Both maps still kill K and differ from the identity by a K-valued map. For the second statement, average the differences x −VsFKVs−1x, each of which belongs to K, and then use conjugation invariance. These two properties imply P 2 K = PK and ker PK = K: apply PK to x−PKx ∈K, and conversely use that same inclusion when PKx = 0. Reindexing the finite group sum proves signed equivariance; the real average proves conjugation equivariance. Its norm is bounded by (CVP2) times the fixed finite average of ∥Vs∥∥Vs−1∥. The real average does not enlarge that bound, because conjugation is isometric in the frame coordinates. At the finitely many cutoffs where the calibrated 283 ## Source page 286 A Wilson Yang–Mills construction J. Rodgers symmetry has not yet been established, use (CVP1) instead. This supplies a uniform bound at every cutoffand the required exact symmetries on the admitted tail. The projection maps are therefore idempotent and uniformly bounded before extraction. Their limits are idempotent as well, at each fixed period and in the outer limit. Their kernel dimensions stabilize in both limits. Indeed, if P, Q are idempotents and ∥P −Q∥< 1, the map u 7→u −Qu sends ker P into ker Q injectively: if u = Qu and Pu = 0, then ∥u∥= ∥(Q −P)u∥≤∥Q −P∥∥u∥, forcing u = 0. Interchanging P, Q gives equal kernel dimensions. The physical detector estimates identify the limiting kernel with the exact equation-null space, rather than merely bounding its dimension. The transported insertion comparison shows that a label in this kernel has vanishing separated smeared correlations: its discrepancy has a uniform separated bound tending to zero, and the step-function integral argument gives the same vanishing for every compact separated test. Conversely, the finite full-rank detector pairs separate the quotient. A label with zero continuum correlations against every spectator has zero detector readings, hence lies in the limiting kernel. Thus {continuum-null labels of degree ≤d} = ker Pd. (CV2) The quotient is finite dimensional, carries the real and signed-motion structures preserved by Pd, and is the same quotient used for the detector normalization and subsequent differentiation. The derivative construction removes the coordinate-projection restriction on the raw table. Transport the finite differences to this quotient using their retained coefficient estimates. Their limits agree with the raw table on generic tuples. The local Taylor and next-difference argument proved earlier identifies their derivatives and extends them smoothly across coordinate ties, as long as no two full four-dimensional positions coincide. The generic set is dense there, so two continuous such extensions must agree. This gives the smooth table T with the original Mp as its fixed-period tables and the same separated smeared limits. No pointwise assertion at a coordinate tie is obtained from raw compactness alone. In the present construction the full fixed-volume mesh limits were already established by the two-level Cauchy argument. The auxiliary mesh subsequence just used therefore converges to those same Mp. To see explicitly that this restores the required smeared order, fix a token word and one compactly supported separated test t. For each suitable period p, let RN,p(t) be its lattice smeared moment. The full-mesh result gives RN,p(t) →rp(t) as N →∞. The extracted table gives, along its mesh subsequence φ(N), a limit Dp(t), followed by the stated volume limit of Dp(t). Cofinality of φ also gives Rφ(N),p(t) →rp(t). Uniqueness of a complex limit therefore identifies Dp(t) = rp(t) at each suitable p. Substitute this equality into the retained volume-limit identity. The result has the full mesh order followed by precisely the same volume subsequence as the final table. This argument restores the smeared convergence assertion; it does not claim that every auxiliary structure originally extracted on a mesh subsequence has been extended to all meshes. Finally, countably many further stages can be imposed without combining incompatible choices. If each successive order refines its predecessor, take its diagonal mesh and period sequences sepa- rately. For a nested sequence Fj of strictly increasing sequences, the diagonal tj = Fj(j) is strictly increasing: writing Fj+1 = Fj ◦gj, one has gj(j + 1) ≥j + 1. For every fixed stage i, the tail of t lies in the range of Fi. Apply this to mesh and period orders. Every eventual bound and every retained limit from any fixed stage survives on the resulting diagonal order. This is why the hierarchy, quotient, numerical limits and later comparison services can all be attached to the same extracted table. 284 ## Source page 287 A Wilson Yang–Mills construction J. Rodgers 11.6 The calibrated equation image and its silent representatives We specify the finite-cutoffchange of coefficients used in the preceding projection construction. This also explains why its kernel directions are silent in continuum tests. Fix a degree, its coefficient frame, and a second sound local section V , such as the marked equation section. Write ej for the coordinate vectors and c ≥1 for the fixed cost of signed motions and coordinate decomposition in this frame. The local insertion transport constructs columns uj satisfying |ej −uj|coeff≤(2c2)−1. (CVR1) The columns are combined by the signed and conjugate averaging already used for local sections, rather than by an unsymmetrized matrix. Explicitly, if χs,j(x) are the averaged coordinates of the inverse motion of x, then Px = X s,j χs,j(x) suj, x = X s,j χs,j(x) sej, X s,j |χs,j(x)| ≤c|x|coeff. (CVR2) Here a conjugating motion conjugates the coordinate as well, so the combined map is complex linear. The fixed motion bound is |sv|coeff≤c|v|coeff. Subtract the two sums and apply (CVR1): |x −Px|coeff≤1 2|x|coeff, |P −1y|coeff≤2|y|coeff. (CVR3) Indeed the first inequality makes P injective; finite dimensionality makes it surjective, and applying the first inequality to P −1y gives the second. These bounds hold uniformly in the period on the eventual mesh tail. For precision about the purchase of (CVR1), put X = V.stencil ∪T.stencil, where T is the original section. At stage N, write hN for the support exponent, qN for the number of columns, and bN ≥0 for the purchased derivative-credit bound of V . The transport estimate gives L1/8 N−1|ej −uj|coeff≤2(1 + qN)YN, YN = 2 + e81hN (bN + qN + 2|X|). (CVR4) The delayed length requirement is precisely 2(1+qN)YN ≤L1/8 N−1/(2c2), which proves (CVR1). It is bought together with bN ≤bβN, not inferred from a limit after selecting the columns. The common error price can be taken to be EN = 4(1 + qN)YN + e81(hN+2)ZNζd,N/uN, ZN = (aV + 2aT ) X j θ−fj N , (CVR5) where aV , aT are the fixed coefficient costs of the two sections, θN > 0 is the field threshold and fj the field-number degree of coordinate j, uN > 0 is the purchased insertion allowance, and ζd,N > 0 its degree-dependent margin. The sum is the finite price of undoing the threshold dilation. Each factor in (CVR5), including u−1 N , has the subexponential bound supplied by the common calendar. Finite sums and products preserve that bound by dividing a requested exponent among their finitely many factors. At any fixed admitted reporting indices m, k, the rooted discrepancy between the insertion of ej through V and that of uj through T, including every signed or conjugate motion, is at most Cd,m,kENe−sN/8, Cd,m,k = exp{(am/ak)4τk + 5} a−d k ζd,k/4esk/8. (CVR6) Here τk is the explicit vacuum treasury derived in the denominator estimates above. These are fixed finite constants after the reporting indices are chosen. Expand any subexponentially priced coeffi- cient row by (CVR2). The finite number of terms and their subexponential coordinates, together 285 ## Source page 288 A Wilson Yang–Mills construction J. Rodgers with (CVR5), absorb at most esN/16 eventually. Thus the entire discrepancy has rooted bound tending to zero, with a remaining exponential factor e−sN/16. Its fixed stencil, gauge invariance and continuity make it eligible for the separated spectator product and smearing argument below. Let T be the terminal coefficient isomorphism and B the marked embedding of free labels into terminal coefficients. The calibrated equation image is exactly KN,p = T PT −1B(Id). (CVR7) If x ∈KN,p, set y = P −1T −1x. Then T y ∈B(Id). The actual insertion through V with coefficient y is therefore exactly zero against physically separated spectators on a sufficiently fine mesh, by the local marking identity. The uniform radius for that assertion was proved by the finite-basis maximum in the equation-lift construction. Formula (CVR3) preserves the subexponential price of y, and (CVR6) compares this exactly silent insertion with the original insertion of x. Consequently every subexponentially priced row in (CVR7) tends to zero against every separated test. This is the kernel cancellation used in (CV2) and (QD3); it follows from the actual calibrated map and a local identity, rather than from dimension counting alone. 11.7 Inclusions and derivatives on the same physical quotient The differential system is constructed using all spectator tests on this quotient. Let Ed be the finite coefficient space and Nd the null subspace in (CV2). A test reading means: put the distinguished label at any chosen slot, put arbitrary other labels in all remaining slots, multiply the correlation by a smooth compactly supported test whose support lies in the separated configuration space, and integrate. Denote it by Lψ(b), including the spectator data in ψ. The family of all these linear functionals has common kernel Nd. For a smooth table this follows from uniqueness of continuous kernels as distributions on the open separated set; a nonzero value has a neighborhood and a compact bump detecting it. At finite mesh aN, a derivative insertion is the actual forward difference δaIb(o) = a−1 N {Ib(o + ea) −Ib(o)}. (QD1) In the finite torus sum, translating the distinguished summation variable gives the exact summation- by-parts identity a4n N X r ψ(aNr) ⟨δaIb(ri) spectators⟩= a4n N X r ψ(aNr −aNei,a) −ψ(aNr) aN ⟨Ib(ri) spectators⟩. (QD2) Translation is a permutation of the finite torus, so this is true even at its seam. Once the physical period contains the compact support and its one-step thickening, both potentially nonzero terms have their actual cover coordinates. Taylor’s theorem makes the test quotient converge uniformly to −∂i,aψ, on one fixed separated compact. The separated correlation bound controls the remainder in the cell integral. Take the mesh limit at that fixed suitable period first, and then the retained volume limit. This proves the derivative test limit in the same mesh-then-volume order as the hierarchy; no uniform count of lattice points over all periods is needed. The coefficient approximations used for both inclusion and (QD1) have subexponential top- coordinate cost: for every ε > 0, their pullback to the terminal frame is eventually bounded by exp(εsN), where sN is the calendar scale. For bounded reference coefficients this follows by multiplying the backward column bounds: their logarithm is P i 0, and word w = ((xi, vi))n i=1 of distinct points inside 0 < xa i < d. A buffer 0 < r ≤1 requires pairwise maximum-coordinate distance at least r and r ≤xa i ≤d −r. For degrees at most D and coefficient norms at most R, we construct ∥Ud(w)∥≤Cn(n/r)An, (BS1) with C > 0, A ∈N depending only on D, R. The empty word has bound one. On a future plant the prescribed action inserts w before that plant translated by dea. Boundedness precedes extension to the Hilbert completion. The centered one-cell estimate gives a threshold F, exponent P, and J > 0, such that one insertion at level k ≥F has price Ja−P k . Include all signed rotations and reflected conjugates by enlarging the coefficient ball by their finitely many operator norms on the finitely many raw degree spaces. These constants consequently work for every reflected copy. The decreasing schedule satisfies ak+1 ≥c0a2 k. At the first m ≥F below r/32, we have am ≥c1r2; decrease c1 > 0 to include m = F. Choose fixed sufficiently small s > 0 and the first k ≥F below s(r/n)2. The same first-crossing argument gives ak ≥c(r/n)4, 20nak < am, k ≥m. (BS2) Indeed 20nak < 20sr2/n ≤20sr2 < c1r2, and strict decrease gives k ≥m. Exterior insertions are paid at still finer scales, chosen after this main scale; their prices may depend on the exterior word. First let d′ be an integer multiple of am. Alternating insertion of w and its reflection produces, in the squared norm after q repetitions, 4q main copies and two exterior copies. The main positions repeat the original 2n position patterns modulo am. Thus cell packing needs only 20nak < am, independently of q. The buffer separates the main train from the translated exterior. The rooted product estimate and the fixed finite-word lattice limit give ∥plant of the repeated word∥2 ≤Bp[Ja−P k ]4nq, (BS3) 288 ## Source page 291 A Wilson Yang–Mills construction J. Rodgers with Bp independent of q. This is the train estimate in Next/PHASE/Trains.travel_price and Seats.replicas. By (BS2), its main price is Q = [Ja−P k ]n ≤[Jc−P ]n(n/r)4Pn. We may use the latter expression for Q. To remove the exterior prefactor, let X send algebraic future words to the reflection Hilbert space, V insert the main word, and V ♯insert its opposite reflected word. Reflection and translation invariance give ⟨XV u, Xv⟩= ⟨Xu, XV ♯v⟩. Put T = V ♯V . Taking square roots in (BS3) and finite linear combinations gives ∥XT qu∥≤BuQ2q. Therefore su = supq Q−2q∥XT qu∥is finite. Moving powers of T across the pairing yields (Q−2q∥XT qu∥)2 ≤∥Xu∥su. Taking suprema gives s2 u ≤∥Xu∥su, hence su ≤∥Xu∥. Finally ∥XV u∥2 = ⟨Xu, XTu⟩≤Q2∥Xu∥2. Insertion thus annihilates null vectors, descends to the quotient, and extends uniquely with norm at most Q. This is OS/Train.producer. For general d, use d′ = am⌊d/am⌋. Since am < r/32, the same word lies in the narrower slab with buffer r/2. Compose its operator on the right with the heat contraction of width d −d′. On plants this is exactly the required insertion at width d, and its norm does not increase. The empty word is the heat contraction. All intermediate choices give the same operator because they agree on the dense plants. Replacing r by r/2 only enlarges C, proving (BS1) for one common family of operators. This is the padding and uniqueness argument of OS/Halves and Work/B01. We next prove operator-norm regularity. Fix labels and width, and write F(x) = Ud(wx) on the open set of distinct slab configurations. For a slot-axis index b = (i, j), replace its label by a representative of its physical j-derivative and call the resulting placement Fb(x). Set B(x)h = X b hbFb(x). (BS4) Near a fixed configuration there is one positive buffer for the original word and these finitely many differentiated words. Equation (BS1) makes B locally bounded in operator norm. Between two plants, the placement is a separated correlation with a fixed exterior word. The scalar slot-derivative law gives its derivative as the matrix element of B(x), also for finite linear combinations of plants. On a convex position ball suppose ∥B(z) −B0∥≤η. The scalar mean-value estimate gives |⟨u, [F(y) −F(x) −B0(y −x)]v⟩| ≤η∥u∥∥v∥∥y −x∥. Density gives the same estimate in operator norm. First set B0 = 0: local boundedness makes every placement locally Lipschitz. Apply this also to each differentiated label array; all Fb, hence B, are norm continuous. Then set B0 = B(x) and shrink the ball. Its oscillation tends to zero, proving DF(x) = B(x) in norm. Induction with differentiated labels gives smoothness of every order on the same position domain. This is the argument of Work/B02/Analysis.norm_remainder and Work/B02/Slab.smooth_place; it establishes the norm regularity used in the waves and compression-valued moments below. 12.2 Surjectivity of the reflection attachment Surjectivity of attachment The reflection Hilbert space is formed from a positive pairing on future words. An attachment map can preserve that pairing and still have a proper closed range, 289 ## Source page 292 A Wilson Yang–Mills construction J. Rodgers so its being an isometry is not the final step. Surjectivity below identifies a dense family of heated word vectors in that range and uses closedness to obtain the whole space. This identifies the reconstructed representation with the intended cyclic word space and is needed when the boundary fields are later placed on their common domain. The attachment J is an isometry from the complete boundary Hilbert space, so its range is closed. First consider a regularized word vector integrated against the Fourier transform of a compact smooth test. The boundary tilt identity identifies it with J applied to the boundary word distribution tested against the correspondingly chilled test. Here is the Hilbert-space step in that identification. If A is the reflection-space regularized vector and B the proposed attached vector, the joint pairing formula gives ⟨A, A⟩= ⟨B, B⟩. Pair A with the regularized approximants tending to B. Their partial-boundary correlations converge by the boundary distribution construction, giving ⟨A, B⟩= ⟨B, B⟩. Therefore ∥A −B∥2 = 0 and A = B. The strict-time wave and independently heated word constructions below establish the boundary tilt and joint pairing identities used here. Compact cutoffs approximate any Schwartz test after Fourier transform. The regularized word map is continuous, by its bound C∥f∥L1, so closedness of the range extends membership to every integrated regularized word vector. For a pointwise regularized word vector, pair with a vector orthogonal to the range. All its smeared pairings vanish. The pointwise pairing is continuous, so equality as a distribution implies that it vanishes everywhere. The pointwise vector belongs to the double orthogonal complement of the range, hence to the range itself. A future Euclidean word with strictly ordered positive times is exactly such a regularized word: choose the regularization below every initial time and consecutive time gap, then recursively insert the local regularized fields and intervening translation and heat operators. This recursion repro- duces the reflection-space plant of the word with its specified insertion representatives. Quotient- field representatives give the same plant because their table expectations agree against all future words. Hence every strictly time-ordered future plant lies in J’s range. For an arbitrary future word, permutation symmetry first sorts its times. Add εi to its i- th sorted time. For ε > 0 the times are strictly increasing and remain positive. The original configuration is separated; the perturbed configurations remain separated and converge to it as ε ↓0. Smoothness of the separated table makes the associated plant vectors continuous in norm, as seen by expanding the squared distance into three table pairings. Since the range is closed, the unperturbed plant also lies in it. Future plants have dense linear span by the GNS construction. Therefore the closed range is the whole reflection Hilbert space, proving surjectivity. 12.3 Explicit tested boundary formula Fix a field word v and let Ψ(y, f) be its holomorphic wave smeared against a joint Schwartz test f at strictly increasing imaginary heights y. Let ai = i and define the continuous test operator D = −i∂dawn(a). The tested Cauchy–Riemann identity is d dtΨ(y + ta, Dkf) = Ψ(y + ta, Dk+1f). Assume the wave bound |W(x+iy)| ≤Ck−m whenever the height gaps are at least k, 0 < k ≤1. For any nondecreasing y and t > 0, y+ta has gaps at least t. Define Pm(y, f; t) = m X k=0 (−t)k k! Ψ(y + ta, Dkf). Differentiating and cancelling adjacent terms yields ∂tPm(y, f; t) = (−t)mΨ(y +ta, Dm+1f)/m!. The wave bound and integration in x give 290 ## Source page 293 A Wilson Yang–Mills construction J. Rodgers |∂tPm(y, f; t)| ≤C m!∥Dm+1f∥L1, 0 < t ≤1. Thus the expression B(y, f) = m X k=0 (−1)k k! Ψ(y + a, Dkf) − Z 1 0 (−t)m m! Ψ(y + ta, Dm+1f) dt. is defined for every nondecreasing y. For strictly increasing y, the fundamental theorem of calculus gives B(y, f) = Ψ(y, f). For y approaching any nondecreasing y0, the finite terms are evaluated at interior heights and are continuous. The remainder integrands are continuous for every t > 0 and share the constant integrable bound just proved. Dominated convergence makes B continuous at y0. Therefore Ψ(y, f) tends to B(y0, f) along every strict-height approach, with no restriction on the ratios of the gaps. Furthermore |B(y, f)| ≤C m X k=0 ∥Dkf∥L1 k! + C m!∥Dm+1f∥L1. This is a continuous Schwartz seminorm. Linearity in f and multilinearity in the labels pass through the interior limits and yield a tempered multilinear boundary distribution. Different choices of the admissible exponent m give the same distribution by uniqueness of these limits. The common character-independent C, m prove both boundary continuity in the character and passage of the character barycenter to the boundary. The construction of the waves and the tested Cauchy– Riemann and polynomial gap estimates used here is given in (SW1)–(SW7) and (CR1)–(CR2) below. 12.4 Character integration at strict imaginary times For a fixed field word v of arity n, the heat-regularized field factors have bounds Qa p−da. Choosing the regularization p = k/8 for imaginary-time gaps at least k, with 0 < k ≤1, gives a product bound |Wχ(v; x + iy)| ≤[ Y a Qa(8)da]k−P a da, uniformly in χ and the real positions x. The comparison of the phase regularized factor norm with the unselected factor norm supplies the same constants in every phase. The factors are constructed in (SW1)–(SW3); the phase norm comparison is (ST1)–(ST4), using the bounded slab construction (BS1)–(BS4). At a strict imaginary-time configuration the phase values are continuous in χ and have the unselected value as their integral. For a joint Schwartz test f and fixed strict heights y, the preced- ing bound gives the integrable dominator Ck−m|f(x)|. Dominated convergence therefore proves continuity of χ 7→ R Wχ(v; x + iy)f(x)dx. The same bound, combined with the probability normal- ization of the phase measure, makes the double integral absolutely integrable. Fubini transfers the character integral inside the x-integral and yields the unselected strict smeared wave. 291 ## Source page 294 A Wilson Yang–Mills construction J. Rodgers The source passes this identity to the Wightman boundary using a common finite-order Taylor representation along an interior height ray. Its polynomial part uses finitely many strict evaluations; its remainder has a common integrable bound on the ray parameter after enough test derivatives. Thus continuity in χ passes separately through that finite sum and the dominated remainder in- tegral. The same representation bounds the smeared strict waves uniformly in χ as the heights approach the boundary. Dominated convergence in χ then gives the boundary barycenter identity. The tested boundary formula and strict-time wave estimates below provide this Taylor representa- tion and its common polynomial-order bound. 12.5 Identification of the heat ground projection Let Es be the positive reflection-space contraction semigroup, and let Π be its strong s →∞limit, the projection onto its fixed vectors. Its lifted joint momentum measure gives for every vector x ⟨x, Esx⟩= Z exp(−sE(p))dνx(p). Since E ≥0 almost everywhere, the integrands are bounded by one and converge to the indicator of E = 0. Dominated convergence gives the limit νx({E = 0}) = ⟨x, P({E = 0})x⟩. The strong semigroup limit gives the same limit ⟨x, Πx⟩. Polarization of these quadratic pairings proves Π = P({E = 0}), without an assumption that this projection has rank one. For the isometric attachment J from the boundary Hilbert space to this reflection space, set Bs = J∗EsJ. The attachment identifies the diagonal spectral measures, so ⟨x, Bsx⟩is the Boltzmann integral for the boundary momentum measure. It also intertwines the zero-energy projections. Applying J∗to the reflection-space strong limit and using J∗J = I gives Bsx →Pboundary({E = 0})x for every x. The vacuum rank is established later by the locality and character argument, not used in this identification. Surjectivity of J, supplied separately by the attachment construction, then realizes the whole reflection heat evolution on the boundary Hilbert space. 12.6 Weak irreducibility from heat evolution With the reconstructed Hilbert space and heat evolution, let Φ be the reconstructed fields on their common word domain, Ωthe cyclic unit vacuum, U(t) the time translations, and B a bounded operator satisfying the weak commutation identity ⟨Φ(a, f)∗u, Bv⟩= ⟨u, BΦ(a, f)v⟩ for word-domain u,v, with the adjoint interpreted by its actual Hilbert-space adjoint restricted to that domain. No preservation of the domain by B is assumed. Repeated application moves a whole word across the pairing: ⟨w∗u, Bv⟩= ⟨u, Bwv⟩. Fix a word vector u = wΩand let v = w∗Ω. The heat-space attachment supplies bounded continuous scalar functions on the closed upper half-plane, holomorphic in its interior, with real boundary values F(t) = ⟨u, U(t)BΩ⟩, G(t) = ⟨B∗Ω, U(t)v⟩. Translation covariance, invariance of Ωand the weak word identity give F(t) = G(−t) for real t. Glue F above the axis to G(−z) below it. The boundary equality makes the glued function continuous. Cauchy’s theorem on rectangles cut along the real axis, with cancellation of the common boundary integrals, shows that the glued function is entire. It is bounded, so Liouville’s theorem makes it constant. In particular 292 ## Source page 295 A Wilson Yang–Mills construction J. Rodgers ⟨u, BΩ⟩= ⟨u, exp(−sH)BΩ⟩, s ≥0. The constructed heat evolution converges strongly to the vacuum projection as s tends to infinity. Hence ⟨u, BΩ⟩= ⟨u, Ω⟩⟨Ω, BΩ⟩for every word vector u. Density gives BΩ= aΩ, a = ⟨Ω, BΩ⟩. Move each word across B once more to obtain ⟨u, BwΩ⟩= a < u, wΩ> for all word vectors u,wΩ. Density and boundedness imply B = aI. Conversely every scalar multiple of I satisfies the weak identity by the field adjoint relation. Thus the weak commutant is precisely the scalars. The heat-contraction, holomorphic-edge and vacuum-projection arguments of this chapter establish the properties used in this weak-commutant argument. 12.7 From low-energy measure bounds to uniform heat decay Low-energy spectral bounds control heat decay without supplying a positive mass gap. The heat- ground projection and its compression algebra are the actual objects used in phase decomposition. Full support of the character measure subsequently upgrades certain continuous almost-everywhere identities to every character, after the required continuity and common bounds have been estab- lished. The shell estimate uses positivity of a time average. Let μ, ν be the scalar spectral measures of a compact word combination and its adjoint word combination, both supported in the forward cone, with total mass sum at most M. Their characteristic-function difference A(x, t) = bμ(x, t)−bν(−x, −t) is the vacuum commutator pairing. Locality gives A(x, t) = 0 whenever |x| > R and |t| ≤1, with R uniform in the character: the words have fixed compact test supports and the label action is common across phases. Define D(x) = Z 1 0 [A(x, t) −A(x, −t)]/(2i)dt, ρ(E) = Z 1 0 sin(Et)dt. For E > 0, ρ(E) = (1 −cos E)/E ≥0, with ρ(0) = 0 and |ρ| ≤1. For 0 ≤E ≤1, the inequality sin(Et) ≥2Et/π gives ρ(E) ≥E/π. Fubini applies since both measures are finite and the integrands are bounded. It expresses D as the spatial Fourier transform of the sum of two positive measures: ρ(E) dμ pushed to spatial momentum, and ρ(E) dν pushed to negative spatial momentum. Moreover, |D(x)| ≤M, and D vanishes outside the radius-R ball. Fix 0 < u ≤1 and a smooth nonnegative spatial bump Bu equal to one on |p| ≤u and supported in |p| ≤2u, with Bu ≤1. Its Fourier transform is bounded in absolute value by the volume of the radius-2u ball, hence by C3u3. Since D is supported in a fixed ball, Fourier inversion and positivity imply Z ρ(E)Bu(pspatial)dμ ≤K(R, M)u3, where the fixed positive Fourier normalization is absorbed into K. On the shell u/2 < E ≤u, the cone support gives |pspatial| ≤E ≤u, and ρ(E) ≥u/(2π). Thus μ({u/2 < E ≤u}) ≤2πK(R, M)u2. Decompose 0 < E ≤u into the countable dyadic shells with upper endpoints u2−n. The corresponding masses sum to at most K’ u2 P n4−n ≤2K′u2. This proves the low-energy bound used below. The uniform total masses follow from continuity of the word norms on the compact 293 ## Source page 296 A Wilson Yang–Mills construction J. Rodgers character space; those norms are table expectations of adjoint words concatenated with words. The word-moment continuity, locality and spectral reconstruction in this chapter give these properties. Let ν be a finite positive spectral measure supported in the forward cone, and let E(p) be its time coordinate. Suppose ν({0 < E ≤u}) ≤Cu2 for u > 0. For s > 0 the identity e−sE = R ∞ E se−st dt on E > 0 and Tonelli’s theorem give Z E>0 exp(−sE)dν = s Z ∞ 0 exp(−st)ν({0 < E ≤t})dt ≤Cs Z ∞ 0 t2 exp(−st)dt = 2Cs−2. The final equality follows by substituting u = st and integrating u2 exp(−u) twice by parts. A bound initially supplied only for 0 < u ≤1 extends to all u by increasing C to at least ν(total), since u2 ≥1 for u ≥1. Applied to the scalar spectral measure of a word vector, this is the estimate for ⟨x, (exp(−sH) −P0)x⟩. It leaves the zero-energy atom separate throughout. Continuity in the character is obtained from translation pairings. The characteristic function of the scalar spectral measure νχ of a fixed sum of words is the inner product of that vector with its translated vector. Both vectors are represented by fixed words with the same transported labels, so the continuum table’s word-pair continuity gives continuity in χ at each translation. The total masses νχ(total) have a common bound M. For a compact smooth momentum test g, Fourier inversion writes R g dνχ as a fixed Fourier normalization times the integral of the characteristic function against the Schwartz function hat(g). The bound M|hat(g)| is integrable. Dominated convergence therefore proves continuity of R g dνχ. For the heat multiplier exp(−sE), use a smooth bump equal to one on the Euclidean ball of radius 2N and supported in the ball of radius 3N. On the forward cone, outside the smaller ball E ≥N. Thus the error in every phase is at most M exp(−sN), tending to zero uniformly in χ. The heat pairings are consequently uniform limits of the continuous compact-test pairings, and are continuous. These arguments supply the heat continuity and shell-to-heat steps used in the removal of exceptional characters. The preceding time-average argument supplies the shell bound; the compression-valued word construction below supplies continuity of the word pairings. 12.8 Removal of exceptional characters in vacuum uniqueness From almost every phase to every phase The measure-theoretic argument first gives the relevant property outside a null set of characters. To obtain it at every character, one also needs continuity of the fixed probe readings and full support of the phase measure. A nonzero continuous violation would persist on a nonempty open set, which full support gives positive measure. The common countable probe family makes the exceptional set compatible before this continuity argu- ment is applied. Almost-everywhere statements alone would not establish the asserted all-phase conclusion. The phase measure has full support: every nonempty open subset of the compact character space has positive measure. Consequently a closed set containing almost every character is the whole space; otherwise its nonempty open complement would have both positive measure and measure zero. For a finite sum of compactly supported words w let xχ be its vector in phase χ and Ωχ its vacuum. The construction supplies a common M ≥0 and, for almost every χ, a unique heat ground state together with the inverse-square estimate for the zero-energy-subtracted heat pairing. In those 294 ## Source page 297 A Wilson Yang–Mills construction J. Rodgers phases the zero-energy projection Pχ is the vacuum projection. Indeed the heat attachment sends its range into the one-dimensional heat ground space; normalization and self-adjointness determine its scalar coefficient as ⟨Ωχ, x⟩. Thus for each s ≥1 almost every χ obeys 0 ≤⟨xχ, exp(−sHχ)xχ⟩−|⟨Ωχ, xχ⟩|2 ≤Ms−2.(4) For fixed s this pairing is continuous in χ, as is the vacuum overlap. The set where (4) holds is therefore closed. Full support extends (4) to every χ, with the same M. This step uses continuity of heat pairings, without first assuming continuity of the ground projections themselves. Fix χ and let s tend to infinity. The spectral theorem gives strong convergence exp(−sHχ) →Pχ for nonnegative Hχ. Equation (4) yields ⟨xχ, Pχxχ⟩= |⟨Ωχ, xχ⟩|2. Since Pχ is an orthogonal projection fixing Ωχ, expansion of the norm gives ∥Pχxχ −⟨Ωχ, xχ⟩Ωχ∥2 = ⟨xχ, Pχxχ⟩−|⟨Ωχ, xχ⟩|2 = 0. Thus Pχ acts as the vacuum projection on these compact word vectors. Approximate each Schwartz test in a word by compact cutoffs. Continuity of the joint smeared word map gives con- vergence of the word vectors, and boundedness of the two projections transfers their equality to every Schwartz word vector. Cyclicity and boundedness then extend it to the whole Hilbert space. This proves rank-one zero energy in every character, supplying the vacuum-projection heat limit used in the weak-commutant argument. The low-energy estimate, character probability construc- tion, heat-pairing continuity and attachment arguments in this chapter provide the ingredients used here. 12.9 Propagation of analytic limits on a complex slice Here is the analytic limit argument needed when continuing strict Euclidean values to real-time slices. Let U ⊂C be open and connected, let Fa be eventually holomorphic on U, and suppose that at each x ∈U some ball B(x, r) has a common bound |Fa| ≤A on a tail of the indexing filter. The tail may depend on the ball. If a holomorphic f is the pointwise limit on a subset accumulating at an interior point of U, then Fa(y) →f(y) at every y ∈U. First derive the requisite local equicontinuity. Choose s > 0 with B(x, s) ⊂U, put t = min(r, s)/2, and D = max(1, A/t). For z ∈B(x, t), the closed disc of radius t about z lies in B(x, r)∩ U. The Cauchy derivative estimate gives |F ′ a(z)| ≤A/t ≤D on the common tail. Integration along the segment from x to y ∈B(x, t) yields |Fa(y) −Fa(x)| ≤D|y −x|. If Fa has pointwise limit g throughout U, passing to the limit gives the identical local Lipschitz estimate for g. Given ε > 0, restrict further to |y −x| < ε/(4D) and use |Fa(x) −g(x)| < ε/4. The triangle inequality proves local uniform convergence to g; hence g is holomorphic. To remove the assumption of an everywhere pointwise limit, pass to an arbitrary ultrafilter finer than the original filter. Each scalar family Fa(z) is eventually in a compact closed disc, so this ultrafilter has a limit g(z) for every z ∈U. The preceding argument makes g holomorphic. At every point of the prescribed accumulation set its value equals f, by uniqueness of scalar limits. The identity theorem gives g = f throughout U. Thus every finer ultrafilter converges to f(y), proving convergence along the original filter. This argument applies equally to the boundary-height filter and to limits of phase characters, and uses only local bounds on tails. 295 ## Source page 298 A Wilson Yang–Mills construction J. Rodgers 12.10 Bounded slab operators in every character chamber The operator estimates needed in reconstruction hold at every character of the compression algebra. Their proof does not require discarding exceptional characters. We give the power argument, including the step that removes a character-dependent constant. Let X : V →H map the algebraic span of future words into its reflection-positive Hilbert completion. Let A, B : V →V satisfy ⟨XAv, Xw⟩= ⟨Xv, XBw⟩. Set T = BA. Suppose D > 0 and, for every v ∈V , there is a finite constant Cv such that ∥XT nv∥≤CvD2n (n ≥0). (ST1) The constant need not be uniform in v. The adjoint relation implies symmetry of T, and then of each power, relative to this Gram form. Put dn = ∥XT nv∥D−2n, s = sup n≥0 dn. The supremum is finite by (ST1). Symmetry and Cauchy–Schwarz give d2 n = D−4n|⟨Xv, XT 2nv⟩| ≤∥Xv∥d2n ≤∥Xv∥s. Taking the supremum shows s2 ≤∥Xv∥s, hence s ≤∥Xv∥. Finally, ∥XAv∥2 = |⟨Xv, XTv⟩| ≤∥Xv∥∥XTv∥≤D2∥Xv∥2. (ST2) Thus A descends through the null space and extends to a bounded operator of norm at most D on the Hilbert completion. Apply this to insertion of a slab word and insertion of its reflected word. Translation invariance of the hierarchy gives the displayed adjoint relation. Starting with a future word v, denote by vn the word obtained by adjoining n successive reflected/unreflected slab pairs. It suffices to have | Gram(vn, vn)| ≤QvD4n. (ST3) For a scalar multiple av, the Hilbert norm bound is |a|√QvD2n. A finite linear combination has (ST1) by the triangle inequality, with constant equal to the corresponding finite sum. Hence a diagonal estimate on individual words really does produce a bounded operator on their entire dense span. This proves the slab operator construction without assuming boundedness beforehand. To obtain (ST3) in a character chamber, let U be the slab operator in the unselected chamber and let Xv be its ground-space word map. Concatenation says Xenter(w,v) = UXv, while reflection gives U∗for the reflected slab. Therefore Xvn = (U∗U)nXv, ∥Xvn∥≤∥Xv∥∥U∥2n. Let χ be the chosen continuous character of the closed compression algebra. The character chamber receives the scalar Gram values of that algebra, and the compressed two-sided word is X∗ vXv. Consequently | Gramχ(v, v)| = |χ(X∗ vXv)| ≤∥χ∥∥Xv∥2. 296 ## Source page 299 A Wilson Yang–Mills construction J. Rodgers For any D > ∥U∥, substitution of vn proves (ST3) with Qv = ∥χ∥∥Xv∥2. The power argument removes this prefactor, producing a character-chamber slab operator of norm at most D. All the operators obtained for different D > ∥U∥agree on the dense future-word span, where their action is prescribed by insertion. They are therefore the same bounded operator. Letting D ↓∥U∥proves the exact estimate ∥Uχ∥≤∥U∥. (ST4) This also covers ∥U∥= 0. Every unselected arbitrary-copy estimate thus transfers with its original constants to every character, uniformly over the character space. For each character construct its reflection-positive chamber, these bounded slab maps, and the derivative calculus inherited by the character hierarchy. The transferred arbitrary-copy estimates give the slab laws required to form its contour and strict waves. Apply the field reconstruction to those data to obtain its Hilbert space and fields, with the common conjugation and rotation labels. At the previously selected character, choose the already constructed chamber, slab maps, contour and fields themselves. This is legitimate because they satisfy the same requirements; consequently no comparison between independently chosen copies is needed there. The resulting family supplies the common phasewise operator bounds used in the strict-wave polynomial estimates and in passage to the tested Wightman boundaries. The remaining contour and field construction is developed below; vacuum uniqueness is a separate conclusion and is not assumed in this step. 12.11 Centered insertions and the strict-time waves Write E(u) for the chamber’s heat contraction, F(ζ) for its holomorphic contraction semigroup on Rζ > 0, and U(x) for spatial translation. Thus F(u) = E(u) for u > 0; translations commute with the heat flow and fix the unit vacuum Ω. For a field label v and p > 0, define its centered, buffered insertion by the bounded slab operator Bp(v) = place 2p, [(0, p; v)] . (SW1) Initially v belongs to one finite field space. Its matrix element between any two future-word vectors is the Euclidean hierarchy with that insertion between the reflected left spectators and the right spectators translated forward by 2p. The spectators and insertion have distinct positions. Multilinearity of the hierarchy in the insertion therefore gives linearity of Bp. If two finite-level labels represent the same element of the common field union, the derivative-plan inclusion identity makes these matrix elements equal. Equality on the dense future-word span then makes the bounded operators equal. Hence (SW1) defines a linear map on the common field union, rather than a map depending on a representative. The arbitrary-copy slab bound applied to this one-insertion word gives, for each fixed union label v, constants Qv > 0 and dv ∈N with ∥Bp(v)∥≤Qvp−dv (0 < p < 1). (SW2) The word has both boundary buffers equal to p, which is why that estimate applies at this width. Width zero is unused and may be assigned the zero map. For w ∈C and s ≥0, define R(w, s, x) = F(−iw −s)U(x), Iw > s. Its norm is at most one, because R(−iw −s) = Iw −s > 0. Let z1, . . . , zn have strictly increasing imaginary parts, let xi ∈R3, and take 0 < p < 1, 4p < Izj −Izi (i < j). (SW3) 297 ## Source page 300 A Wilson Yang–Mills construction J. Rodgers The buffered matrix element is Mp(z, x; v) = ⟨Ω, Bp(v1)R(z2 −z1, 2p, x2 −x1)Bp(v2) · · · R(zn −zn−1, 2p, xn −xn−1)Bp(vn)Ω⟩. (SW4) For n = 0 it is ⟨Ω, Ω⟩. All factors are defined under (SW3), and this expression is multilinear in the n union labels. There are two exact identities behind the construction. Slab stretching and concatenation give Bp+u(v) = E(u)Bp(v)E(u). (SW5) When the larger width is admissible, the semigroup identity and commutation with spatial trans- lation give R(w, 2p, x) = E(u)R(w, 2(p + u), x)E(u). (SW6) Insert (SW5) into the larger-width version of (SW4). Each intervening pair of heat contractions combines with its larger-width propagator according to (SW6); the two exterior contractions disap- pear against Ω. Thus Mp+u = Mp. Comparing the smaller and larger of any two admissible widths proves complete width independence. At purely imaginary times zi = iyi, the same product is exactly the original Euclidean hierarchy at the points (xi, yi). To see this directly, build its ket recursively from the final insertion backwards. The centered insertion adds the next plant at time p; the positive heat interval yi+1 −yi −2p and spatial translation move the remaining word into its prescribed relative position. Slab insertion and chamber translation act on planted word vectors by those very operations. Taking the vacuum inner product gives the expectation of the complete word, translated by a common vector. Translation invariance removes this vector, and the quotient-label identity identifies the result with the hierarchy on the common union. This proves the Euclidean identity for every admissible p. Define Fn(z, x; v) = Mp(z, x; v) on the strict-time region. An admissible p exists because there are finitely many positive imaginary gaps; the choice is irrelevant by width independence. A single such p remains admissible on a neighborhood of any fixed strict-time point. On that neighborhood all propagators in (SW4) vary holomorphically in the complex time variables and continuously in the spatial variables. Bounded composition therefore proves the same properties for Fn. In particular this argument requires no continuity of the arbitrarily chosen width as a function of z. Repeated contraction bounds and Cauchy–Schwarz show |Fn(z, x; v)| ≤∥Ω∥2 n Y i=1 ∥Bp(vi)∥. If 0 < k ≤1 and every imaginary gap is at least k, choose p = k/8, which satisfies (SW3). From (SW2), |Fn(z, x; v)| ≤Cvk−mv, mv = X i dvi, Cv = ∥Ω∥2 Y i (Qvi8dvi). (SW7) The bound is uniform in all real times and spatial positions. At arities zero and one the pairwise gap condition is empty; the same choice of p is still allowed. No uniformity in arity or in the labels is asserted. 12.12 Tested Cauchy–Riemann equations For a real spacetime configuration X, let fv,y(X) = Fn (ti(X) + iyi)i, (xi(X))i; v , Ψv(y; φ) = Z fv,y(X)φ(X) dX, 298 ## Source page 301 A Wilson Yang–Mills construction J. Rodgers where y1 < · · · < yn and φ is a joint Schwartz test. A neighborhood of any such y has one common positive lower gap k. This follows by decreasing the finitely many strictly positive gaps slightly; equivalently subtract a sufficiently small multiple of the increasing height vector. Equation (SW7) therefore bounds fv,u, uniformly in X and nearby u, by a fixed constant. The integral defining Ψv is absolutely convergent, and dominated convergence proves continuity in y. For a height direction a, write ea for the spacetime direction whose i-th time coordinate is ai and whose spatial coordinates vanish. The one-complex-variable slice hX,u(ζ) = fv,u+(Iζ)a(X + (Rζ)ea) is holomorphic wherever its heights remain strictly ordered. A single disc |ζ| ≤r is admissible for every X and all u in a sufficiently small height neighborhood. The same common bound C holds on these discs. Cauchy’s estimate gives |∂eafv,u(X)| ≤C/r. (CR1) The radius is independent of the noncompact real configuration X; this is essential for integration. Holomorphy of the slice also gives d dsfv,y+sa(X) = i ∂eafv,y+sa(X). Bound (CR1) permits differentiation under the joint test integral. Integration by parts along the real direction ea then yields d dsΨv(y + sa; φ) = −i Ψv(y + sa; ∂eaφ). (CR2) Both the original integrand and the derivative integrand are absolutely integrable by the Schwartz decay and the uniform bounds above. These hypotheses justify integration by parts directly on the whole real space. Thus (CR2) holds for arbitrary joint tests and arbitrary height directions staying in the strict cone, not just for tensor tests or a single approach ray. Together with (SW7), it supplies the tested Taylor and tempered boundary arguments used earlier. 12.13 The frequency identity and the time spectral cone Use the Fourier normalization of the construction, denoting its Schwartz transform by H ; it in- cludes the fixed coordinate rescaling that removes the 2π from the following derivative identity. If d = ey is the pure-time spacetime vector belonging to a height vector y, that identity is −i∂d(H g) = −H (⟨d, ·⟩g). (FT1) For compactly supported smooth g, multiplication by es⟨d,q⟩defines a Schwartz test gs, with the same compact support, for every real s. Moreover d dsgs = ⟨d, ·⟩gs in the Schwartz topology. Indeed the divided difference of the scalar exponential is smooth at zero. Its products with g, and every derivative of those products, are continuous jointly in the parameter and the point on a common compact support. Uniform convergence of every weighted derivative follows from compactness; outside that support all those derivatives vanish. This proves the asserted Schwartz derivative, including at an arbitrary parameter value. 299 ## Source page 302 A Wilson Yang–Mills construction J. Rodgers Let Wv be the tempered boundary functional of the strict waves. For strictly increasing y, consider A(s) = Ψv(sy; H gs), s > 0. The tested Cauchy–Riemann identity and (FT1) show A′(s) = 0: the derivative due to the heights is Ψv(sy; −i∂dH gs), and the derivative of the test is its negative. This product rule is valid also near the boundary. To justify moving tests there, the tested Taylor bound supplies one continuous Schwartz seminorm p with |Ψv(u; φ)| ≤p(φ) for every strict height u in the boundary approach region. Thus a test difference tending to zero in Schwartz topology contributes a quantity tending to zero uniformly in u. On the interior, the ordinary locally dominated derivative argument applies. Constancy of A on (0, ∞) and the boundary limit at s ↓0 give c Wv(g) := Wv(H g) = Ψv  y; H e⟨ey,q⟩g(q)  . (FT2) This equality uses absolute frequency coordinates; it is not an assertion only in relative coordinates. Now let y be weakly increasing and let g have compact support in {q : ⟨ey, q⟩< −1}. Fix a strictly increasing height vector r with unit gaps, and apply (FT2) at Y (t) = ty + r, t ≥0. These heights retain gaps at least one. The tests gt(q) = e⟨eY (t),q⟩g(q) tend to zero in Schwartz topology. For completeness, each derivative of order j is bounded in a weighted Schwartz seminorm by a finite sum of terms j l ! e−t(t∥ey∥)lpi,j−l (0 ≤l≤j), where the constants include the fixed factor e⟨er,q⟩g(q). Every term tends to zero. The unit-gap version of (SW7) bounds the right side of (FT2) by a fixed continuous Schwartz seminorm of H gt, so it tends to zero. Its left side is independent of t, and hence c Wv(g) = 0. If a frequency point q has ⟨ey, q⟩< 0 for some weakly increasing y, rescale y so that this pairing equals −2. An open neighborhood of q then has pairing less than −1, and the preceding compact-test argument makes the distribution vanish there. It follows that supp c Wv ⊂ \ y1≤···≤yn {q : ⟨ey, q⟩≥0}. (FT3) The weak ordering, including constant height vectors of either sign, is essential: it enforces total time-frequency conservation as well as the ordered partial-energy inequalities. The passage from this time cone to the relativistic spectrum also uses the rotation and covariance arguments developed next. 12.14 Removing the outer angular flux and obtaining all Euclidean rotations Continuous Euclidean rotations are obtained from shell Ward identities and control of the outer angular flux. They are not inserted as an extra invariance assumption. The corresponding label representation must descend to the physical quotient and agree with the finite-degree free compar- ison. Only after this step can the analytic tube identities be converted into the real boost action used in reconstruction. 300 ## Source page 303 A Wilson Yang–Mills construction J. Rodgers The lattice symmetries give only signed coordinate permutations. Continuous rotations of the physical quotient require the conserved stress and its shell maps constructed below. Write pα for the translation shell map and Y (K) for the angular shell map, where K is a real skew 4-by-4 matrix. Before their normalization, the angular Ward expression is AK(v; x) = X l {S(v1, . . . , Y (K)vl, . . . ; x) + (Kxl)αS(v1, . . . , pαvl, . . . ; x)} . (RT1) Its vanishing cannot be inferred by simply enlarging a spherical cutoff: the correlations have sub- power infrared bounds, rather than the decay that such an argument would require. The source removes the outer flux through tested spectral estimates. We give the details of that passage here. The tests in this argument have the following precise meaning. Fix a nonnegative even smooth compactly supported function c, equal to one on [−1, 1], and let cw(p) = w−1c(p/w), w > 0. The source uses this central frequency profile and real Schwartz profiles with the same normalization. Its j-th moment is R pjf(p) dp; substitution gives the moment of fw as wj times that of f. In the Fourier convention used here, ∂j x bf(0) = (−i)j Z pjf(p) dp. Differentiation under the integral is justified by the integrable Schwartz majorant |p|j|f(p)|. If moments below m vanish, repeated integration of the derivative along a segment bounds | bf(x)| ≤ C|x|m on a fixed ball. This is the origin-jet cancellation used in the compensated profiles. For four one-dimensional tests fa, the tested stress pair is the product-test extension of Rab,cd(f) = Z ⟨Jab(x)Jcd(0)⟩ Y i fi(xi) d4x. (RT0a) When a factor vanishes sufficiently far in Taylor order at zero this is the ordinary integral. For arbitrary Schwartz factors, subtract the finite product Taylor polynomial near zero; the polyno- mial short-distance bound makes the remainder integrable. This defines the extension without presuming rotation invariance. Pair exchange, tensor symmetry and signed covariance follow by the corresponding changes of variables in the remainder integral. Its tested conservation defect is ∆b,cd(f) = X i Rib,cd(f0, . . . , f′ i, . . . , f3). Let gj(x) = xjc(x)/j!, and let f(a) have factor g1 at a and g0 elsewhere. Set wab,cd = ∆b,cd(f(a)), R◦ ab,cd(f) = Rab,cd(f) −wab,cd Y i fi(0). (RT0b) The subtracted term is supported at the single contact x = 0. Its differentiated value on f(a) is exactly wab,cd, because g(l) j (0) = 1{j=l}. The resulting defect therefore vanishes on all first-order product jets. It vanishes on the order-zero and order-two jets by total inversion: the three remaining tensor indices give an odd sign whereas an even total test order gives an even sign. These finite contact subtractions preserve the pair and signed symmetries; reflection in a single axis kills a coefficient unless its four tensor indices occur in pairs. Thus the compensated conservation defect 301 ## Source page 304 A Wilson Yang–Mills construction J. Rodgers has no jets of total degree below three. This precise contact remainder, rather than an assumed exact conservation law at coincident points, enters the spectral band estimates. For a fixed exterior word on coordinate line l, put Fdb(y) = ⟨Jdb(y) Y l Bl(xl)⟩. Realize that exterior word as a finite word vector and pair it with the one-stress vector. The hierarchy’s permutation and conjugation laws identify this pairing with Fdb whenever the points are distinct. This is why the spectral bounds apply to an arbitrary exterior word, not only to the stress two-point function. The actual outer term is Z (∂dφab,l λ )(y)Fdb(y) d4y, d ̸= a, (RT0c) for the compensated angular primitive φab,l λ . Its frequency widths are λ−100 in axes a, b, λ−150 on the remaining line axis l, and λ−300 otherwise; it carries the primitive amplitude λ100. The band subtractions described next give the actual estimate Cλ−1 for (RT0c), including its integrability. The finite excluded insertion set has Lebesgue measure zero and does not change this integral. Compact shell conservation identifies the difference between the summed local charges and this outer integral. Its vanishing therefore proves (RT1). For each coordinate i, reflection positivity supplies its own positive joint spectral measures νA i , supported on E = pi ≥0, for finite linear combinations A of reflected words. They include atoms at E = 0. Mixed readings are obtained by polarization and obey the weighted Cauchy–Schwarz inequality |⟨B, f(P)A⟩| ≤1 2  t Z X dνB i + t−1 Z Z dνA i  if |f|2 ≤XZ, t > 0. (RT2) This follows by applying 2uv ≤tu2 + t−1v2 to the two spectral vectors; approximation by simple spectral functions proves it for the displayed measurable weights. No identification of spectral measures for different axes is needed. All transverse moments needed below are finite after positive time damping. To prove this, take the characteristic function of the finite damped measure and apply k pairs of forward and backward differences in a transverse direction j. Its integral multiplier, up to the cancelling sign of the difference pair, is eihpj −1 h 2k . The fundamental theorem of calculus bounds the difference expression by the corresponding 2k-th derivative of the smooth separated correlation. Fatou’s lemma as h →0 gives the moment of |pj|2k. The same argument for moved word clusters uses their actual positive time gaps. For a cluster collapsed in the i direction this yields, for some finite D, C depending on the word and k, Z e−tE|pj|2k dνA i ≤Ct−D, 0 < t ≤1. (RT3) Thus moment orders can be chosen before shrinking the spectral bands. Here is the interpolation that controls low energy. For distinct nodes zj = (j + 2)2, let Lj be their Lagrange basis polynomials, 0 ≤j < m. Polynomial interpolation gives 1 x + y − X j 0. At zero this is the continuous value of the full response, including the profile’s zero-energy contribution; one does not divide a bare factor 2E by E and discard an atom. The numerator factors in (RT4) stay of one sign because the high nodes lie outside the primary profile band. Their fixed ratios give cm; their denominators give the stated reciprocal bound. The reason this positive response is small is a sign identity for the same tested position corre- lation. Suppose A has reflection parities εi, εj of opposite sign on axes i, j. Write Pi(h; f) = Z Li(h)(pi) Y l∈Q\{i} fl(pl) dνA i . Here fl are nonnegative even Schwartz frequency profiles and Li is the real even Laplace response of the inverse Fourier profile in direction i. The two spectral formulas for that one tested correlation give Pi = −Pj. If some transverse axes are absent, insert profiles equal to one on larger and larger intervals; bounded dominated convergence gives the same identity for that subset Q. Split each active profile into its Lagrange interpolant and residual. Both residuals have nonnegative response. Consequently the primary residual on axis i is bounded by the sum of the two cross terms containing an interpolant. The interpolants are finite sums of profiles in a high band. For clarity the choice of orders in this repeated operation is important. Put λ = eN, and scale each frequency profile as f(N) l (p) = λ−wlfl(p/λwl), wl ≤0. A bound Cλγ will be called a bound of exponent γ in this paragraph. If a high band has exponent a′ > 0, a transverse 2k-th moment from (RT3) bounds its far remainder with exponent at most −2ka′ + K|Q \ {i, j}| + max(K, dD), (RT6) where −K ≤wl and d is the earlier chosen band exponent. Given any η > 0, choose k so that the negative term exceeds the other fixed losses, and then choose d > 0 so small that dD < η. At each switch the coordinate just treated is removed from the list still requiring a bound. If its parity agrees with the next required coordinate, insert an opposite-parity coordinate first. A finite induction on that list terminates; the terminal case has only nonshrinking transverse widths and is bounded by (RT3). Choose moment orders and admissible band exponents backwards through this finite list. This proves an exponent η, for every η > 0, for the residual response. Equation (RT5) therefore gives Z 0≤pi≤1 Q l∈Q\{i} f(N) l (pl) pi + λq dνA i ≤Cηλη, q < 0, (RT7) when the collapsed cluster has both parities on Q. A word on a coordinate line l is collapsed on the other axes but can have a genuine time gap on l. Subtract there a shifted positive-time profile. Its response is positive, dominates a fixed multiple of (E + λq)−1 on 0 ≤E ≤1, and its damped transverse moments cost at most λ−q, by (RT3) at 303 ## Source page 306 A Wilson Yang–Mills construction J. Rodgers the actual gap. The same switch pays this loss by increasing k in (RT6). Thus (RT7) holds on i = l. On i ̸= l, bound the additional l-profile by its supremum Cλ−wl. The resulting bound has exponent η −wl. This proves the line estimate even when the time-axis cluster is not collapsed. For a diagonal stress reading a further estimate is necessary. Subtract its finite position Taylor jet with a cutoffequal to one at zero. The remainder vanishes to the required order: repeated integration along the segment from zero bounds its k-th derivative by C|x|m when derivatives k, . . . , k + m −1 vanish. Tensor products of these subtractions preserve reflection parities. Conser- vation, followed by integration by parts on these tests, gives the three tested frequency expressions B = p2 i A + R1, C = p2 i B + R2. (RT8) On profiles of width at most W = λa, a ≤0, the remainders have respectively bounds O(W 4) and O(W 6), with an arbitrarily small exponent loss. On profiles with the prescribed zero moments the remainder is exactly zero: it is a finite contact Taylor jet and each of its moments is zero. Apply (RT4) at widths W, 2W to A, B, C. After division by W 2k at levels k = 0, 1, 2, denote their six band readings by a, b, v, V, r, R, and their residual differences by f, g, h. The positive spectral responses have signs f ≥0, g ≤0, h ≥0. Equations (RT8) and the two-band subtraction give small bounds for v −a, V −4b, r −v, R −4V, f −(a −b), g −(v −V ), h −(r −R). These imply a bound for f with the same exponent. Indeed if the sum of the seven absolute errors is e, these equalities within error e imply a −b ≥−e, a −4b ≤3e, a −16b ≥5(−e); eliminating a, b bounds f by 100e. In the exact case they force a = b = f = 0; the numerical factor safely bounds the finite error elimination. The constant-moment term of each interpolant cancels between W and 2W; every other term has at least a factor W 2. Combining this bound with the positive residual lower bound gives Z 0≤E≤1 EW 4 (E + W)4 Y j̸=i f(N) j (pj) dνJii i ≤Cηλ4a+η. (RT9) This deduction uses positivity of spectral measures and of the indicated scalar responses. It does not differentiate a positive matrix or posit positivity of its derivatives. Now approximate the plane Killing field xaeb −xbea by compactly supported smooth vector fields. Choose frequency widths λ−100 on axes a, b, λ−150 on the remaining line direction if present, and λ−300 on the other axes. An odd primitive in direction a, multiplied by λ100, and its partner with a, b interchanged give the two components. Their derivatives in the a, b pair agree exactly. Since Jab = Jba, these two terms in P i,j ∂iFjJij cancel before any integral is split. The remaining derivatives are supported outside the insertion neighborhoods, where no contact term is present. Split each such tested spectral integral into E ≤1 with transverse frequencies inside the box widened by λ, and its complement. On the complement, the Schwartz bound of order k gives an arbitrary factor λ−k in a transverse direction. For E > 1 the actual exterior time gap gives e−E/λw; for every fixed R, M this is at most λ−Me−RE, eventually. The finite damped moments from (RT3) then pay the whole complement. Inside the box use (RT2) with the two reciprocal weights from (RT7) for shear components, and (RT7),(RT9) for a diagonal component. Split the latter once more at E = λ−3000. Below that threshold (including zero), the large gain in the line estimate pays a crude stress bound; above it (RT9) applies. With the stated widths these estimates, including the primitive amplitude λ100, leave a bound Cλ−1. Thus the exterior flux tends to zero. Here are explicit definitions of the responses and contact expressions in this argument. For a real Schwartz frequency profile f, write Dsf(p) = s−1f(p/s), mq(f) = R pqf(p) dp, and bf(t) = 304 ## Source page 307 A Wilson Yang–Mills construction J. Rodgers R e−iptf(p) dp, with the Fourier constants used in (RT2)–(RT4). Choose an even nonnegative smooth u of integral one supported in 1 ≤|v| ≤2, and set uk(v) = v−2ku(v), which is smooth because its support misses zero. Write Lj(X) = P k ljkXk for the node polynomial in (RT4). The actual finite interpolant is Bm,sf = X j 0). (RT12) It is the real part of R e−|t|E \ Rm,sf(t) dt, not a rational expression assigned independently at E = 0. The latter integral gives its continuous value there, namely pref f(0), since the interpolant vanishes near zero. The odd-parity response uses R sgn(t)e−|t|E bf(t) dt = −2i R pf(p)/(p2 + E2) dp for E > 0. Its subtraction is obtained by applying the even interpolant to pf(p) and dividing the result by p; division is on support away from zero. Even and odd residuals have the required first 2m moments zero. For f ≥0 supported in [−s, s], every product factor in (RT12) lies between zero and one, because |v| ≥1 and j + 2 ≥2. Each is at most (2(j + 2))2(s/E)2; if s ≥1, it is at least 1 2(1 + E)−2. The lower estimate follows from (sv(j + 2))2 ≥4s2 and p2 ≤s2. For the central profile f = Dwχ, where χ = 1 on [−1, 1], integration just over 0 < p < wE/(E + w) gives 2E Z Dwχ(p) p2 + E2 dp ≥(E + w)−1. Consequently Rm,s,Dwχ(E) ≥2−m(E + w)−1(1 + E)−2m. (RT13) Continuity proves the same inequality at zero; on 0 ≤E ≤1, (RT5) follows with cm = 2−3m. The fixed subtraction band s ≥1 and the shrinking primary width w are distinct. At high energy the product upper bound gives, for q ≤2m, a bound Cm,q(w−1 + sq)(1 + E)−q. These are the precise positive majorants used in the parity switch. For an arbitrary profile of either parity, a nonnegative even compact majorant and the odd division construction bound its subtracted response by a constant times an even response of this form. To specify the contacts in (RT8), let Rab,cd denote the compensated product-test distribu- tion (RT0b), let Tab,cd be its value after Fourier transforming all four test slots, and let Qbcd = i Fourier(P j ∂jRjb,cd). Writing Pj for multiplication of frequency slot j by pj, the source identity is Qbcd = P j PjTjb,cd. Thus the quantities called A, B, C in (RT8) are A = Tii,ii, B = X j,a̸=i PjPaTai,ji, C = X j,a,b,c̸=i PjPaPbPcTcj,ab. (RT14) Their contact remainders are exactly R1 = X j̸=i PjQiji −PiQiii, R2 = X j,a,b̸=i PjPaPbQjab −Pi X j,a̸=i PjPaQaji. (RT15) Expand Q = P PT, use the exchange symmetries of the two tensor pairs, and commute the scalar multipliers Pj; this gives B = P 2 i A + R1 and C = P 2 i B + R2 exactly. The compensated Q has no 305 ## Source page 308 A Wilson Yang–Mills construction J. Rodgers contact jet of total order below three by (RT0b). On widths ≤λa this costs 3a with arbitrarily small exponent slack. Multiplication by one frequency costs a and by three costs 3a; hence (RT15) costs 4a and 6a, respectively. This proves the powers asserted after (RT8). Vanishing on profiles with the required moments follows directly by evaluating the finite contact polynomial on those moments. The exterior-flux powers can also be made explicit. Put E = pd, HN(p) = Q j̸=d Dλ1+wj χ(pj), and Dq(p) = 1E≤1 HN(p) E + λq , Zz(p) = 1E≤1 Eλ4z (E + λz)4 HN(p). (RT16) The exterior-word integral of D−3000 has exponent u ≤150: u = 1 if d = l, and u = −(1 + wl) + 1 otherwise, by the line estimate. For a shear stress, the stress integral of Dwd has exponent one. The squared response is at most C2λ2(3+wd)D−3000Dwd, by e−E/λwd ≤λwd/(E + λwd). Weighted Cauchy–Schwarz therefore gives exponent 3 + wd + (u + 1)/2. If d = l, then u = 1 and wd ≤−150, giving at most −146. If d ̸= l, a, b, then wd = −300 and u ≤150, giving at most −221.5. These exhaust the shear case d ̸= a, b; both bounds are stronger than exponent −101. If d = b, split instead at E = λ−3000. On the upper part use Z−99. Its stress integral has exponent 1 + 4(−99) = −395, by (RT9) with slack one, and the squared response is at most C2λ6D−3000Z−99. The resulting exponent is at most 3+(150−395)/2 < −101. On the lower part use the crude stress weight D0, whose integral has exponent at most 3·300 = 900. The squared response is at most C2λ2(3−1500)D−3000D0, giving exponent at most 3−1500+(150+900)/2 < −101. These pointwise estimates are valid on the nonnegative spectrum and include E = 0. The complement of the low box has arbitrary negative exponent by the damped moments and Schwartz tails discussed after (RT9). Thus the spectral reading has exponent −101. Multiplying by the angular primitive’s λ100 leaves exponent −1, as claimed in (RT0c). Decomposition into the finitely many reflection parities preserves the bound; parity-mismatched terms integrate to zero by reflecting their odd axis. The compact cutoffflux identity is therefore (RT1) equal to zero for every separated tuple on a coordinate line. Translation along that line changes Kxlby the fixed vector Kel and leaves the correlation unchanged. Subtracting the two angular identities gives the corresponding sum of the translation shell maps equal to zero. Empty tuples give empty sums throughout. 12.15 The order of choices in the spectral sign walk We record the quantitative induction that supplies the small mixed-band bound in (RT6). Let Q be the finite set of axes being tested, with two signs assigned so that each axis has an opposite-sign partner. Let U ⊆Q be the axes whose width exponent is still allowed to be negative, and let i be the active axis. A finite route alternates signs and removes the currently visited axis from U. Such a route exists by induction on |U|: if the active axis is in U, remove it and move to an opposite partner. If it is not, choose an axis still in U; reach it directly if its sign is opposite, and otherwise take one opposite-sign detour. This reduces the nonempty set after at most two such moves. The induction statement is uniform in a width vector w with −K ≤wj ≤a, wi ≤d, and wj ≥0 outside U, for every 0 ≤d ≤a. It asserts a residual budget δ > 0 after sufficiently high moment order, for nonnegative even compact profiles and all fixed support factors. If U is empty, the direct spectral moment bound of order zero has a finite price exponent D. Choose a > 0 with aD < δ. The nonnegative widths then pay its residual by that budget. At an induction step from axis i to an opposite-sign axis j, first obtain the future bandwidth a′ > 0 from the induction hypothesis for U \ {i}. Put p = K|Q \ {i, j}| and z = max(K, δ). Choose an integer k > (p + z)/a′. (RSW1) 306 ## Source page 309 A Wilson Yang–Mills construction J. Rodgers Only now purchase the moment estimate of order k, with its finite price exponent D, and choose 0 < a < a′ with aD < δ. The positive profile majorant constructed by the finite interpolant bounds the baked profile and keeps the remainder nonnegative on the spectral half-line. The far part of the two-axis comparison has exponent at most −2ka′ + p + max(−wi, dD) ≤−2ka′ + p + z < 0. (RSW2) The near part is paid by the future hypothesis after replacing the active profile by that majorant and its exponent by the nonnegative d. The two residuals have nonnegative spectral integrands. The signed comparison inequality from (RT5) therefore bounds their sum by the two paid positive quantities; no spectrum is transferred from one axis to another. Moment orders and thresholds can be enlarged to their maximum. Induction along the finite route gives the bound for arbitrary initial nonpositive widths, with one sufficiently small positive a. This explains why the future bandwidth is chosen before the present moment order and why the contact-price exponent is chosen before the present final bandwidth. It supplies the precise quantifiers behind the mixed-band estimate used to derive the collinear Ward identities. 12.16 Normalized two-band diagonal estimate For clarity, the six normalized readings used in (RT8)–(RT9) have the following common normal- ization. Let W = λa, a ≤0, and let q(pW) k denote the order-k profile from (RT4) at width pW, with all other profiles fixed. Put a0 = Re A(q(W) 0 ), b0 = Re A(q(2W) 0 ), v = W −2 Re B(q(W) 1 ), V = W −2 Re B(q(2W) 1 ), r = W −4 Re C(q(W) 2 ), R = W −4 Re C(q(2W) 2 ). (DB1) The normalization on the second band is still by W, not by 2W. Multiplication of the order-k + 1 profile by the frequency square gives (pW)2 times its order-k predecessor. Consequently (RT8) gives small errors for v −a0, V −4b0, r −v, R −4V . The first contact error has unnormalized order W 4, the second order W 6; their respective normalizations leave order W 2, including the arbitrary small exponent slack. The zero-moment contact subtraction has the same bound: its constant interpolation coefficient cancels between the two bands, and every nonconstant even coefficient has at least a factor W 2. Let f, g, h be the real spectral gap readings at these three orders, normalized by 1, W −2, W −4. Their signs are f ≥0, g ≤0, h ≥0. The difference between a gap profile and its two-band difference is exactly the finite interpolation debris just bounded. Hence the additional errors f −(a0 −b0), g − (v −V ), h −(r −R) have the same small budget. Let K be the sum of the absolute values of these seven errors. From g ≤0 we get a0 −4b0 ≤3K; from h ≥0 we get a0 −16b0 ≥−8K. Therefore 12b0 ≤11K, and 0 ≤f ≤a0 −b0 + K ≤3b0 + 4K ≤7K ≤100K. (DB2) Finite sums and fixed scalar multiples preserve the budget exponent, so (DB2) proves the small diagonal gap bound. This is the algebraic step in (RT9); it needs positivity of the three response signs and the contact bounds, and makes no positivity claim for the six individual band readings. 12.17 Ward kernel transfer and the rotation action We explain how these identities extend to general tuples and to a group action. Differentiating a shell integral is allowed because its fixed compact annulus remains separated under small move- ments of the distinguished point. Differentiation acts both on the correlation and on the centered 307 ## Source page 310 A Wilson Yang–Mills construction J. Rodgers angular coefficient. The latter derivative is the relevant constant translation coefficient. Hence every position derivative of a line Ward test is again a finite sum of line Ward tests for successor- degree labels. The all-order scaling comparison (WS7) transports these tests to the free quotient, where the translation and angular actions are explicit derivatives of polynomials. Finitely many separated reading maps suffice at each finite degree, by the detector argument in (WS4)–(WS5). Their matrices converge to injective free matrices. This transfers the free rank statements for the spaces of possible internal defects and forces any such defect to vanish. Thus the translation shell map is one positive scalar c times the already constructed derivative, and the angular rows, divided by c, satisfy the Ward identity at every separated tuple. The finite-dimensional rank arguments just used have two different domains. For translation rows, the domain at cap d is the real subspace of Q4 i=1 HomC(Fd, Fd+1) consisting of rows that kill the unit, preserve each lower filtered image, commute with conjugation, and transform as a vector under signed permutations. A reading is a position derivative at a fixed separated collinear tuple of the Ward sum P lS(v1, . . . , Tivl, . . . , vn). The free readings have common kernel precisely the real line spanned by the derivative row. Select finitely many such readings by minimizing the dimension of their common kernel. Adjoining a further reading cannot lower that minimal dimension, so this finite kernel already equals the full kernel. Pullback by Uk turns the physical readings into matrices on this one fixed real space; their convergence is (WS7). Eventual rank and the two known rows in their kernels then give the translation normalization stated above. For the internal angular defect, the domain is instead the real subspace Id ⊂HomR(so(4), HomR(so(4), EndC Fd)) . (RT17) Its elements T(K, L) are alternating in K, L, kill the unit, commute with conjugation, preserve every lower filtered image, and intertwine simultaneous conjugation of K, L by orientation-preserving signed permutations with their action on labels. The reading for a separated tuple, labels vl, and K, L, is P lS(v1, . . . , T(K, L)vl, . . . , vn). The free internal Ward separation says the common kernel of these readings is zero at the required sufficiently large caps; lower caps follow by inclusion. Select finitely many readings by the same decreasing-kernel-dimension argument. Their joint map is injective. Transport by T 7→UkTU −1 k , retaining both Lie algebra slots, identifies the actual and free domains; reality, signed covariance, unit preservation and lower-image preservation follow from (WS1). On fixed free basis inputs the actual reading matrices converge to the free ones by order- zero joint comparison. Linear independence of finitely many matrix columns is open, so they are eventually injective. This permits testing the unknown, scale-dependent pulled-back defect without assuming it is bounded. Its readings vanish by the Ward identity; injectivity makes the defect itself zero. The free internal separation used here has a concrete angular argument. Identify fields with their Wick polynomials as in (FM3)–(FM7). For a fixed pair K, L, the assumed zero Ward sums say that the corresponding polynomial endomorphism T is silent in every moment; in particular its two-point pairings obey ⟨Tp(x), q(0)⟩+ ⟨p(x), Tq(0)⟩= 0. (RT18) It preserves the degree-four subspace by the lower-image condition. The explicit curvature contrac- tion (FM2), expanded on its degree-four invariant polynomials, gives at equal radii x, y ̸= 0 the identity ⟨p(x), q(0)⟩= ⟨(ρ(HyHx)p)(y), q(0)⟩, (RT19) where Hz = I −2zzT/|z|2 is reflection in the hyperplane normal to z, and ρ acts on all tensor legs. This follows by substituting the orthogonal factor I −2zzT/|z|2 in the degree-four Wick pairings; 308 ## Source page 311 A Wilson Yang–Mills construction J. Rodgers the radial factors agree, and contractions use the invariant color inner product. The explicit low- degree evaluations and the extension to every filtered cap are (FS6)–(FS14), with (WE1)–(WE4) and (HC1)–(HC4) supplying their concrete evaluation and collision steps. Apply (RT18) at x and y, use (RT19) in both terms, and subtract. Positive definiteness of the reflected polynomial pairing at x = −rei forces T to commute on degree four with ρ(HxHy). Taking y = ei + ej, with the matching radius, makes this product a signed quarter-turn in the ij-plane. Signed covariance of the two Lie algebra slots now makes T(K, L)p, for each degree-four p, invariant under simultaneous quarter-turn conjugation of K, L. An alternating bilinear map with this invariance is zero. To check this without representation theory, use the plane basis Eij of so(4). Equal arguments vanish by alternation. For two planes sharing one axis, the square of a quarter-turn in the remaining two axes preserves one plane and negates the other. For disjoint planes, two quarter-turn substitutions exchange the arguments with the sign which, combined with alternation, negates the value. Every coefficient is therefore its own negative. This proves that the internal endomorphism kills the whole degree-four subspace, including the free stress. The local angular Ward identity for the free stress then propagates this to higher degrees: insert the stress in the silent moment sum, apply the compact angular shell integral, and use that its T-term is zero. The remaining identity is the moment pairing of the commutator of T with the free angular generator. Nondegeneracy of the Wick polynomial tests, proved by the reflected Gram argument following (FM7), makes this commutator zero. Finite-dimensional ODE uniqueness exponentiates that commutation to each quarter-turn at the higher cap. Signed covariance and the same plane-basis calculation again force T = 0. This proves the separation of Id for every d ≥4 used in (RT17), rather than assuming an additional internal symmetry classification. For the translation detector there are two further points behind its free one-dimensional kernel. A separated target tuple can be joined to a tuple on any prescribed coordinate line by a segment staying in configuration space: order the line coordinates by the target coordinates on that axis, break ties by distinct arbitrarily small increments, and interpolate. Strict inequalities prevent a collision along the segment, and tied target coordinates remain separated until the endpoint. All position derivatives of the Ward sum vanish at the line tuple when all the scene readings vanish. Analyticity of the explicit Wick kernels gives zero in its neighborhood and then along this segment. Thus scene vanishing is the full silent-moment identity. The color-vector, reality and filtration restrictions of (RT17)’s translation analogue then give precisely the derivative line: the free Wick coefficient calculation identifies its degree-four charge as a real scalar translation charge; subtract that scalar derivative. The remaining silent row kills the free stress. The radial shell identity makes its commutator with the engineering-weight derivation a ghost. The fixed exterior-cap separation (FS13) kills that commutator, and the inversion-parity calculation (FS14) kills the row on every higher cap. The complete charge and polynomial-separation calculations are (FS1)–(FS14), built on the hierarchy (FM1)–(FM7) and the invariant spaces (FC1)–(FC15). The zero scalar case is included. Apply two orbital derivatives to that identity and subtract in opposite orders. The vector fields x 7→Kx have bracket with the opposite sign to the matrix bracket, while the substitutions in a multilinear correlation have the same opposite sign. The resulting defect is [Y (K), Y (L)] − cY ([K, L]), tested in the internal defect space just eliminated. It is zero. This proves the Lie relations on the actual quotient. There is a useful additional detail in fitting this representation to the free one: the deformed matrices need not be normal. Put D = c−1Y , and work at one finite degree in the space of linear maps between the two quotients. Its relative generators are RK(T) = D(K)T −TYfree(K). Their sum of squares commutes with every RK: expand the commutator, use the plane matrix brackets, and pair the two opposite ordered-plane terms. On the free side this sum is negative semidefinite 309 ## Source page 312 A Wilson Yang–Mills construction J. Rodgers in the invariant inner product. A real polynomial in it projects onto the zero eigenspace, by interpolation on its finite real spectrum. Evaluate that same polynomial on the nearby deformed sum of squares. The resulting matrices Pk converge to the free projection P, commute with all the deformed generators, and obey P 2 k −Pk →0; they are not yet projections. Correct them by iteration of φ(Z) = 3Z2 −2Z3. If e(Z) = Z2 −Z, direct polynomial multipli- cation gives φ(Z) −Z = (1 −2Z)e(Z), e(φ(Z)) = (4e(Z) −3)e(Z)2. (RT10) For bounded Z and sufficiently small initial defect, the defect decreases at least geometrically by 1/2, and the increments are bounded by a fixed constant times that defect. The iterates are Cauchy and converge to an exact idempotent Qk, with Qk −Pk →0. Polynomial iteration preserves every commutation and every closed real symmetry constraint. It remains to show that the relative generators vanish on its range. Each plane generator is the bracket of two other plane generators. Since Qk commutes with them and is idempotent, the restricted matrices RiQk satisfy those same perfect bracket identities. They tend to zero because the free generators kill P. If M is the largest of their finitely many norms, the bracket identities give M ≤2M2. Eventually M < 1/2, hence M = 0. Apply Qk to the comparison map corresponding to the identity: its limit is the identity, so the result is invertible for large k and is an exact intertwiner. Reality and signed permutations preserve it, because signed conjugation permutes the plane generators with signs whose squares are one. This proves the required compatibility without assuming a spectral theorem for the deformed matrices. Transport the free orthogonal representation by this intertwiner. Its exponential derivative is c−1Y (K). Along xl(t) = etKxland the simultaneously moved labels, the ordinary product rule and the Ward identity show that the derivative of the correlation is zero. The product rule is valid jointly in positions and labels: expansion in fixed finite bases is a finite sum of products of smooth position functions and polynomial coefficient functions. Thus the correlation is constant along every plane rotation. Plane rotations align any unit vector with a coordinate axis; every hyperplane reflection is a conjugate of a coordinate reflection by such rotations. Reflections generate the orthogonal group, so covariance holds for all of O(4). All actions preserve the inclusions. This is first true for their generators by the quotient deriva- tive and Ward identities; the two sides of an inclusion intertwining equation then solve the same finite-dimensional linear differential equation along every exponential, with the same initial value. Uniqueness proves it on the exponentials and the group generators. Finally average the normalized stress c−1J, with its two tensor indices moved as well, over normalized Haar measure on compact O(4). Conservation, reality and each shell identity are finite-dimensional linear identities, so they pass through the integral. This produces a symmetric conserved covariant stress with the normal- ized translation and angular charges on the same hierarchy. The next subsection uses precisely this Euclidean action. 12.18 From Euclidean rotations to the tested boost identity Let Ωa be the Euclidean skew generator in the time–spatial-a plane, with vector field xa∂t −t∂a. Write La for its action on a finite field space, and put Ka = −iLa for the corresponding Lorentz boost generator on labels. Differentiating Euclidean covariance at a configuration of distinct points gives D(ΩaXi)iSv(X) = − X i Sv[i←Lavi](X). (BC1) Indeed move the points by esΩa, transfer this rotation to inverse rotation of all labels, and differenti- ate at zero. Each label belongs to a finite-dimensional field space, so the multilinear scalar reading 310 ## Source page 313 A Wilson Yang–Mills construction J. Rodgers is differentiable by its finite basis expansion. Smoothness of the Euclidean hierarchy on distinct configurations justifies the positional derivative. The extension of (BC1) to strict complex times is made after joint testing; no spatial differen- tiability of the strict waves is assumed. First take a compactly supported joint test φ. For fixed ordered y, introduce the complex scaling parameter z by ζi(z, X) = z ti(X) + iyi. Let U be the set of z for which these times are strictly ordered in imaginary part for every X ∈ supp φ. The set U is open by compactness of the support and continuity of the finitely many gaps. It is convex, since each gap inequality is affine in Iz. It contains the entire real axis, in particular zero and one. For z = iw, the waves in this integral are Euclidean values at the folded points Qw(X)i = (xi(X), yi + wti(X)). Consider the real vector field on joint configuration space Vw(X) = X i  xi,a∂ti −w(yi + wti)∂xi,a  . Its divergence is zero. Its image under the linear part of Qw is exactly w times the Euclidean rotation vector field at Qw(X). This identity avoids division by w and remains valid at w = 0. Equation (BC1), the chain rule, and integration by parts applied to Sv(Qw(X))φ(X) give Z   Fv(ζ(iw, X), x(X)) X i {xi,a∂ti + iw ζi(iw, X)∂xi,a}φ(X) −iw X i Fv[i←Kavi](ζ(iw, X), x(X))φ(X)  dX = 0. (BC2) The product being differentiated is smooth near the compact support: its folded points have strictly ordered heights and hence distinct positions. It extends by zero smoothly away from that support. The zero divergence makes its integrated directional derivative vanish. Replace iw in the integrand of (BC2) by arbitrary z ∈U. The integral is holomorphic on U. To justify this, regard the integrand as a continuous function on the compact support. Its dependence on z is holomorphic at every point, and it is jointly continuous; locally the Cauchy formula is uniform on that compact set. Integration is a bounded linear functional on the continuous-function space and therefore preserves holomorphy. Equation (BC2) makes this holomorphic integral vanish at imaginary parameters accumulating at zero. The identity theorem on the connected set U makes it vanish at z = 1. Thus X i Ψv[i←Kavi](y; φ) = Ψv(y; Ba,yφ), Ba,y = X i  xi,a∂ti + (ti + iyi)∂xi,a  . (BC3) For a general Schwartz test, use compact cutoffs converging to it in Schwartz topology. The polynomial-coefficient differential operator Ba,y is continuous on that space, and the fixed-height tested functional is continuous. Passing to the limit proves (BC3) for all joint Schwartz tests. 12.19 Strong motion of tests and finite boosts The Poincaré action on joint tests is inverse placement, (g · φ)(X) = φ(g−1X). 311 ## Source page 314 A Wilson Yang–Mills construction J. Rodgers It is continuous in the Schwartz topology. Here is the bound that accounts for this assertion. For an invertible linear map A, translation b, and seminorm pi,k(φ) = supX ∥X∥i∥Dkφ(X)∥, the chain rule and ∥X∥≤(∥A−1∥+ ∥b∥)(1 + ∥A(X −b)∥) give pi,k φ(A( · −b))  ≤(∥A−1∥+ ∥b∥)i∥A∥k 2i max r≤i s≤k pr,s(φ). (BC4) The derivatives vary continuously on compact sets as A, b vary, and (BC4), with higher weighted seminorms of the fixed test, controls the complement. This proves convergence in every Schwartz seminorm. For a boost gs in direction a, pointwise differentiation of inverse placement gives d ds(gs · φ) s=0 = −Ba,0φ. (BC5) This is also a derivative in Schwartz topology. In fact its pointwise derivative at arbitrary s is gs·(−Ba,0φ), a strongly continuous Schwartz family by (BC4). The fundamental theorem of calculus after evaluation at X writes the difference between the difference quotient and this derivative as Z 1 0 D(us) −D(0) (X) du. The same equality holds after each spatial derivative. Taking each weighted supremum bounds its seminorm by sup0≤u≤1 pi,k(D(us) −D(0)), which tends to zero by continuity on the compact parameter interval. This proves (BC5) strongly without integrating in the Schwartz space itself. Pass (BC3) to the boundary. Its extra height terms are the finite sum i X i yi Ψv(y; ∂xi,aφ); each tends to zero, because the tested boundary limit exists for that fixed derivative test and yi →0. Therefore X i Wv[i←Kavi](φ) = Wv(Ba,0φ). (BC6) Choose one finite field space containing all labels, using the directed inclusions. Consider H(s) = W(ρ(gs)vi)i(gs · φ). Differentiate at zero. The label derivative is the sum on the left of (BC6), and the test derivative is its negative by (BC5). Their cancellation gives H′(0) = 0. This product rule needs no topology on the entire field union: expand the multilinear distribution in a basis of the one finite field space, differentiate the finitely many scalar coordinate products and the pairings with fixed basis labels, and sum. At an arbitrary parameter value repeat the same argument with the already moved labels and test. The group law gives H′(s) = 0 everywhere, hence W(ρ(gs)vi)i(gs · φ) = Wv(φ). (BC7) Real translations and spatial rotations follow first at strict heights from their Euclidean identities by one-complex-parameter uniqueness, and then at the boundary by change of variables, whose Jacobian has modulus one. The transformations satisfying the boundary covariance identity form a subgroup: composition follows from the representation law and test pushforward law, and inverses follow by applying the identity to inversely transformed labels and tests. Spatial rotations and coordinate boosts generate the specified Lorentz group, so (BC7) gives full covariance once that generation statement is established. 312 ## Source page 315 A Wilson Yang–Mills construction J. Rodgers 12.20 Dual cone and a smooth frequency cutoff The absolute-frequency cone has a useful elementary dual description. Write p = (p1, . . . , pn), with total momentum zero, and Pj = Pn i=j pi. Its defining remaining condition is that each proper suffix momentum belongs to the closed forward cone. If y has all successive differences in the corresponding closed dual forward cone, summation by parts gives ⟨p, y⟩= n X j=2 ⟨Pj, yj −yj−1⟩≥0. (DC1) The endpoint terms vanish because the total momentum and the empty suffix are zero. Self-duality of the forward cone proves the inequality term by term. The identity also covers the empty tuple. There is a quantitative interior form. If y is in the open tube-height cone, choose r > 0 such that B(y, r) is contained in that cone. For ∥u −y∥< r/2 and nonzero p in the frequency cone, the point u − r 2∥p∥p lies in B(y, r). Apply (DC1) there to obtain ⟨p, u⟩≥(r/2)∥p∥. (DC2) For p = 0 the estimate is immediate. This gives one positive exponential damping margin on a whole neighborhood of any interior height. A smooth cutoffnear the frequency cone is constructed explicitly. Convolve the indicator of its open distance-two neighborhood with a nonnegative, unit-integral smooth bump supported in the ball of radius 1/2. Denote the result by η. It equals one on the distance-one neighborhood of the cone: every point sampled by the convolution then has distance less than 3/2 from the cone. Its support is contained in the distance-5/2 neighborhood, and its closed support lies in the distance-three neighborhood. All derivatives are globally bounded, since derivatives can be put on the compactly supported bump and the indicator has absolute value at most one: ∥Dkη∥∞≤∥Dkb∥L1. The smoothing and support assertions use the actual Euclidean metric on the full absolute-frequency space, including its dimension-zero case. For a closed tube height y and a frequency x in this closed support, choose a cone point z within distance three of x. Equation (DC1) and Cauchy–Schwarz imply −⟨y, x⟩≤⟨y, z −x⟩≤3∥y∥. Hence on ∥y∥≤R, ∥Dl xe−⟨y,x⟩∥≤e3RRl. Leibniz’s formula and the cutoffderivative bounds show that η(x)e−⟨y,x⟩has uniformly controlled derivatives for bounded closed heights. More generally the polynomial-growth cutoffestimates used in the source give ∥Dk x(ηe−⟨y,·⟩)(x)∥≤Ck,R(1 + ∥x∥)Nk. (DC3) Outside the closed support all its derivatives vanish. Thus these multipliers act continuously on Schwartz tests, with bounds uniform over bounded closed heights; interior heights additionally have the exponential margin (DC2). These are the cutoffand damping estimates needed for the subsequent tube construction. 313 ## Source page 316 A Wilson Yang–Mills construction J. Rodgers 12.21 Holomorphic tube functions from the frequency distribution The tube is an open complex-time domain with strictly ordered imaginary heights. Positive heights provide the damping needed for the frequency distributions and Hilbert vectors. Removing them is a tested tempered boundary limit; it is not substitution of a pointwise value at zero height. The Gram argument below shows that independently heated word vectors give one positive boundary form on the common word space. Here the radian Fourier transforms on RN, N = 4n, are explicitly H g(q) = Z e−i⟨x,q⟩g(x) dx, Pf(q) = (2π)−N Z ei⟨q,x⟩f(x) dx. They are inverse continuous maps of the Schwartz space. For a closed tube height y, set Dyg(q) = η(q)e−⟨y,q⟩g(q). The estimates (DC3) and Leibniz’s rule bound each seminorm of Dyg, uniformly for y in bounded subsets of the closed height cone, by finitely many seminorms of g. Joint smoothness of the multiplier gives convergence of every jet on compact frequency sets as y varies. The same bounds with one extra polynomial weight control the complements. It follows that y 7→Dyg is continuous in the Schwartz topology even at closed-cone boundary heights. For a full complex configuration Z = X + iY in the open tube, define the lens λZ(q) = η(q)ei⟨Z,q⟩= η(q)ei⟨X,q⟩−⟨Y,q⟩. It is a genuine Schwartz test. More precisely, (DC2), together with the distance-three support bound, gives constants c > 0, A such that −⟨Y, q⟩≤A −c∥q∥ (q ∈supp η); A = 3(∥Y ∥+ c) is sufficient. Each frequency derivative of the exponential contributes at most a power of its linear functional norm. Derivatives of η have polynomial growth. Leibniz’s formula and (1 + t)keA−at ≤k!a−keA+a (t ≥0, a > 0) therefore prove, for every k, j, a finite bound sup q (1 + ∥q∥)ke(c/2)∥q∥∥Dj qλZ(q)∥< ∞. (TL1) The exponential margin retained in (TL1) justifies complex differentiation of the lens as a Schwartz- valued function. For completeness, if L is a small real-linear, complex-valued frequency functional, with ∥L∥≤ min(a, 1), the scalar exponential remainder and its derivatives satisfy ∥Dj(eL(q) −1 −L(q))∥≤∥L∥2(1 + ∥q∥)2ea∥q∥. (TL2) For j = 0 this is the exponential-series remainder. For j = 1 differentiate once and bound eL −1. For j ≥2, derivatives of the affine terms vanish and the bound follows from ∥L∥jeRL(q), using ∥L∥≤1. Multiply (TL2) by the lens and use (TL1) and Leibniz’s rule. Every Schwartz seminorm of λZ+H −λZ −i⟨H, ·⟩λZ is at most Ck,j∥H∥2. Dividing by ∥H∥proves the complex Fréchet derivative dλZ(H)(q) = i⟨H, q⟩λZ(q) (TL3) 314 ## Source page 317 A Wilson Yang–Mills construction J. Rodgers in the actual Schwartz topology, jointly in all complex coordinates. For the tempered frequency functional c Wv, define Fv(Z) = (2π)−N c Wv(λZ). (TL4) Continuity and linearity of c Wv transfer (TL3) to complex differentiability of Fv. Finite-dimensional complex analysis makes it holomorphic on the full open tube. At this point no spectral support assertion is needed for holomorphy itself; the support condition will make the result independent of the cutoff. The identification with a damped distribution requires an integral argument in Schwartz topol- ogy. At a fixed interior Y , put g(q) = η(q)e−⟨Y,q⟩, a Schwartz test by (TL1), and gX(q) = ei⟨q,X⟩g(q). Its Fourier transform is the translate (H g)( · −X). For any Schwartz function h, pi,j h( · −X)  ≤2i h pi,j(h) + ∥X∥ip0,j(h) i . (TL5) This follows from ∥q∥≤∥q−X∥+∥X∥, splitting according to which summand is larger. Continuity of a tempered distribution bounds it by finitely many Schwartz seminorms. Thus X 7→c Wv(gX) is continuous and has at most polynomial growth. Its product with any joint Schwartz test φ(X) is absolutely integrable. For compactly supported φ, integrate gXφ(X) over its compact support. Uniform control of every jet permits the continuous functional c Wv to pass through this integral. Fourier inversion gives c Wv DY (Pφ)  = Z Fv(X + iY )φ(X) dX. (TL6) For arbitrary Schwartz φ, use its smooth compact cutoffs. The left side converges by continuity on Schwartz space; the right side converges by (TL5), polynomial growth, and dominated convergence. This proves (TL6) for full joint tests, with an absolutely convergent right side. It identifies the holomorphic emission with the actual damped boundary distribution at every fixed interior height, without a fictitious norm on Schwartz space or an exchange of unbounded integrals. 12.22 Identification of the tube continuation and its covariance Spectral support now removes the auxiliary cutoff. A distribution supported in the closed frequency cone annihilates a Schwartz test that vanishes on a neighborhood of that cone. For compactly supported g, the two tests η(q)e−⟨Y,q⟩g(q), e−⟨Y,q⟩g(q) agree on the distance-one neighborhood of the cone. Their difference has closed support disjoint from it; hence the supported distribution gives them the same value. This argument uses the compact support of g when defining the second test. It does not assert that an unrestricted exponential is a Schwartz multiplier. At a fixed strictly ordered temporal height y, the strict restriction fv,y is bounded and continuous by (SW7). Its integral against a joint test therefore defines a tempered distribution uy. In (FT2) use the compact test e−⟨ey,q⟩g(q). Its exponential lift cancels exactly. The preceding cutoffremoval then gives c Wv(Deyg) = buy(g) for all compactly supported smooth g. Both sides are continuous on Schwartz space, so smooth compact cutoffs extend equality to every Schwartz test. Fourier inversion and (TL6) show Z Fv(X + iey)φ(X) dX = Z fv,y(X)φ(X) dX. 315 ## Source page 318 A Wilson Yang–Mills construction J. Rodgers Both functions are continuous. Equality against all compact smooth tests gives almost-everywhere equality, and full support of Lebesgue measure upgrades it to pointwise equality: Fv(X + iey) = fv,y(X). (TI1) Thus the constructed full-tube function continues the very strict waves previously constructed. Let r be a real Lorentz transformation in the specified group, and r∨its dual for the Euclidean frequency pairing, so that ⟨rX, r∨q⟩= ⟨X, q⟩. The tube-height cone is invariant under r, and the frequency cone is invariant under r∨. Compare the two Schwartz lenses λZ(q), λrZ(r∨q). Their exponential factors agree everywhere. Their cutofffactors need not agree globally, but both equal one near every cone point p: use the intersection of the unit ball about p and the inverse image of the unit ball about r∨p. Consequently their difference has closed support disjoint from the cone and is annihilated by c Wv. Fourier transformation of boundary covariance, followed by this lens comparison, gives F(ρ(r)vi)i(rZ) = Fv(Z). (TI2) This proves real Lorentz covariance inside the tube without requiring an invariant cutoff. The same local support argument compares any two permitted cutoffs and proves that their emissions agree. 12.23 Analytic germs at an adjacent spacelike pair Fix adjacent argument slots i, j = i + 1, and write σ for their transposition. Choose height profiles y(m) →0 with exactly this adjacent gap flattened: y(m) i = y(m) j , and all other ordered gaps strictly positive. One explicit choice is y(m) k = ak/(m + 1), where ak = k except aj = i. At two arguments this is the zero profile. Both the original and transposed imaginary configurations lie in the closed tube-height cone, and their complex configurations are in the closures of the respective tubes. At a real configuration with spacelike difference v = Xj −Xi, set T = ∥v∥> |v0|, and choose the spatial unit vector e = v/T. The infinitesimal boost Cev = (v0e, e · v) is strictly future timelike, since its spatial norm is |v0| and its time component is T. Let Se project onto the spatial-e and time plane. The complex boost is Ae(z)Z = Z + sinh z CeZ + (cosh z −1)SeZ. For z = ib, its imaginary difference at any adjacent pair is Y + sin b V + (cos b −1)W, (LG1) where Y is the original imaginary difference, V = Ce applied to the real difference, and W = SeY . For every unflattened adjacent gap, Y is already strictly timelike, so (LG1) remains so in a neighborhood of b = 0 and of the given configuration. At the flattened pair, V is strictly timelike. Use the exact identity sin b V + (cos b −1)W = sin b  V − sin b 1 + cos bW  . 316 ## Source page 319 A Wilson Yang–Mills construction J. Rodgers The vector in parentheses is strictly timelike throughout a neighborhood of (b, Z) = (0, Z0), and sin b > 0 for sufficiently small b > 0. Adding it with that positive coefficient to a closed forward- cone vector Y enters the open cone. Since there are finitely many adjacent gaps, one product neighborhood and one small positive angle work for all of them, uniformly down to angle zero. Apply the same calculation to the transposed configuration with the opposite imaginary boost; at the flattened pair the two reversals of sign cancel. Thus the original tube and the transposed tube each have a small imaginary-boost path into their interiors, with this path property holding for nearby points of the corresponding tube, not just at the boundary point itself. The labels also have an analytic boost action. The following elementary Banach-algebra ar- gument supplies it. For a continuous one-parameter group u(t), put V (t) = R t 0 u(s) ds. Then V ′(t) = u(t) and V (t)/t →1. Choose nonzero a so that g = V (a)/a is invertible. The group law gives u(t)V (a) = V (t + a) −V (t), u(t) = a−1[V (t + a) −V (t)]g−1. The right side proves differentiability. If B = u′(0), differentiating the group law gives u′(t) = u(t)B, whence d dt[u(t)e−tB] = 0. Evaluation at zero yields u(t) = etB. For each finite-dimensional field space this gives the entire complex extension u(z) = ezB of its boost representation. Fix the finite field space containing the finitely many labels. For a boundary configuration Z0, take the small imaginary-boost endpoint it just constructed and define near Z0 G(Z) = F(u(it)vk)k(Ae(it)Z). (LG2) The endpoint lies in the tube, so (LG2) is holomorphic on a full neighborhood of Z0. It agrees with the original emission on nearby points already in the original tube. To prove this agreement, for such a point Z consider QZ(z) = F(u(z)vk)k(Ae(z)Z). This is holomorphic in a neighborhood of the entire imaginary-boost path. For moving labels, expand in the finite field-space basis; each coefficient product is holomorphic and each fixed-label emission is holomorphic. At real z, (TI2) makes QZ(z) equal to the original emission at Z. The one-variable identity theorem propagates this constant value along the connected path from zero to it. This proves the claimed agreement and hence the analytic germ. For the transposed tube repeat the argument with the opposite path and with labels permuted by σ. The permutation map is complex linear and commutes with real boosts, so the same argument applies. We have thus constructed both holomorphic germs at every spacelike adjacent pair, with the common spectator-height sequence tending to zero. The next argument identifies the two germs and passes their equality to the boundary exchange relation. 12.24 Identifying the germs and realizing their partial boundaries First consider a real configuration whose time coordinates are zero, with the adjacent pair spacelike, and the spectator profile y chosen above. The associated Euclidean configuration q = ((xk, yk))k has no collisions: every pair other than the flattened pair has different heights, and the flattened pair has different spatial positions because it is spacelike. Use the totally real Wick slice M(q) = (qk, iq0 k)k. Its image spans the full complex configuration space over C. Indeed each complex configuration is M(u)+iM(v), by taking spatial real and imaginary parts and, in the time coordinates, respectively the imaginary part and the negative real part. 317 ## Source page 320 A Wilson Yang–Mills construction J. Rodgers Let G+ and G−be the two holomorphic germs previously constructed at M(q). On the open Euclidean side with strictly ordered heights, G+◦M equals the original smooth Euclidean hierarchy Sv. On the different open side where the transposed heights are strictly ordered, G−◦M equals the same function: the strict-wave identification gives the permuted Euclidean hierarchy there, and permutation symmetry identifies it with Sv. Both open sides accumulate at q, since the tied heights can be separated in either direction. For each order m, differentiate on the relevant open side and approach q. Continuity of the m-th derivatives of the analytic pullbacks and of the smooth hierarchy proves Dm(G+ ◦M)(q) = DmSv(q) = Dm(G−◦M)(q). The chain rule identifies these real derivatives with restrictions of the complex m-linear derivatives of G± to arguments in the image of M. Complex multilinearity and the spanning property extend equality to all complex arguments. Thus all complex derivatives agree. The convergent analytic Taylor series give G+ = G− on a neighborhood of M(q). (GE1) Notice that equality was not assumed on a common open part of the two Euclidean sides; equality of their limiting jets is what proves (GE1). We also need to know what the germs represent as distributions. The following local argument supplies that fact. Suppose E is an interior tube function, G is a holomorphic germ through X +iY , and the interior slices have a weak distributional limit Q as their heights approach Y . Choose a small ball of real configurations about X and a small height neighborhood such that G is defined and bounded in their product, and agrees with E at the interior points there. For a joint compact smooth test φ supported in that ball, dominated convergence with the majorant C|φ| gives Q(φ) = Z G(X′ + iY )φ(X′) dX′. (GE2) The height approach is nonempty because the base point belongs to the closure of the tube. Holo- morphy gives continuity of G, and the already established damped-distribution boundary theorem supplies the weak limit. Thus (GE2) identifies that particular partial boundary; it does not merely define a new distribution. Apply this argument separately to the original and transposed tubes. In the second application reindex all real integration variables by the adjacent transposition. This map preserves Lebesgue measure, is its own inverse, and sends the test to its corresponding pullback; it also permutes the imaginary heights. Therefore the transposed partial boundary is represented locally by the second germ in precisely the same original variables. This proves the local representation of both partial distributions by their respective germs. Combined with (GE1), it proves their local equality at the zero-real-time spacelike configurations. The extension from these configurations to general spacelike pairs and the final zero-height limit are the next steps. 12.25 Independently heated word vectors and their Gram limit The positivity argument uses actual vectors in the reflection-positive chamber. For a positive ordered height profile y, choose a width p smaller than its initial height and its gaps. The buffered shot vector is ξp,y,v(X) = R(z1, p, x1)Bp(v1)R(z2 −z1, 2p, x2 −x1)Bp(v2) · · · R(zn −zn−1, 2p, xn −xn−1)Bp(vn)Ω, 318 ## Source page 321 A Wilson Yang–Mills construction J. Rodgers where zi = ti(X) + iyi. At arity zero it is Ω. Recursive contraction estimates give ∥ξp,y,v(X)∥≤∥Ω∥ Y i ∥Bp(vi)∥. (HV1) It is strongly continuous in X: the heat factors vary in operator norm while their parameters remain in the open half-plane; the spatial translations are strongly continuous; induction handles their application to the recursively varying ket. If the real times vanish, the shot is exactly the planted future word at the Euclidean positions (xi, yi). This follows by the insertion and translation recursion used for (SW4), now with the initial reference width zero. Its inner product with a second shot, of independent width q and positive profile z, is the Euclidean hierarchy of the first word reflected and reversed, followed by the second. The first block of heights is (−yn, . . . , −y1), the second is z; the first labels are Jvn, . . . , Jv1, with J the induced adjoint on union labels. The same identity holds for arbitrary real times. Multiply the first array of real times by ̄c, and the second by c, in a complex parameter c. Antilinearity of the first Hilbert slot makes the scalar inner product holomorphic in c. There is an open horizontal strip containing both zero and one where all initial-height and gap inequalities for these fixed widths hold. The hierarchy side is holomorphic on that strip as well. At purely imaginary c both shots are Euclidean planted words, with the corresponding shifted heights, so the preceding Gram identity holds there. The identity theorem gives it at c = 1. Thus ⟨ξp,y,v(X), ξq,z,w(Y )⟩= f(Jvn,...,Jv1,w1,...,wm), (−yn,...,−y1,z1,...,zm)(Xn, . . . , X1, Y1, . . . , Ym). (HV2) In particular no equality of the two widths was needed. The real coordinates in the reversed first block are reversed without a further geometric reflection; the imaginary heights have already changed sign. For a smeared word b = ((vi, φi))n i=1, let φb(X) = Q i φi(Xi), take yi(t) = it with slots numbered from one, and take p(t) = min(t, 1)/8. Define ηb(t) = Z φb(X)ξp(t),y(t),v(X) dX, t > 0. This is an absolutely integrable Hilbert-valued integral by (HV1) and Schwartz integrability. Trans- lating every real argument by a common a translates the shot by the chamber’s real-time translation unitary. For the empty word this uses invariance of Ωunder both the spatial group and the real- time edge group: invariance under heat first extends to the holomorphic half-plane by analytic uniqueness, and then to its real-time edge. For nonempty words the initial propagator changes by that edge translation, whereas all relative propagators are unchanged. Change of variables in the integral therefore gives ηa·b(t) = U(a)ηb(t). (HV3) Apply (HV2) under the two integrals. Their absolute integrability follows from the same fixed- height bounds. The map (X, Y ) 7→(Xn, . . . , X1, Y1, . . . , Ym) is a real linear isometry and preserves Lebesgue measure. The product test becomes φb(X)φc(Y ), exactly the test of the reversed adjoint word followed by c. Hence the all-height boundary theorem gives the independent two-parameter limit lim t↓0 u↓0 ⟨ηb(t), ηc(u)⟩= W(b∗c). (HV4) The joined profile is strictly ordered for all t, u > 0 and tends to zero as the pair tends to zero, which justifies using that boundary theorem. The independence of the parameters is stronger than a diagonal limit and is what will make the shot vectors Cauchy. 319 ## Source page 322 A Wilson Yang–Mills construction J. Rodgers 12.26 Density of product tests Finite sums of product Schwartz tests are dense in the joint Schwartz space. Here is the concrete approximation used. First take a compactly supported joint test φ, and choose a product cutoff χ(X) = Q i χi(Xi) equal to one on its support. Approximate its Fourier transform by compactly supported Fourier cutoffs. Inverting and multiplying by χ gives Schwartz convergence to φ. Each approximant is an integral over a compact frequency set of the tests χ(X)ei⟨q,X⟩= Y i h χi(Xi)ei⟨qi,Xi⟩i . This is a continuous family in Schwartz topology: all derivatives vary jointly on a common compact spatial support, giving uniform control of every seminorm. Quadrature on the compact frequency set approximates its integral in each seminorm by finite sums of these product tests. Thus each approximant is in the closure of the product-test span, and so is φ. Finally smooth compact spatial cutoffs converge to any Schwartz test, proving density in general. At arity zero the product test is the constant one on the one-point configuration space and already spans that space. This density permits subsequent joint-test conclusions to be recovered from word-vector identities. 12.27 Spacelike exchange on joint tests and on words Two germs anchored to the same tube at a point of its closure agree. To prove this, take a ball on which both extensions are holomorphic and agree with the tube function in that ball’s intersection with the tube. This intersection contains a point because the base point belongs to the tube closure; it is open because the tube is open. Analytic uniqueness on the connected ball makes the extensions equal there. This also proves propagation along a continuous path of base points in the closures of two tubes. At each base point take the two germ extensions on a ball. At nearby base points their germs agree with the germs selected there, by the preceding uniqueness statement. Whether the two extensions agree is therefore locally constant along the path: agreement propagates across the ball by analytic uniqueness, in either direction. On a connected parameter interval it is constant. Given any real configuration X with the designated pair spacelike, contract all its real time coordinates to zero: Xk(s) = (xk, stk), 0 ≤s ≤1. The pair remains spacelike, since s|tj −ti| ≤|tj −ti| < ∥xj −xi∥. With the fixed spectator heights, the path X(s) + iey belongs to both tube closures. Both germs exist throughout it. Their equality at s = 0, proved in (GE1), therefore propagates to s = 1. Thus the two local extensions agree near every spacelike configuration. Equation (GE2) now identifies the two partial distributions locally on a sufficiently small real ball. Their difference vanishes there on every compact smooth test. Consequently its distributional support misses every point of the spacelike-pair open set, and it vanishes on all Schwartz tests whose closed support is contained in that set. Apply this to every profile y(m). The damped distribution at ey(m) converges, on each fixed test, to the undamped boundary distribution: the closed-height multiplier is continuous in Schwartz topology, and its zero-height cutoffis one near the frequency support. The same is true after the adjacent transposition. Since the two partial distributions agree for every m, uniqueness of their scalar limits gives Wv(φ) = Wv◦σ(φ ◦σ) if supp φ ⊂{X : Xj −Xi is spacelike}. (LC1) This is the joint-test exchange identity with all spectator variables present. 320 ## Source page 323 A Wilson Yang–Mills construction J. Rodgers For product tests φ = Q k φk, its closed support is contained in the inverse image of supp φk under every coordinate projection. If the supports of φi, φj are mutually spacelike, that joint support lies in the set in (LC1). Permuting the joint product permutes exactly those two test factors. Place these factors between any two spectator word lists to obtain W a (v, φi)(w, φj) b  = W a (w, φj)(v, φi) b . (LC2) No support restriction is placed on the spectator tests. This is the word relation that will give locality on the common field domain. 12.28 The label quotient remains faithful at the boundary Suppose a union label x is a boundary marker: every word expectation containing its smearing, with arbitrary words before and after, is zero. Then every joint distribution with that label in any prescribed slot is zero. It vanishes first on product tests, since a word can be split just before that slot, and then on all joint tests by product-test density and continuity. Represent x in a finite quotient field space by a raw label w. The full-tube emission for any distribution containing x is zero, so (TI1) makes its strictly time-ordered Euclidean reading zero. This remains true after any permutation of the arguments, because the marker may occupy any slot. For a general Euclidean configuration of distinct points, sort its time coordinates. They are weakly ordered, possibly with ties. Approach the sorted heights by strictly ordered heights while keeping spatial coordinates fixed. The configurations remain distinct near the given one. Smoothness of the hierarchy there makes their readings converge to the given reading, which is therefore zero. Undoing the sorting permutation proves vanishing at every distinct configuration. The Euclidean null-space criterion now says that w belongs to its finite-level null subspace. The selected hierarchy has exactly the retained null subspaces of the original hierarchy, so its class in the original quotient is zero. Hence x = 0 in the common field union. This proves faithfulness of the boundary label quotient; it is not an additional field assumption. 12.29 Boundary positivity and the common field domain The common domain is the complex span of finite smeared field words applied to the vacuum. Field multiplication, adjoints and spacetime covariance are asserted on that domain. Its density follows from the cyclic construction, and invariance follows from adjoining another smeared field or moving every test by the spacetime action. No extension to a common bounded-operator algebra is claimed. For each word b, (HV4) with c = b makes ηb(t) Cauchy as t ↓0. Indeed expand ∥ηb(t) −ηb(u)∥2 = ⟨ηb(t), ηb(t)⟩−⟨ηb(t), ηb(u)⟩−⟨ηb(u), ηb(t)⟩+ ⟨ηb(u), ηb(u)⟩. Every term has the same limit by the independent two-parameter convergence, so the expression tends to zero. Completeness gives a vector ηb. Taking independent limits in (HV4) yields ⟨ηb, ηc⟩= W(b∗c), η∅= Ω, ηa·b = U(a)ηb. (FR1) Consequently the kernel K(b, c) = W(b∗c) is Hermitian and positive: for any finite coefficients, X α,β zαzβK(bα, bβ) = X α zαηbα 2 ≥0. 321 ## Source page 324 A Wilson Yang–Mills construction J. Rodgers The empty expectation is one because the chamber vacuum has norm one. Form the vector space of finite formal sums of smeared words, with sesquilinear form K, linear in its second argument. Divide by its zero-norm subspace and complete. Denote the resulting Hilbert space by HW , the word classes by [b], and their algebraic span by D. This span is dense, and ΩW = [∅] is a unit vector. The common operator domain will be exactly D. For a letter l= (x, φ), left insertion on formal sums is Ll[b] = [lb]. The word involution gives ⟨Llp, q⟩= ⟨p, Ll∗q⟩, l∗= (Jx, φ). (FR2) This identity also proves insertion preserves the null space. If [p] = 0, choose a formal sum q representing the vector Llp in D; then (FR2) makes ∥Llp∥2 = ⟨[p], Ll∗[q]⟩= 0. Thus Φ(x, φ)[b] = [(x, φ)b] (FR3) is a well-defined linear operator D →D. It is bilinear in x, φ. To prove this directly, pair the proposed additive or scalar relation with every word vector. The corresponding word expectation is linear in the inserted label and test, since the joint distribution is multilinear in its labels and the product test is multilinear in its factors. All pairings vanish, and density proves the relation. There is no boundedness claim for the fields themselves. Equation (FR2) is the adjoint relation on the common domain. Repeated application of (FR3) gives the word vectors, and their vacuum expectations are the prescribed W’s. Cyclicity follows because D is exactly their span. The smearing map φ 7→Φ(x, φ)ψ is norm-continuous for each ψ ∈D. For a word vector, continuity of the diagonal and mixed joint-distribution pairings implies continuity of the norm difference by its squared-norm expansion; finite linear combinations give the assertion for ψ. The jointly continuous two-smearing Gram pairing used here follows from continuity of product tests in Schwartz topology and continuity of each fixed-label joint distribution. The word exchange identity (LC2), paired against all word probes, gives [Φ(x, φ), Φ(y, ψ)] D = 0 when the two test supports are mutually spacelike. Finally if Φ(x, φ) = 0 on D for every φ, all expectations with x in any slot vanish: pair its zero insertion against the reversed adjoint left spectator word. Involutivity of J removes that double reversal. The boundary marker criterion already proved gives x = 0. Thus the field representation retains the full prescribed label quotient. 12.30 Unitary spacetime action on the reconstructed space For a Poincaré transformation g, transform each letter by its finite-dimensional label action and in- verse placement of its test. Boundary covariance preserves K(b, c); the adjoint on letters commutes with this transformation. Thus U(g)[b] = [g · b] extends isometrically to HW . Its inverse is U(g−1), and the word transformation law gives the representation law on the dense span and hence everywhere. Its strong continuity can be proved without a topology on the full label union. Pin the finitely many labels in one finite field space and expand the multilinear distribution in a fixed basis there. Each moved expectation is a finite sum of products of continuous label coordinates and continuous fixed-label distribution pairings with moved tests. Strong continuity of test motion was proved in (BC4). Hence all mixed word-vector pairings vary continuously. The squared-norm identity gives norm continuity of U(g)[b], and then of finite sums. To extend to any vector, approximate by such a 322 ## Source page 325 A Wilson Yang–Mills construction J. Rodgers sum within ε/3; isometry controls the two approximation errors uniformly in g. This proves strong continuity on the Hilbert space. The domain D is invariant, the empty word fixes the vacuum, and insertion transforms by U(g)Φ(x, φ)U(g)−1 = Φ(ρ(g)x, g · φ) on D. (FR4) The isometric attachment to the original reflection-positive chamber is also concrete: map [b] to ηb in (FR1), extend by continuity, and use equality of all Gram pairings to see that it is an isometry. It sends the vacuum to Ωand intertwines translations by (FR1). The spectral and vacuum conclusions can therefore be transferred through this attachment by the arguments below. 12.31 Coordinate boosts, the spectral cone, and attachment We supply the group-generation fact used above. For a strictly future timelike vector x, its time component T is positive and every spatial component xa satisfies |xa/T| < 1. The coordinate boost of rapidity s = −artanh(xa/T) sets that component to zero, since T sinh s+xa cosh s = 0. It preserves all other spatial components and keeps the vector in the strict future cone. Three successive coordinate boosts therefore bring any future unit timelike vector to the unit time vector e0. For any proper orthochronous Lorentz transformation g, apply these boosts to ge0, obtaining kge0 = e0. A Lorentz transformation fixing e0 preserves the time coordinate, by invariance of the Lorentz pairing. Since it also preserves the Minkowski quadratic form, it preserves the Euclidean norm. It is therefore an orthogonal spatial turn, with the original determinant. Thus kg is such a spatial turn and g = k−1(kg) is a finite product of coordinate boosts and spatial turns. This proves the generation statement used to obtain full boundary covariance. For precision, the absolute-frequency cone is initially defined as the intersection, over all Lorentz transformations r, of the inverse images under r∨of the time cone in (FT3). Boundary covariance and the Fourier change-of-variables formula show that the Fourier distribution is supported in every one of these inverse images: pull a test disjoint from one inverse image through r∨, use the time-cone support for the transformed labels, and transform back. Hence it is supported in their intersection. This intersection has the suffix-momentum description used in (DC1). For any suffix, test the time cone against the weakly increasing step height that is zero before that suffix and one on it. Its time component is nonnegative. Doing this after every Lorentz transformation makes the suffix vector future causal: a vector with nonnegative transformed time component under every proper orthochronous Lorentz transformation is in the closed forward cone. To see the converse characterization of a single vector, if its time is negative the identity already excludes it; if its spatial norm exceeds its nonnegative time, a boost toward its spatial direction makes its time negative. Constant negative as well as positive heights force total time momentum zero. The total suffix is future causal and has zero time, so its spatial part is zero too. Conversely, zero total momentum and future-causal proper suffixes imply all the time-cone inequalities by telescoping and imply them after every Lorentz transformation. This proves the description, with no change to relative coordinates. For the field Hilbert space let P be the joint projection-valued measure of its strongly continuous translation representation, with convention ⟨v, U(a)v⟩= Z ei⟨p,a⟩dμv(p), μv(B) = ⟨v, P(B)v⟩. 323 ## Source page 326 A Wilson Yang–Mills construction J. Rodgers The translation attachment in (FR1) is an isometry I into the reflection-positive chamber and intertwines translations. Equality of these scalar characteristic functions and uniqueness of Fourier transforms of finite measures give equality of the diagonal measures μv with the chamber’s diagonal translation measures at Iv. The latter have nonnegative time support, as established by the semigroup argument below. Lorentz covariance of the unitary representation gives μU(r)v = (r∨)∗μv. (SP1) This follows by conjugating translations and comparing characteristic functions, using preservation of Hilbert inner products. If p is in the support of μv, then r∨p is in the support of the right side of (SP1), and hence has nonnegative time component for every r. The characterization just proved makes p future causal. Thus every diagonal measure is supported in the closed forward cone, and polarization gives P(V c +) = 0. (SP2) This is the spectral condition for the actual field translation representation, not merely for the hierarchy distributions. 12.32 Reflection-positive heat contractions The chamber heat flow is obtained from future-word translation, rather than postulated. On finite sums p of future words let Xp be the reflection-positive word vector and Ttp its forward translate through t ≥0. Translation and reflection of the Euclidean hierarchy give ⟨XTtp, XTuq⟩= ⟨Xp, XTt+uq⟩. (OS1) Its fixed-word correlation bound implies supt≥0 ∥XTtp∥< ∞for every finite sum: first bound each individual translated word norm by the square root of its reflected two-word correlation bound, then sum the finitely many coefficient-weighted bounds. Put c = supt≥0 ∥XTtp∥. Equation (OS1) and Cauchy–Schwarz imply ∥XTtp∥2 = |⟨Xp, XT2tp⟩| ≤∥Xp∥c. Taking the supremum and using c ≥0 yields c ≤∥Xp∥. Thus forward translation preserves null vectors and extends to a contraction E(t) on the completed reflection-positive Hilbert space. Equation (OS1) makes E(t) self-adjoint. Translation parallel to the reflection plane preserves the Gram form, extends to an isometry, and its opposite translation is its inverse. These give the commuting spatial unitary group. The semigroup and commutation laws hold on planted words and hence everywhere. Strong continuity follows first on planted words: the squared norm of the difference of two translated vectors is a sum of continuous Euclidean correlation functions. Finite sums inherit continuity; their density and the uniform contraction bounds extend it to the Hilbert space by approximation. The same argument proves strong continuity of the spatial group. This constructs the self-adjoint contraction semigroup and commuting spatial translations required throughout the wave construction. 12.33 Holomorphic heat powers and the real-time edge Let E(t) be the self-adjoint contraction semigroup just constructed. Each E(t) is positive, since E(t) = E(t/2)2, so its spectrum is contained in [0, 1]. Put A = E(1). For Rz > 0, define F(z) = Az 324 ## Source page 327 A Wilson Yang–Mills construction J. Rodgers by continuous functional calculus, with 0z = 0. The function λ 7→λz is continuous on [0, 1], including at zero. On compact subsets of the right half-plane it varies continuously in uniform norm; point evaluations are holomorphic in z. Equivalently, uniform estimates for λRz| log λ|k on smaller half-planes justify its derivatives. Thus F is holomorphic in operator norm there. Functional calculus gives ∥F(z)∥≤1, F(z + w) = F(z)F(w), F(z)∗= F( ̄z). (HE1) It also commutes with the spatial translations. These powers coincide with the supplied heat flow. For positive integers p, n, let B = E(1/n), so A = Bn and E(p/n) = Bp. Functional calculus on the nonnegative spectrum and the scalar equality (λn)p/n = λp give F(p/n) = E(p/n). Approximate any positive t by positive rational numbers; operator-norm continuity of F and strong continuity of E, applied to each vector, give F(t) = E(t). Every E(t) is injective. If E(t)v = 0, the semigroup law gives E(s)v = 0 for s ≥t. The holomorphic vector function F(z)v consequently vanishes on a real interval in the right half-plane, hence everywhere there by analytic uniqueness. Strong continuity at zero then gives v = 0. Self- adjointness implies dense range: the orthogonal complement of the range is its kernel. For real s, (HE1) gives ∥F(r −is)v∥= ∥E(r)v∥ (r > 0), because both squared norms are the quadratic form of E(2r). Define an isometry on the dense range of E(1) by V (s)E(1)v = F(1 −is)v. It extends uniquely to the Hilbert space. Checking on that dense range with (HE1) gives V (s)F(z) = F(z −is) = F(z)V (s), V (s)V (t) = V (s + t), V (0) = 1. Therefore V (s) is unitary, with inverse V (−s). It commutes with spatial translations. Its strong continuity holds on the range of E(1) by operator-norm holomorphy of F, and on all vectors by density and the isometric bound. Together these operators form the chamber’s strongly continuous real spacetime translation group. The sign is F(r −is), consistent with the spectral convention ei⟨p,a⟩. 12.34 The Gaussian half-plane support argument We prove the precise support fact used for the chamber diagonal spectral measures. Suppose a finite positive measure μ, a continuous real function ω, and a bounded function h, continuous on the closed upper half-plane and holomorphic in its interior, satisfy h(t) = Z eitω(q) dμ(q), |h(z)| ≤C. For b ∈R, ε > 0, multiply by e−ibz−εz2. Cauchy’s theorem on rectangles between imaginary heights zero and s ≥0, followed by sending their vertical sides to infinity, gives Z R h(t)e−ibt−εt2 dt ≤C q π/ε ebs+εs2. (PA1) 325 ## Source page 328 A Wilson Yang–Mills construction J. Rodgers The vertical side integrals vanish because their norms are bounded by a constant times e−εT 2; the horizontal integrals are absolutely integrable Gaussian majorants. Fubini and the scalar Gaussian Fourier integral also give Z R h(t)e−ibt−εt2 dt = q π/ε Z e−(ω(q)−b)2/(4ε) dμ(q). (PA2) If a support point q0 had b = ω(q0) < 0, the open set U = {q : (ω(q) −b)2 < b2/4} would have finite positive measure a. Choose s = −b/(2ε) > 0 in (PA1). Equations (PA1)–(PA2) and positivity imply a e−b2/(16ε) ≤ Z e−(ω−b)2/(4ε) dμ ≤Ce−b2/(4ε). Thus 0 < a ≤Ce−3b2/(16ε), impossible as ε ↓0. Hence supp μ ⊂{ω ≥0}. Apply this to the chamber’s diagonal translation measure with ω(q) = q0. Its required upper- half-plane extension is h(z) = ⟨v, V (Rz)E(Iz)v⟩, Iz ≥0. It is continuous to the edge, holomorphic inside because it equals ⟨v, F(−iz)v⟩, and bounded by ∥v∥2. At real z it is precisely the characteristic function of the time marginal of the joint translation measure. The support argument proves nonnegative time energy. Combined with (SP1), it completes the forward-cone spectral conclusion (SP2). 12.35 The ground compression algebra and its probability space The phase space is the spectrum of the commutative ground compression algebra. Its probability measure is obtained from the vacuum state on that algebra. Character evaluation gives the phase moments; their barycenter returns the unselected hierarchy. The countable common family of probes and the uniform phase bounds are what allow positivity, regularity and vacuum uniqueness to be asserted simultaneously. Let H0 be the heat-fixed subspace of the unselected chamber and P0 its orthogonal projection. Compress each slab operator to that subspace: m(w, d) = P0 place(d, w) H0. Take the norm-closed unital star algebra generated by these operators. Once its commutativity and countable norm density are established below, it is a separable commutative C∗-algebra A. Its character space Θ is compact and metrizable. To see metrizability explicitly, take a dense sequence ak ∈A. The map χ 7→(χ(ak))k is continuous and injective: equality on the dense sequence extends to all of A by continuity of each character. A continuous injection from the compact character space into a countable product of complex planes is an embedding. The vacuum vector state has the representation ⟨Ω0, aΩ0⟩= Z Θ χ(a) dν(χ). (PH1) Here the Gelfand representation identifies A with continuous functions on Θ, and the representation of a positive functional by a finite measure supplies ν. It is a probability measure since ∥Ω0∥= 1. It has full support. The vacuum is cyclic for A on H0, as follows from the compression identity below. Commutativity then makes it separating: if aΩ0 = 0, then a(bΩ0) = b(aΩ0) = 0 on a dense set, so a = 0. If a nonempty open subset of Θ had measure zero, choose a nonzero continuous bump 326 ## Source page 329 A Wilson Yang–Mills construction J. Rodgers supported there. Its squared integral is zero, so the represented operator kills Ω0, contradicting this separating property. Thus every nonempty open set has positive measure. The compressed operator of a distinct word is independent of how it is placed in a slab. Increas- ing the slab width adds heat evolution on its right, which fixes the input ground vector. Increasing its initial time adds heat evolution on the left, which disappears under P0. Spatial translation dis- appears on both sides because ground vectors are spatially invariant. Positive or negative common time translations reduce to these identities by comparing the two placements. Denote the resulting algebra element by M(w). A component hierarchy Sχ is specified by Sχ(w) = χ(M(w)). (PH2) For a future word b, let Xb : H0 →H be its slab insertion with ground input. This is independent of the trailing slab width, and XbΩ0 = [b], M(b)Ω0 = P0[b], M(b∗c) = X∗ b Xc. (PH3) The last equality is slab concatenation and the reflected adjoint identity. Since the future-word vectors span densely, their ground projections do too; the middle equality gives cyclicity of Ω0 for A. Equations (PH1)–(PH2) also give the barycenter of every distinct Euclidean word: Z Θ Sχ(w) dν(χ) = S(w). (PH4) Choose any common slab placement to prove this equality, apply (PH1), and use the vacuum slab expectation and translation invariance. The source constructs component hierarchies satisfy- ing (PH2); their existence and regularity must use these same compression elements, rather than arbitrary disintegrated point values. 12.36 Almost-everywhere one-dimensional heat-fixed space For a future word b in a component chamber, set aχ = P0,χ[b]χ. The heat limit gives ∥aχ∥2 = lim s→∞∥Eχ(s)[b]χ∥2. Use integer s. Each expression before the limit is a continuous character reading of the reflected translated word. It is bounded by its value at s = 0, a continuous integrable function on the compact character space, because heat is contractive. Dominated convergence and (PH4) therefore give Z Θ ∥aχ∥2 dν = ∥P0[b]∥2. On the other hand, multiplicativity of a character and (PH1) imply Z Θ |Sχ(b)|2 dν = ∥M(b)Ω0∥2 = ∥P0[b]∥2. The vacuum pairing satisfies ⟨Ω0,χ, aχ⟩= Sχ(b). Since the component vacuum has norm one, ∥aχ −Sχ(b)Ω0,χ∥2 = ∥aχ∥2 −|Sχ(b)|2 ≥0. Its integral is zero by the preceding identities, so it vanishes almost everywhere for this word. Choose one countable sequence of future words whose linear spans of planted vectors are dense in every component chamber; the source obtains such a sequence from the countable probe con- struction, rather than assuming separability of arbitrary label sets. Intersect the countably many 327 ## Source page 330 A Wilson Yang–Mills construction J. Rodgers full-measure sets just obtained. For each character in that intersection, the projections of all those word vectors lie in the vacuum line. Their projected span is dense in the heat-fixed space, because the ground projection is continuous and onto. The vacuum line is finite dimensional and closed, so the heat-fixed space equals it. This proves the almost-everywhere vacuum property and sup- plies an admissible selected character. The earlier full-support continuity argument upgrades the corresponding final reconstructed uniqueness statement to every character. 12.37 Spatial invariance of the heat ground space Write P0 = limt→∞E(t) strongly. The existence of this limit and its identification with the heat- fixed projection follow from the positive contraction spectral calculus: on the spectrum of E(1), the integer powers λn decrease to 1{1}, and strong dominated convergence applies; general times follow from the semigroup law and contraction bounds. Its range is exactly the vectors fixed by all E(t). For any two distinct Euclidean words p, q, place their reflected and untranslated copies in the positive half-space. Their correlation with separation x, for sufficiently large time component, has the form S(p τxq) = ⟨v, U(x)E(x0 −r −s)w⟩ (GI1) for fixed future-word vectors v, w and fixed offsets r, s. This is an exact translation and reflection identity. Along any sequence with x0 →∞, adding a fixed purely temporal displacement ue0 changes (GI1) by a quantity tending to zero: both heat vectors converge to P0w, and the intervening, possibly varying spatial unitaries preserve the norm of their difference. The same limiting insensitivity holds when adding any fixed b with b0 > 0, along the pure- time separation te0. Choose a Euclidean orthogonal transformation sending b to ∥b∥e0 whose transformed e0 has positive time component. Such a transformation is obtained by reflection in the bisecting hyperplane between b and ∥b∥e0; preservation of their inner product gives the positive time component. Rotate both clusters and their displacements. The previous purely temporal comparison applies because the rotated large separation still has time tending to infinity. Euclidean covariance then gives S(p τte0+bq) −S(p τte0q) −→0. (GI2) Given a spatial displacement a, apply (GI2) to b = a + e0, and subtract the purely temporal comparison for adding e0 to the sequence a + te0. This proves that adding a has no effect on the limiting correlation. Taking heat limits in (GI1) yields ⟨v, U(a)P0w⟩= ⟨v, P0w⟩ for all future-word vectors v, w. Density implies U(a)P0 = P0, so every ground vector is spatially invariant. This argument does not assume spatial invariance while comparing the varying large separations. 12.38 Long cluster products and commutativity Let w1, . . . , wr be distinct-point cluster words, and let their displacements dk(N) have pairwise ordered time separations tending to +∞. Then S r Y k=1 τdk(N)wk ! −→⟨Ω0, M(w1) · · · M(wr)Ω0⟩. (GI3) 328 ## Source page 331 A Wilson Yang–Mills construction J. Rodgers To prove this, first place each word in the positive half-space by a fixed shift, compensated in its displacement. Induct from the last cluster backwards. Between successive cluster insertions there is a heat duration tending to infinity and an arbitrary spatial translation. If vectors vN →v and durations tN →∞, contractivity and the just-proved spatial invariance give U(aN)E(tN)vN −→P0v. Indeed its difference from P0v has norm at most ∥vN −v∥+ ∥E(tN)v −P0v∥. Apply the fixed bounded insertion operator and continue the induction. Vacuum pairing gives (GI3). The large time separations also ensure that the combined word has distinct points eventually. The four-cluster construction proves the needed exchange of two compressed factors. In the plane of Euclidean time and one spatial coordinate choose displacement slopes (0, 0), (1, 2), (2, 1), (3, 3) for clusters a, b, c, d, respectively, and multiply all slopes by N. Their time order is a, b, c, d. In- terchange those two Euclidean axes: the time order is now a, c, b, d. At finite N, permutation symmetry and Euclidean covariance identify the two combined expectations. Taking (GI3) gives equality between the first compressed product and the rotated second product. Remove the simul- taneous rotation using another application of (GI3), placing the four clusters along a direction with both axis components positive, such as e0 + e1; its time order remains the same before and after the interchange. Therefore ⟨Ω0, M(a)M(b)M(c)M(d)Ω0⟩= ⟨Ω0, M(a)M(c)M(b)M(d)Ω0⟩. (GI4) Choose the exterior words in (GI4) as the reversed reflected word of a future probe and as another future probe. Equation (PH3) and the adjoint relation turn (GI4) into equality of ma- trix elements of M(b)M(c) and M(c)M(b) between projected future-word vectors. Those vectors span densely in H0, so the two bounded operators are equal. Compressed generators are closed under adjoints: reflection in the slab sends one generator to its adjoint. Thus all generators and their adjoints commute. Algebraic star polynomials in them commute, and norm continuity of multiplication extends this equality to their norm closure. This proves commutativity of the actual compression algebra used in (PH1). The resulting spatial ground invariance, cluster limit and commutative compression algebra will now supply one compact phase space and one countable family of future probes for all its characters. 12.39 One countable family of future probes for every phase The countable family used in heat-vacuum selection can be chosen before any character chamber is given. For each arity n and tuple of finite label degrees d = (d1, . . . , dn), use the parameter space consisting of distinct positions in the positive half-space and raw coefficient vectors in the finite-dimensional spaces of those degrees. This is a separable metrizable space; choose a countable dense subset. There are only countably many n, d. The union of the realized future words from these subsets, together with the empty word, is therefore a single countable list. For any smooth symmetric hierarchy and any of its reflection-positive realizations, the planted- vector map on each such parameter space is norm-continuous. Its Gram pairing at two parameters is the hierarchy evaluated on their reflected and joined configurations. These configurations remain distinct because the two blocks are on opposite sides of the reflection plane. To see joint continuity in the coefficients, expand in fixed bases of the finite label spaces. The reading is a finite sum of products of continuous coordinate coefficients and smooth fixed-label hierarchy readings. The 329 ## Source page 332 A Wilson Yang–Mills construction J. Rodgers adjoint coefficients in the reflected block are continuous as well. Thus both the diagonal Gram value and the mixed pairing with a fixed planted vector are continuous; expanding the squared norm difference proves norm continuity. Every future word has some arity, degree tuple and parameter in this list of parameter spaces. Its vector is therefore a norm limit of vectors from the corresponding countable dense subset. Since all future-word vectors have dense linear span, the chosen common countable list has dense linear span in every such chamber. This proves exactly the simultaneous density needed when intersecting the countably many full-measure conditions in phase selection. 12.40 Character readings, regularity, and positive quadratic forms The source constructs the component moment at a distinct configuration X by first forming the compression-algebra-valued multilinear moment md(X; v), and then applying the character: Sχd(X; v) = χ(md(X; v)). (CH1) Outside the distinct-point set it uses the specified zero extension. The algebra-valued moment is the same whole-word compression M(w) used in (PH2), not a product of separate point insertions. Its permutation, unit insertion, common translation, adjoint and quotient-label identities therefore transfer through the character. Once individual Euclidean rotation and reflection invariance of these compression elements is established, those identities transfer too. Smoothness in X is also a consequence of the actual algebra-valued moment. A continuous character on the closed compression subspace extends to a bounded linear functional on the ambient bounded-operator space by the norm-preserving extension theorem. Compose that extension with the smooth ambient operator-valued moment. On the compression subspace it agrees with the original character, proving smoothness of (CH1) on distinct configurations. This argument avoids imposing an extra differentiability structure on the character algebra. Reflection positivity follows by applying χ to one positive quadratic expression. For finitely many future words bi and coefficients ci, set F = P i ciXbi. Equation (PH3) gives the algebra element a = X i,j cicjM(b∗ i bj), a H0 = F ∗F. It is positive in the ambient bounded-operator algebra. Spectral permanence for a closed unital C∗- subalgebra makes its spectrum in the compression algebra nonnegative as well. Since χ(a) belongs to that spectrum, it is real and nonnegative. Expanding χ(a) by linearity gives the component reflection-positive Gram sum. This proves positivity at each character, with no exceptional set. For derivative estimates, suppose positional differentiation is represented by replacing a label vi by its prescribed derivative label ∂avi. At distinct configurations it then holds for iterated derivatives as well, by differentiating the local identity: ∂Xa i DjSχ,v = DjSχ,v[i←∂avi]. If each fixed-label scalar reading has a separation bound Cr−A, induction gives such a bound for every fixed derivative order. Specifically the operator norm of a linear map on joint position space is at most the sum of its norms on the coordinate unit vectors. At the induction step take the maximum of the finitely many exponents supplied by the replacement labels and the sum of their constants. Since 0 < r ≤1, all terms are bounded by that common exponent. Quotient compatibility transfers the result to the raw component labels. All constants remain dependent on the fixed arity, labels and derivative order; no unproved uniformity in those data is introduced. 330 ## Source page 333 A Wilson Yang–Mills construction J. Rodgers The next sections construct the compression-valued moments, prove norm separability and individual Euclidean invariance, and transfer the derivative and shell identities. These supply the premises used in the character formulas above. 12.41 Constructing the compression-valued multilinear moments Fix a distinct configuration and choose one translation and one strict slab containing it. The same translation and slab contain all sufficiently nearby configurations. When varying a label at a fixed position, they also work for every coefficient vector of that label degree, since the slab condition depends on positions alone. For future spectator words p, q, the matrix element of this slab operator is the hierarchy reading of the reflected left word, the inserted slab word, and the translated right word. All these positions are distinct. Move a distinguished insertion to the front by permutation symmetry. Its reading is then an exterior-field functional, with those actual two spectator words as exterior data. By definition of the all-spectator null quotient it depends only on the projected field label and is linear in that label. Equality of every such matrix element implies equality of bounded slab operators, because the future-word vectors have dense span on both sides. Thus the slab operator itself depends linearly on the projected distinguished label. Compression is a bounded linear operation, so the same is true for its element of A. Move any chosen slot to the front to obtain this result at every slot. It proves the multilinearity of md(X; v). Replacing labels one at a time proves that equal projected tuples give equal compression elements. Offthe distinct-point set the zero convention preserves these algebraic identities. The identical matrix-element argument proves permutation symmetry and deletion of a unit label before compression. The empty compressed word is the identity because its slab operator is heat evolution and acts identically on ground inputs. Reflection of the slab gives the adjoint of its compressed word. Common-translation invariance was proved by placement independence. These establish the structural identities used in (CH1) without assuming commutativity or applying a character. Smoothness in position uses the locally fixed translation and slab. On that neighborhood the compression-valued moment is exactly X 7−→P0 place d, τsw(X)  H0. The slab regularity estimate gives smoothness in operator norm at fixed d; the position translation is smooth and the compression map is bounded linear. The resulting smooth function agrees locally with the canonical compression moment by placement independence, proving its smoothness on the entire distinct-point set. 12.42 Norm separability and common phase bounds For each arity and degree tuple, fix basis labels in every slot. The image of the distinct configuration space under its compression moment is separable in operator norm, since that space is separable and the moment map is norm-continuous. The union of these images over all basis tuples, degrees and arities is still separable. Multilinear expansion puts every compressed word in its complex linear span. Finite products of a separable set form a separable set, by continuity of multiplication and countable union over the product length. Include adjoints and finite complex linear combinations, and take the norm closure. These operations preserve separability and generate exactly A. This supplies the countable norm-dense sequence used for metrizability of its character space. 331 ## Source page 334 A Wilson Yang–Mills construction J. Rodgers Characters have the uniform bound |χ(a)| ≤∥a∥. A word with separation r, 0 < r ≤1, can be translated into a slab with one unit of padding on both sides; its mutual separation remains r. Compression has norm at most that of the slab operator. Applying the unselected arbitrary-copy slab estimate therefore bounds the entire character reading with precisely the same copy constants, uniformly over characters and all receiving hierarchies with the same specified word values. For fixed arity n and bounded degrees and coefficient norms, the source’s radius estimate is of the form |Sχd(X; v)| ≤KnnBnr−Bn, with its empty-word convention understood. This gives the fixed-label scalar separation bound required for the derivative induction above. Thus the component bounds come from whole-word operator norms, not from separate uncontrolled character norms. 12.43 The reflection square from bounded lattice approximants For a fixed generic grid word w with real coefficient labels, let wθ be its geometrically time-reflected word with the prescribed reflected labels, and set D = M(w) −M(wθ), a = R⟨Ω0, D2Ω0⟩. The grid construction supplies real measurable approximants f(k) i to widely translated copies of this reflection difference, with |f(k) i | ≤Rk, and pair correlation errors at most εk →0. Fix the approximation index k, a finite number q + 1 of copies, and their displacement parameter before taking the ordered mesh/volume limit. The four terms of each difference-product converge to the corresponding continuum two-cluster readings. Next send the displacement parameter to infinity. Equation (GI3) makes each off-diagonal comparison tend to a + εk. At finite lattice parameters positivity of an ordinary real square gives 0 ≤E q X i=0 f(k) i !2 . Bound the diagonal terms by R2 k, and the two symmetric off-diagonal terms by the prescribed pair comparisons. Passing through the limits in the stated order yields 0 ≤(q + 1) R2 k + q(a + εk) . (RS1) Only finitely many pair conditions are needed at each q, so their eventual tails can be intersected before the limit. If a + εk < 0, choosing q large contradicts (RS1). Thus a + εk ≥0 for every k, and a ≥0. For real labels, the adjoint and permutation identities give M(w)∗= M(wθ), so D∗= −D. It follows that ∥DΩ0∥2 = −R⟨Ω0, D2Ω0⟩= −a. Both sides’ nonnegativity and a ≥0 force DΩ0 = 0. The separating vacuum property of the commutative compression algebra then gives D = 0. To extend the identity beyond the generic grids, choose increasingly fine grids from the common funded approximation sequence and approximate a fixed generic configuration by their canonical grid addresses. The grid points are eventually generic and converge to it. Norm continuity of both compression moments gives equality there. Generic configurations have full measure in the distinct- point region; continuity and full support of Lebesgue measure extend equality throughout that open region. Finally expand the multilinear compression moment in real coordinate basis labels, each fixed by coefficient conjugation. Equality on those basis tuples proves equality for arbitrary complex labels. Thus individual geometric time reflection fixes each compressed observable. 332 ## Source page 335 A Wilson Yang–Mills construction J. Rodgers 12.44 Individual Euclidean invariance of compressed words Long two-cluster limits first give simultaneous rotation invariance of pair stages ⟨Ω0, M(p)M(q)Ω0⟩. If a Euclidean orthogonal map r sends e0 to a vector with positive time component, use clusters escaping along e0 in (GI3). If that component is zero, use instead the direction e0 + r−1e0, which has positive time component both before and after rotation. If it is negative, first compose r with time reflection; individual time-reflection invariance removes the added reflection from each com- pressed factor afterwards. These three cases prove simultaneous pair invariance for every Euclidean orthogonal map. Conjugate the individual time reflection by an arbitrary orthogonal map r. Rotate a pair stage by r−1, reflect its second compressed factor only, and rotate both factors back. The first factor returns to itself and the second has been transformed by rθr−1. Hence its pair readings with every first word agree with their originals. Test with the adjoint word of the second factor and of its transformed copy. The Gram identity then makes the norm of their vacuum-vector difference zero. The separating vacuum makes the compressed elements equal. Every hyperplane reflection is a conjugate of θ, except the identity case of a zero normal vector. Orthogonal transformations in finite dimension are finite products of hyperplane reflections. The individual invariance just proved is closed under multiplication by the label representation law. Therefore M(r · w) = M(w) (r ∈O(4)). This completes the Euclidean rotation and reflection identities used for every component hierarchy in (CH1). The reflection-square argument uses its bounded Wilson approximants before taking limits, and the rotation argument uses the same compressed whole words throughout. 12.45 Inheritance of derivatives and stress-window identities Fix a character component T = Sχ. Extend χ, as a bounded complex-linear functional only, from the closed compression algebra to all operators on the ground space, then compose with compression. This gives a single bounded linear functional φ on chamber operators such that S useAt T(X; v) = φ(place(D, τsw(X; v))) (HW1) whenever the translated distinct configuration lies strictly in the slab (0, D). No positivity or multiplicativity is used for φ outside the original algebra. One common s, D can contain any given finite distinct configuration together with any specified compact set of potential insertion points: take lower and upper bounds for their time coordinates and add a strict margin. This placement persists on a neighborhood of the finite configuration. Inclusion identities for finite field labels hold for the slab operator itself, by pairing with arbitrary future words on both sides and using the original all-spectator inclusion identity. Applying φ proves inclusion compatibility in T. Likewise norm differentiability of the fixed-width slab insertion gives ∂Xa i place(D, τsw(X; v)) = place(D, τsw(X; v[i ←∂avi])). The common translation has derivative the identity on position space. Differentiate (HW1) through the bounded linear functional, keeping that width and translation fixed throughout its neighbor- hood. This proves the component derivative identity used in the preceding separation estimates. 333 ## Source page 336 A Wilson Yang–Mills construction J. Rodgers For stress identities, let χ0 be a compact smooth window equal to a constant near the distin- guished insertion and zero near every spectator insertion. Pack its entire closed support in the same slab. For a continuous scalar weight u, form the operator integral Ia,u = Z (∂aχ0)(y) u(y) place(D, τsw(y, X)) dy. (HW2) This is a norm-integrable operator-valued function. Its support is compact. On the support of its scalar factor, y cannot collide with the distinguished point, because the window is locally constant there; it cannot collide with any spectator, because the window vanishes near each of them. The insertion is therefore norm-smooth locally on that support. Offit the complete integrand is locally zero, even if the bare insertion at a collision was defined arbitrarily. These observations prove continuity and compact support of the full integrand. To prove the required identity for (HW2) as an operator, pair it with two arbitrary future- word vectors. The added exterior spectators lie strictly below or above the whole chosen slab, and hence outside the entire window support. The original stress-window identity therefore applies with these genuine additional spectators. Passing the bounded matrix-element functional through the integrable operator expression proves the operator identity by density. Pass φ through the same integral and use (HW1); this gives exactly the numerical identity in T. In particular, if Tab denotes the prescribed degree-four stress label and YbB the translation derivative of a distinguished label B, the identity is − X a Z (∂aχ0)(y) T(Tab(y), B(X0), v(Xsp)) dy = T((YbB)(X0), v(Xsp)). (HW3) For a skew Euclidean generator g, use ub(y) = [g(y −X0)]b in (HW2) and sum over a, b. The corresponding identity is − X a,b Z (∂aχ0)(y)[g(y −X0)]b T(Tab(y), B(X0), v(Xsp)) dy = T((ZgB)(X0), v(Xsp)). (HW4) The geometric window and the same stress and generator labels are retained. Thus both translation and rotation stress laws transfer at every character. 12.46 Smeared reflection positivity of the components The temporal point Gram positivity already proved transfers to any coordinate reflection plane by a signed coordinate permutation taking that axis to time. This permutation intertwines the two word reflections, preserves the positive side, and preserves the component readings by their real-spin identity. It therefore transports the complete finite Gram sum. Point positivity also implies the required positivity for joint smearing tests. Fix an axis and construct the Gram Hilbert space from its positive future-word kernel. For a compact joint test f(X) supported on positive, mutually distinct configurations, let ζf(X) = f(X)[w(X)] there, and define it as zero elsewhere. Its diagonal and mixed Gram pairings are the corresponding smooth smeared hierarchy kernels. They are continuous, including across the cutoffsupport bound- ary. The squared-norm criterion proves strong continuity of ζf; its compact support then gives an integrable Hilbert-valued function. Fubini and the Gram formula yield Z Z Ksmear(f, g; X, Y ) dX dY = Z ζf(X) dX, Z ζg(Y ) dY  . 334 ## Source page 337 A Wilson Yang–Mills construction J. Rodgers Every finite smeared reflection-positive sum is consequently the squared norm of a finite linear combination of these integrated vectors. This proves nonnegativity for joint tests directly from point positivity, without assuming a full hierarchy in order to prove one of its clauses. 12.47 Assembling the bounded reflection witnesses The real bounded witnesses in (RS1) are obtained from the same local observable approximants used in the continuum comparison. For a real label observable O, suppose a local complex approximant O′ has |O′| ≤A and rooted discrepancy at most ε. Replace it by RO′. The observable O is real for the prescribed real label, and RO′ −O = 1 2(O′ −O) + 1 2(O′ −O). The rooted seminorm is subadditive, homogeneous and invariant under conjugation. Its discrepancy is therefore still at most ε. Local dependence and measurability are preserved, and |RO′| ≤A. This supplies genuinely real approximants, without requiring the cutoffobservables themselves to be uniformly bounded. For a fixed finite list of degree-bounded labels, take products of their real local approximants at the prescribed translated lattice sites. The product comparison applies to the concatenated list of two clusters. Distinct clusters with integer displacement slopes i < j are eventually disjoint: any coordinate difference has a fixed finite contribution plus t(j −i), which tends to infinity. At the finite period, injectivity after wrapping also holds on the period tail. Thus the same product estimate bounds each of the four combinations of an ordinary cluster and its reflected cluster, on the same extraction order. Let f(k) i and g(k) i be the resulting real product approximants for these two clusters. Set the reflec- tion witness to f(k) i −g(k) i . Its uniform bound is the sum of their two product bounds. Expand the pair product into its four terms. If their individual comparison costs are ckA00, ckA01, ckA10, ckA11, the triangle inequality gives total error εk = ck(A00 + A01 + A10 + A11) −→0. All four integrals exist by the local approximant estimates and finite lattice moment bounds. This is precisely the common witness family used in the finite-square argument. Fixed approximation index, fixed finite number of copies, large displacement, and the prescribed ordered mesh/period limits retain the ordering stated there. The source obtains the individual local approximants by first fixing a reporting scale and an admissible finite atlas, then choosing a sufficiently late auxiliary scale. The gain-chart denominator makes the rooted approximation error small: its decaying scale factor dominates the subexponential column and stock factors. It then takes the real part as above and transfers the resulting estimates to the original mesh and period tails, using locality on the common chamber. The following coarse- lens and rooted-approximation argument supplies the gain-chart estimate, with the same local chamber and transported coefficients. 12.48 Coarse lenses, coefficient recovery, and rooted approximation At a fixed coarse scale k, the lens of a label L is an explicit bounded local function: Lk,L(U) = akL1 + 1Healthy(θk,F) Ck,L(U). (AP1) Here ak is the prescribed field scaling factor, L1 its constant coefficient, F the local stencil footprint, and Ck,L the scaled candidate polynomial of its nonconstant coefficients. The other regular activity 335 ## Source page 338 A Wilson Yang–Mills construction J. Rodgers terms of the primary lens are identically zero in this specialization. The candidate is bounded on the healthy compact chart region, uniformly in the finer terminal cutoffand period, and is multiplied by zero outside that region. Thus the bound for (AP1) is |akL1| + Ak,L. Its read set is contained in the coarse window of radius equal to the stencil radius plus four: each inspected edge is within four sites of a stencil point, so the triangle inequality gives the inclusion. This proves the lens locality and uniform bound used for the real witnesses. The finite coefficient inversion in the approximation is quantitative. In the coefficient norm |·|1, suppose the recovered coarse coefficient vectors Jbi satisfy |ei −Jbi|1 ≤1 4 for every coordinate basis vector ei. Let Av = P i viJbi. Then |v −Av|1 ≤1 4|v|1, |v|1 ≤2|Av|1. (AP2) The first inequality follows by expanding the difference and summing coordinate norms, and the second by the triangle inequality. Hence A is injective and, in finite dimension with equal domain and codomain, surjective. For a terminal label L, solve this linear system and use the injective coefficient map J to recover L = X i vi terminal(bi), |v|1 ≤2ecolumnLogs(k)|L|1. (AP3) The last factor is the earlier backward coefficient budget. Denote its right side by stockk(L). Each basis lens, lifted back from scale k to the fine lattice, approximates its recovered terminal observable in the root seminorm, with common error at most stakesJ,m/w. Their linear combination with (AP3) is a fine-lattice approximant B to OL, with ∥B∥∞≤stockk(L)Ak, ∥B −OL∥root ≤stockk(L)stakesJ,m w . (AP4) Back transport preserves the sup bound and sends the coarse window into the prescribed common fine chamber, by the atlas room condition. Subadditivity and homogeneity of the root seminorm prove the second bound after the exact terminal-label decomposition. Thus both locality and the approximation concern the same B. The basis root bound comes from an actual many-copy insertion. Use the separated atlas copies of the observable difference, with the compulsory mark at each copy, and extract the coefficient containing every copy once. Exact RG transport preserves its Wilson integral. The insertion estimate bounds that coefficient by |M| wq ≤eqc, c = (aJ/am)4 treasurym + 5, where q is the number of atlas cells and M is the corresponding Wilson moment of the product of copied differences. The term (aJ/am)4 follows from the exact cell census, and the remaining five units are the insertion and vault costs in this application. Taking the q-th root gives the root estimate ec/w = stakesJ,m/w. The coefficient identity cancels the same positive Wilson partition function on both sides; it is not an estimate in a different normalization. The coefficient construction requires w |am|4(1 + columnsk) gainChart(k, m) ≤1 2ζd,m. (AP5) 336 ## Source page 339 A Wilson Yang–Mills construction J. Rodgers Fix the reporting data J, m, d first. Choose w at the upper value allowed by (AP5). Then the error in (AP4) is a fixed constant times the product of the stock and column factors divided by the gain chart. Those factors have subexponential growth, whereas the inverse gain chart contains the strictly decaying scale factor. Their product tends to zero; hence a sufficiently late k, also satisfying the atlas room condition, makes (AP4) smaller than any prescribed positive tolerance. Fixing that k fixes the sup bound as well, uniformly on all later mesh and period parameters. Taking the real part as above supplies the real bounded approximation needed for the reflection witnesses. 12.49 Transporting the approximation difference with its paid mark The comparison used in (AP4) retains a single common source history. Starting from the primary coefficient L at scale k, the backward coefficient construction chooses a reference label b and a residual history Zi, m ≤i ≤k. Its identities at the upper comparison scale are exp(Zk) = exp(primary L) + exp(history(−b)), forced(Zk) = forced(history(−b)). (AP6) These are exact identities for the evaluated exponential entry and the forced activity profile. They follow by linearity of the history map and cancellation of the reference label in the primary/reference decomposition. In particular the primary contributes no forced profile here. Its residual coefficient is L −Jkb, with norm at most Q/4; the backward cone estimate gives this bound, while the terminal/reference coefficient is bounded by 2E. The remaining residual cost at scale i obeys cost(Zi) + |Zcoeff i |1 ≤E/gainChart(k, i). Transport the mark price from w at m by the common multiplicative schedule. The scalar reserve recurrence and (AP5) imply, throughout m ≤i ≤k, that the transported price times the scaled residual cost is at most ζd,i/2. The same bound pays for the reference-label contribution at k, using its coefficient bound 2E. These are the two half-reserves in the construction. Now insert, at each separated atlas copy, the negative terminal reference observable, transport its marked polynomial exactly from N to k, and multiply there by the corresponding coarse lens insertion. Equation (AP6) identifies the resulting first marked slot and forced slot with the residual- history slots. The paid insertion transport theorem then carries those slots from k down to m, along the same migrating copy positions. The separation and occupancy bounds are obtained from the chosen atlas at each intermediate scale. The resulting insertion at m has the served-source bound with support exponent of that scale and the two vault bounds 2, 1 used in the root extraction. The residual-history estimates pay both bounds; no new history or mark schedule is chosen during this transport. The coefficient identity relating this insertion to the actual fine-lattice discrepancy is algebraic. For a finite census, the new coarse insertions multiply as Q i(1 + ziQi). Extracting a prescribed multi-index coefficient sums over the subset of slots taken from this product; every other slot is the coefficient transported from the negative fine observable. Coefficient extraction commutes with each finite RG push by linearity and integrability. The resulting subset sum is exactly the joint fine/coarse difference polynomial. Integrating the push does not change its integral, and restoring the fine/coarse coupling gives the positive Wilson partition function times the discrepancy moment. This is the identity used before taking the all-copy coefficient in (AP4). Thus the comparison bound, its normalization, and its locality all refer to the same transported difference. The backward-cone estimate used here follows from the one-step coefficient and remainder bounds, as follows. Use the coefficient norm |u|1, including its unit coordinate. At a step of block 337 ## Source page 340 A Wilson Yang–Mills construction J. Rodgers size L, put q = L1/8 and e = q−3. If v is the finer coefficient, let yα = Ld−deg(α)vα, Y = |y|1. All exponents are nonnegative, so |v|1 ≤Y . The step supplies an invertible linear map B, commut- ing with deletion of the unit coordinate, with ∥B∥1, ∥B−1∥1 ≤K, K ≥1. Its coefficient error and remainder costs r, t ≥0 satisfy |u −By|1 ≤e(| core v|1 + t), r ≤K| core y|1 + et. Applying the inverse bound, the triangle inequality, and | core v|1 ≤|v|1 ≤Y , gives Y ≤K |u|1 + e(Y + t) , r ≤KY + et. (AP7) Let Ci ≥1 be the column allowance at scale i, and put ai = 32(1 + Ci)2. The room and bank choices give ai ≥4K2, 4Ke ≤1, 4Kai+1e ≤1, (4K2 + 1)qe ≤1. (AP8) For precision, the bank condition is q ≥100000 M6(1 + Ci+1)2, where M ≥K is the neighboring price bound. It implies q ≥128K2(1 + Ci+1)2 ≥1. Use e = q−3 ≤q−1 and qe = q−2 ≤q−1 to get the last three inequalities in (AP8). The room condition Ci ≥64K2, with the stipulated monotonicity of stage prices, gives the first. Suppose ai|u|1 ≤r. By (AP7) and K ≥1, 4K|u|1 ≤Y + et. Substitute this bound on |u|1 back into (AP7). The first smallness inequality in (AP8) absorbs the resulting Y -term and gives Y ≤4Ket. Consequently ai+1|v|1 ≤ai+1Y ≤t, qr ≤q(KY + et) ≤t. (AP9) The same argument with e = 0 also holds directly. Thus the cone ai|u|1 ≤r propagates from the coarser endpoint to the finer endpoint, while the remainder cost gains the factor q in the opposite direction. To obtain the history to which this estimate applies, start with the given terminal formatted insertion and its remainder bound. Descend the finite interval one scale at a time, using the same one-step marked transport and its linear coefficient decomposition. At each step update only the newly reached component of the history. Previously obtained components and their step identities are retained. This finite backward induction constructs the entire matched history; its terminal density is exactly the prescribed insertion plus the reference history. Subtract the reference coefficient determined at the coarsest endpoint. Linearity makes the residual coefficient at that endpoint zero, while preserving every density and step identity. Its nonnegative remainder cost therefore satisfies the cone there. This explains why the cone is available for the residual history, rather than being an assumption on an arbitrary primary history. Let ci be its remainder cost on the interval [l, k], and vi its residual coefficient. Iterating (AP9) gives ai|vi|1 ≤ci, gainChart(k, i)ci ≤ck, gainChart(k, i) = exp scale(k) −scale(i) 8  . (AP10) 338 ## Source page 341 A Wilson Yang–Mills construction J. Rodgers The identity used in this iteration is gainChart(k, i) = gainChart(k, i+1)L1/8 i . The terminal pairing bound is ck ≤(1 + Ck)w + Ck|vk|1. Since the cone implies 2Ck|vk|1 ≤ck, it follows that ck ≤2(1 + Ck)w. Using |vi|1 ≤ci in (AP10) yields the claimed residual estimate in its explicit form: gainChart(k, i)(ci + |vi|1) ≤4(1 + Ck)w. (AP11) The constant E used in the approximation argument is this terminal bound with its fixed coefficient and stock factors. The numerical choices required for (AP8) can be made before selecting a history or a volume. At each stage choose its column allowance above the room threshold, and choose its block lower bound so that L1/8 exceeds the displayed bank threshold. This is possible since L1/8 →∞; enlarging either the prescribed block lower bounds or the later delays preserves the choice. Equations (AP6)– (AP11), followed by the paid marker transport and root extraction, therefore establish the bounded local approximants on the original common family. 13 Separated and nested operator products An operator-product expansion is an approximation at a fixed finite degree horizon. One homoge- neous coefficient reader is chosen on the physical filtration and used for every phase and truncation. Lower-degree terms are subtracted with that same reader, which is crucial when an expansion is in- serted into a second expansion. The estimates keep the cluster separation and spectator separation parameters explicit. 13.1 The separated expansion and its fixed homogeneous channels Let Ed be the finite-dimensional field space of degree at most d, with the coherent inclusions already constructed. A reader is chosen once for this filtration, its real structure, and its Euclidean rotation action. For finite input degrees di, write ∆= P i di. The expansion proved below has the following precise form. If ∥xi −o∥≤Mr, ∥xi −xi′∥≥ηr (i ̸= i′), and the spectator points satisfy ∥o −yj∥≥c, ∥yj −yj′∥≥c (j ̸= j′), where M ≥0 and η, c > 0, then for each fixed truncation D and slack ε > 0, all sufficiently small r > 0 satisfy Ss(a(x)b(y)) − D X j=0 X α∈Cj Cj,α(x, a; o) Ss(ej,α(o)b(y)) ≤rD+1−∆−ε. (OE1) Here s may be the unselected hierarchy or any character component. The threshold precedes every allowed x, o, y, s. The same coefficients serve every D, spectator and viewer. For each fixed channel, |Cj,α(x, a; o)| ≤rj−∆−ε (OE2) on its corresponding small-radius tail. These are fixed finite-order asymptotic statements; no convergence as D →∞is asserted. 339 ## Source page 342 A Wilson Yang–Mills construction J. Rodgers To define the homogeneous pieces, begin with the real, rotation-averaged finite reader at level m. Apply it to one field at the origin to obtain linear maps Lm,q : Eq →Em. They respect inclusions in the second index, and for q ≤m reproduce that included field. Define recursively Km = 1 − X i 0 the norm is eventually at most e(λ+ε)T . Suppose the cluster is of size e−aT , is internally separated at that rate, and a > 0. If ZH = P≤H(x, a; o), the reader comparison gives An(x, a; o) −An(o, ZH; o) rate at most (|d| −H −1)a. (OE4) This statement holds with every varying rotation, not just each fixed rotation. Integration therefore preserves the rate uniformly: otherwise choose a violating rotation at each violating parameter. The conjugate comparison supplies the second half of the real average. Applying the fixed linear map Kn yields hn −πn,HZH rate at most (|d| −H −1)a. (OE5) In particular, at H = n, all old grades below n are annihilated by πn,n. The remaining old grade n has rate (|d| −n)a; the error (OE5) has the better rate (|d| −n −1)a. Finite summation proves the same rate (|d| −n)a for hn, and hence for every coordinate Cn,α. For completeness, the scalar remainder bookkeeping does not lose a grade. Suppose cluster-to- exterior separation has rate b, exterior internal separation rate c, with 0 ≤b ≤c and b < a. A centered field of degree j, paired with the exterior labels of total weight |e|, costs jb + |e|c. For a varying field of rate t, its cost is t + jb + |e|c: expand in a fixed finite basis and use the fixed-field bounds on each coordinate. Reconstruct ZD by (OE3). The exact identity for the new remainder is W(x, a; y, b) − D X j=0 W(o, hj; y, b) = [W(x, a; y, b) −W(o, ZD; y, b)] + D X j=0 W(o, πj,DZD −hj; y, b). 341 ## Source page 344 A Wilson Yang–Mills construction J. Rodgers The old remainder has rate ξ = |d|a −(D + 1)(a −b) + |e|c. By (OE5), the j-th correction has rate (|d| −D −1)a + jb + |e|c ≤ξ. The finite sum thus has rate at most ξ, uniformly for every varying viewer. This last quantifier is what permits passage to the ground-algebra norm, rather than merely pointwise character bounds. In the separated regime T = −log r, a = 1, b = c = 0, these rates are precisely (OE1) and (OE2), after spending the arbitrary positive slack ε. To make the threshold uniform over all admissible geometries, a failure would select a bad geometry at arbitrarily small radii; restricting to those radii contradicts the just-proved rate along that selected path. This also explains why the coefficient family is common to every truncation and phase. The finite readout comparison in (OE4) follows directly from testing the old scalar remainder. Each readout is a finite sum of fixed exterior configurations with fixed scalar weights. After a varying rotation, these exterior configurations remain a fixed positive distance from the shrinking center and from one another, and their rotated labels and loaded output vectors have rate zero, by compactness of the rotation group. The old scalar remainder at these moving tests has rate (|d| −H −1)a. To allow their moving labels, expand every exterior label in a fixed finite basis; the coordinates have rate zero and multilinearity preserves the rate in a finite sum. Summing the fixed detector weights and output vectors proves the rotated readout comparison. Conjugating the whole correlation and using the hierarchy’s reality identity proves its conjugate version with the same norm. This proves (OE4), including the necessary arbitrary varying rotations. We detail the grading argument supplying the old estimates. At logarithmic scale T, choose lattice levels with mesh exponents 0 < b < a < x < d, where the actual cluster has exponent x. Put τ = |g|d + 4 |I|(d −a). The physical approximation supplies, for an auxiliary horizon q, a centered label Q of degree q, an approximation amplitude, and a survival statement for their actual continuum limits. Its label budget has exponent τ −qa, and its mismatch against all allowed exterior tests has exponent at most τ −(q + 1 16)(a −b). (OE6) The same mismatch is used at every detector; no refitting of the centered field to a detector or phase is allowed. For a mode of degree h ≤q, applying its physical readout to Q costs only τ −ha. For h = 0 this follows from the whole-label budget. For h > 0, the approximation-to-lower-degree estimate gives a label s belonging to filtration degree h −1, with distance from Q bounded by its degree-(h −1) fare times the original budget. The degree-h readout vanishes on s, by coherence and inclusion compatibility. Its value on Q therefore equals its value on Q −s; the bounded linear readout and the fare’s exponent (q −h)a give the asserted exponent. This subtraction is essential: using just the norm of Q would lose all of this graded improvement. Choose q ≥h large enough that (h + 1)a ≤(q + 1 16)(a −b). Testing (OE6) at the finite detector list and adding the preceding readout bound proves that the actual degree-h coefficient has exponent at most τ −ha on the lattice screen. For a truncation at 342 ## Source page 345 A Wilson Yang–Mills construction J. Rodgers degree N, choose instead q ≥N with (N + 1)a ≤(q + 1 16)(a −b). Let the exterior clearance exponent be c < b, its internal exponent be u ≥c ≥0, and its total weight be |r|. The centered degree-h correlation costs hc + |r|u. Reconstructing the centered Q through all its readouts, split its finite expansion into degrees h ≤N and N < h ≤q. For the first group subtract the true coefficients; (OE6) pays each difference and the centered exterior correlation. For the second group use the graded budget τ −ha + hc + |r|u ≤τ −(N + 1)(a −c) + |r|u. The original product-to-Q mismatch has this same upper exponent by the choice of q. The exact sum of these three differences is the desired product minus its degree-N truncation. Thus the lattice remainder has rate at most τ −(N + 1)(a −c) + |r|u. (OE7) All statements retain arbitrary varying viewers throughout. The passage from these lattice screens to arbitrary points uses the smooth interpolation estimate, applied to the coefficient, or to the product minus its fixed truncation. Its hypotheses consist of the rough correlation bound, the screen bound just proved, and the positional derivative bounds for these same functions. For coefficients the rough rate is x|g|, since the detector exteriors stay at positive distance. For remainders it is at most x|g| + |r|u + Nc, obtained by separately bounding the original product and the finitely many truncated terms. The derivative bounds are those of the same hierarchy and finite detector sums. Interpolation gives (OE7) with the displayed τ at arbitrary approaching configurations. Finally choose a = x −δ, d = x + δ, and an intermediate b strictly between the exterior exponent and a. Mesh levels with each of these positive exponents exist on the supplied clock. As δ ↓0, the coefficient exponent tends to (|g| −h)x, and (OE7) tends to |g|x −(N + 1)(x −c) + |r|u. For any strict requested upper exponent choose δ > 0 small enough first, then the mesh levels and auxiliary horizon. This proves the old rate estimates with their exact quantifier order. Translation to a moving center changes no distances or readouts. The empty cluster requires no approximation: its coefficient is the readout of the degree-zero unit, its positive-degree coefficients vanish, and its truncation is exactly the identity insertion in every exterior word. This covers the cases excluded when an anchor in the cluster was chosen above. Here are the details of the smooth interpolation step. Consider a family fT (p, z) of scalar functions, whose label list p has total degree |p|, and suppose every positional coordinate derivative replaces one label by its prescribed derivative, raising total degree by one. Assume that in a corridor of configurations of size and separation exponent a, fT (p, z) has rate ≤a|p| + c, while on screened points of mesh exponent b > a it has rate ≤b|p| + d. Choose an integer n such that (b −a)n ≥c −d. We prove the better rate for every degree by descending induction with the statement that it holds whenever |p|+k ≥n. At k = 0 the rough bound suffices. For the induction step move z to a screened mesh configuration z′ at distance of rate −b. Every point in the ball of that radius remains in the same corridor because b > a. Each derivative there is a function with 343 ## Source page 346 A Wilson Yang–Mills construction J. Rodgers label degree |p| + 1, so the preceding induction statement bounds its rate by b(|p| + 1) + d. The finite coordinate basis bounds the operator norm of the derivative by the sum of its coordinate norms. The mean-value inequality on the ball gives fT (p, z′) −fT (p, z) rate ≤b(|p| + 1) + d −b = b|p| + d. Adding the screened value proves the induction step. Taking k = n includes every original degree. The required uniform derivative bound on this moving ball follows from the pathwise hypotheses without assuming uniformity: if it failed arbitrarily late, select a violating point in each such ball, and use that selected path in the derivative hypothesis. The same selection argument supplies eventual differentiability at every point of the ball. Screened approximation itself is elementary finite-dimensional rounding with finitely many forbidden coincidences: a fixed number of mesh shifts suffices to avoid them, with distance bounded by 2K times the mesh size, where K depends only on the number of cluster positions. Thus the displacement rate is indeed −b. Finally, the derivative replacement identity holds for the actual functions in question. It holds for full correlations by the hierarchy’s derivative law, for a readout by its finite sum of full correlations, and for the remainder by subtraction and the finite product rule (the centered exterior factors do not vary with the cluster positions). No differentiability of a chosen approximating label is used. We also specify the compactness and exterior-price passage in the physical matching step. A finite plot at levels k ≤m < m + 1 ≤j satisfies 8∥xi∥< am+1, ∥xi −xi′∥> 2ajB2 j (i ̸= i′). At sufficiently large periods its wrapped positions retain their actual locations and are separated by B2 j lattice units, while every position belongs to the same central block through level m + 1. The matching estimate therefore produces, along the original extraction order, degree-q labels whose products differ from the original cluster by a paid local insertion of amplitude A, and whose quota at level m is at most p. The quota comparison bounds their norm in the fixed finite-dimensional reference space by Kp. Finite-dimensional bounded compactness gives a nontrivial refining filter and a limit label Q. Both the paid mismatch and the quota bound persist on that filter. This is the precise survival statement used above: it retains the actual refinement filter, geometry, paid insertion, reference horizon and quota, in addition to convergence of the labels. To test that same mismatch against an exterior word, first fix arbitrarily small positive slack ρ. The bounded-copy construction for an exterior field of degree ej gives a per-field envelope μj = (1 + Yj)a−ej−ρ l + 1. At a mesh exponent ξ, its product has rate (|e| + ρ|J|)ξ. Choose auxiliary mesh exponents strictly between the exterior clearance exponent and the matching support exponent, then a further pair strictly above the exterior internal exponent. Their strict gaps ensure all finite support widths and separation requirements for the paid-insertion estimate; any fixed number of fields costs no exponent. The common survival filter makes the paid insertion estimate apply to the very same Q. After the continuum limit the compressed exterior mismatch has rate at most δ + (|e| + ρ|J|)ξ, where δ is the amplitude’s rate. Choose these auxiliary exponents with ξ ↓u, then ρ ↓0, before requiring any eventual bound. This gives exactly δ+|e|u. Contractive evaluation gives the identical bound for arbitrary varying phase viewers. The two-group full-correlation bound follows by the 344 ## Source page 347 A Wilson Yang–Mills construction J. Rodgers same argument with the product of the cluster and exterior envelopes, giving engineering rate |g|a + |e|u. These arguments justify all exterior costs used in the grading and interpolation proof. For the approaching plots, the strict exponent inequalities ensure their two geometric conditions eventually: the cluster size is smaller than am+1/16, its internal spacing exceeds 2ajB2 j , and the selected level indices are ordered. Here Bj is subpower relative to the mesh, so it consumes none of the strict gaps. Consequently the compact matching limit exists at each approaching parameter. Selecting one such limit label pointwise gives the family Q used in (OE6), with the same amplitude and quota rates. The matching accuracy is funded as follows. Let the chosen block level have exponent a, the input mesh exponent d > a, and the exterior support level exponent b < a. The individual input budgets have product rate −|g|d. The many-copy initial amplitude includes the volume rescaling factor of rate −4|I|(d −a), hence is a positive quantity xT of rate −τ, with the same τ = |g|d + 4|I|(d −a) as above. Its degree-q terminal cost is explicitly eT = 4(1 + columnsm) 2aq m+1 blingm 4 xT . The column count, the displayed chart factor, and the copy controls have rate zero; the adjacent level m + 1 has the same mesh exponent as m. Thus eT has rate τ −qa. The output quota is 2eT . Choose zT = exp (q + 1 16)(a −b) −τ T  . The stock-opening level has the same exponent b as its starting level, and its stakes have upper rate zero. The error amplitude is AT = 2 stakes zT , which has the upper rate in (OE6). The two affordability conditions in the matching construction hold eventually. For the march condition, the available exponent is the full gain (q + 1)(a −b); for the chart condition it is (q + 1 8)(a −b). The chosen expenditure uses only (q + 1 16)(a −b). Both leave strict positive margins. All numerical constants, stocks, columns and chart prices have rate zero, so they are absorbed by those margins. The remaining room condition also follows from b < a: the clock satisfies Jm+1 ≤Jm + 1, while each block scale increment is at least log 3. Hence Jm + 5 ≤J0 + 5 + scalem / log 3 and log(Jm + 5)/ scalem →0. A fixed readout radius and this subpower extra width are dominated by the strict mesh gap. These choices are inserted in the finite matching construction before the continuum limit. It yields a centered degree-q label with quota 2eT , a mismatch supported in the prescribed central chamber, and root-test bound AT . The input positions are required to remain in that chamber through level m + 1, exactly the plot conditions already checked. The finite matching construction itself is the sweep-and-replica construction; its remaining formulas are addressed below. There is also no hidden independence assumption in the exterior test passage. On a common finite lattice the original cluster register and its centered replacement are multiplied by precisely the same exterior register. The algebraic identity is (registerx −O(Q, 0)) registery = registerx⊔y −O(Q, 0) registery . The paid insertion bound applies to its left side as one main insertion and the given number of exterior insertions. Taking the original extraction limit along the survival filter gives the two terms on the right with the actual limiting label Q. On screened configurations this proves the norm bound for their compressed difference. For arbitrary exterior positions, approximate by screened lattice 345 ## Source page 348 A Wilson Yang–Mills construction J. Rodgers configurations avoiding the finite set consisting of the center and all cluster positions, coordinate by coordinate. Their separation inequalities persist eventually, and the hierarchy is smooth at both noncoincident limiting configurations. For each character the scalar inequalities therefore pass to the limit. Taking the supremum over characters gives the compressed algebra norm bound. The same approximation, without a replacement, proves the corresponding bound for full products. Thus the extension to arbitrary exterior points uses actual continuity of the already constructed hierarchy and one fixed replacement, not a fresh replacement chosen after the test. We describe the geometry and algebra of that finite matching construction more precisely. Make one copy of the input list in every cell of a fixed coarse tiling, with the prescribed reflection and translation of that cell. Write Fv,i for these copied insertions and qv = Q i Fv,i. The anchor copy is unchanged. At level m + 1 all members of the original list have parent zero, so the parents of the copied list are exactly the tiling centers. At every finer level the copied position has the same coarser parent as its center and lies within half the intervening block length of the embedded parent. Two distinct centers are separated at the parent level by at least B2 m+1+1. If the intervening enlargement factor is L, their embedded separation is at least L(B2 m+1 + 1); the two deviations cost strictly less than L. The propagated budget satisfies B2 i ≤LB2 m+1. The triangle inequality therefore gives the required cross-copy separation at every intermediate level. Within a copy the given finest-level spacing supplies the remaining separation condition. Represent the copied insertions by a polynomial with one formal marker per occurrence (v, i). Descending integration preserves this polynomial and its integrability. Grouping markers within a copy into one marker replaces Fv,i by qv. Grouping in stages or all at once has identical coefficients: the coefficient of a specified marker set is the product over its full inverse image. Since grouping is a fixed linear map on a finite-dimensional polynomial space, it commutes with each conditional integral. This proves the conservation identity between the descended polynomial of the products qv and the grouped descended polynomial of the individual Fv,i. The descending sweep starts with positive single-field allowances wi = δjφjai/8. At each level it joins precisely the fibers that have acquired a common parent, retaining one anchor in each fiber; thus the final number of fibers is at most the original arity. Its available normalization is bounded below by the compensation of Q i wi, including the four-dimensional volume ratio between levels m + 1 and j, and the licenses used in these joins. The initial allowance is α|I| compensation |I|, m + 1, j, 0, α, (δjφj/8)|I| Y i ai ! . The chosen α ≤1 is no larger than any intermediate license, so fewer surviving fibers only increase this lower bound. This is the finite quantity whose rate was denoted xT above. The licenses can be chosen uniformly down to every preceding level while preserving rate zero. If ui > 0 is the license and log ui/ scalei →0, set αi = min(1, u0, . . . , ui). Then 0 < αi ≤1 and αi ≤uj for all j ≤i. For any t < 0, the rate-zero hypothesis gives uj ≥et scalej for large j. Monotonicity of the scale implies αi ≥min(αi0, et scalei). Dividing logarithms by the scale and letting t ↑0 proves that αi still has rate zero. The halo purchase bounds the logarithm of the geometric part of each license by 16 q bβi, whose ratio to the scale tends to zero. The other license factors have the same rate-zero property. Thus this minimum introduces no power loss. At the last join, let z > 0 be at most the surviving normalization and let x ≥1 be the purchased local coefficient constant. The terminal normalization estimate gives a local analytic coefficient with 346 ## Source page 349 A Wilson Yang–Mills construction J. Rodgers remainder bound 4/z. Prune its insertion allowance to I′ v = z/(16x). This is no larger than the available normalization. It satisfies the two elementary fusion inequalities 2I′ v 2(4/z) ≤1, I′ v |λ−1| x(4/z) ≤1, where λ = (lm)q ∈C, and lm ≥2 is the integer block size at this step. This λ is a scalar multiplying every coefficient in this degree-q fusion; it is not an arbitrary linear change of frame. Since q ≥0, |λ| = lq m ≥1, |λ−1| = |λ|−1 = l−q m ≤1. For q = 0 both moduli equal one. Equivalently, scalar multiplication by λ−1 has operator norm at most one on the coefficient space. Thus the first smallness expression is 1/x ≤1, and the second is |λ−1|/4 ≤1/4. A lower bound on the norm of a general operator would not imply this inverse bound; the scalar identity is what proves it here. The target march allowances satisfy Wv ≤δmI′ v, so their products meet the insertion substitution bound. Applying the local fusion estimate yields a coefficient Q0 with ∥Q0∥coeff≤p′ = aq m+1x 4/z and the same remainder budget, representing the descended products simultaneously in every copy. All transformations of one copy into another are inherited from those of the original Fv,i; the anchor is at zero. Restoring the polynomial coefficient, then applying the bounded-copy comparison to its actual local section, gives a physical label with quota at most 2e and root-test error 2 stakes /b0, where b0 is the chosen final march amplitude. These are exactly the two outputs used in funding, with b0 = zT . The common cluster purchases used in this construction are made for every finite arity on the original family. They comprise the same positive activities, smallness inequalities (2Bi + 1)4 scurryi ≤1 2, escurryi −1 ≤1, the normalization lower bound, and the local fusion and analytic remainder estimates at every sufficiently late degree reference. The degree reference ensures the clock has reached at least degree plus three. These clauses are supplied by the already chosen scheduling and local analytic estimates; they do not entail replacing the Wilson family when the arity or output degree changes. For the paid-product limit, a finite linear combination of screened words is encoded with its actual moving lattice labels. Its continuum compression and its lattice expression use the same scalar coefficients. Products of registers agree identically with the register of the concatenated list. Differences use the coefficients 1, −1. Applying the paid-product inequality to this exact expression and passing to the selected continuum limit gives the compressed norm bound by the product of its insertion allowances. In particular it applies to the single mismatch times all exterior factors. This explains why the subtraction stays intact through the limit; estimating its two terms separately would discard the approximation accuracy. The compensation inequality underlying the sweep can be written without any scale shorthand. If H is the original number of fields, s is the number of remaining groups, 0 < α ≤1, and w > 0, define K(i, j, s) = aj ai 4H α 8 H(H−s) w. (OE8) 347 ## Source page 350 A Wilson Yang–Mills construction J. Rodgers The sweep maintains K(i, j, s) ≤Qs γ=1 Wγ. At the finest level, s = H, so it is equality with w = Q Wγ. When the groups stay separated at the next coarser level, their allowances become Wγ/L4 i , where ai = Liai+1 and Li ≥2. Since s ≤H, K(i, j, s) = L−4H i K(i + 1, j, s) ≤L−4s i Y γ Wγ. This pays the entire one-step transport without a loss that accumulates separately in each tiling copy. If separation fails, select a nontrivial grouping, preserving the distinguished anchor, whose representatives are separated at the coarser level. Let its size be s′ < s, and let li be the purchased cover bound for its groups. The local grouping estimate gives the exact product of new group allowances as the old product times license(H, li, i + 1)s; before transport each group also spends a factor 8. Since α is no larger than this license and H(H −s′) ≥H(H −s) + H, αH ≤αs, 8s′ ≤8H, we obtain K(i + 1, j, s′) ≤K(i + 1, j, s)(α/8)H ≤8−s′license(H, li, i + 1)s Y γ Wγ. The preceding transport inequality then pays the coarser step. This proves (OE8) by finite descend- ing induction, in both cases. A regrouping strictly decreases the number of groups, so the exponent H(H −s) pays at most the fixed arity’s worth of grouping events, independently of the number of scale steps and tiling cells. Grouping the marker polynomial and then pushing it is identical to pushing first and grouping afterward, by the finite-dimensional integral identity already proved. Integrability is likewise retained under each such linear grouping. Every step also retains the actual representatives, not just their count: their parent relation is inherited from the original positions, their copy transformations are inherited from the original collars, and their anchor representative is the original anchor. If a group is selected at a coarser level, its cover is bounded by the purchased li. These are exactly the data needed for the local grouping estimate and the final normalization. Thus the induction does not require consistency choices beyond the existing finite marker grouping. For clarity about the norm used throughout, a compressed word at noncoincident positions is its ground-space multiplication operator M; it is assigned zero elsewhere only to make the notation total. For a character χ, χ(M) is its phase correlation. For the unselected hierarchy its evaluation is ⟨Ω, MΩ⟩, because translating the entire finite word into a positive-time chamber leaves its correlation unchanged and the compression formula realizes that translated word. Both evaluations are contractive: the latter by ∥Ω∥= 1, the former by the character norm. Conversely, the Gelfand norm is the supremum of the character values, so a common nonnegative scalar bound on all viewers is equivalent to the compressed operator norm bound. This identifies the all-phase norm in the OPE with the ground algebra already constructed. The representative selection in the grouping step is finite and explicit. For the positions in the anchor copy choose a subset of maximum cardinality containing the distinguished anchor and having pairwise lattice separation at least B2. Every omitted position is within separation < B2 of one of the chosen positions, since otherwise it could be adjoined. Assign each position to such a representative, assigning representatives to themselves. This gives a surjection with a section fixing the anchor. When separation failed, the subset is proper, so its cardinality strictly decreases. Repeat this same relative selection in every copy. Reflections and translations preserve lattice 348 ## Source page 351 A Wilson Yang–Mills construction J. Rodgers separation, and the cross-copy separation was already proved; hence the entire selected configuration is separated simultaneously. The necessary finer-level two-point cover bound follows from the parent relation. If x, y have parents at separation < B2 m, the distance from either point to its embedded parent is less than half the block length Lm. Thus sep(x, y) ≤Lm(Bm + Bm+1 + 1)2 =: cm. A pair at distance at most cm is contained in a halo of its first point. Covering this halo by Bm+1-blocks costs at most (2cm + 1)4 ≤16(1 + cm)4. This is the purchased carpet bound. It applies separately to each point and its representative; the construction does not assume that the entire group lies in a single small halo. We give the remaining moment argument for the compressed norm passage. Let E be a fixed finite linear combination of screened words and let its labels vary along a refining regulator filter, converging in their fixed finite-dimensional label spaces. For each regulator pair, freeze the labels at that pair and use this ordinary static word in the finite even-moment expansion. For an even order 2h, the expansion is a finite sum over choices of one word from E at each of the 2h occurrences. Its coefficients are the products of the scalar coefficients and their conjugates, and its terms are the corresponding translated and reflected full lattice correlations. This is an exact expansion of the even-moment expression, including every subtraction in E. Each such lattice correlation converges along the refining filter with the moving labels. Indeed expand each label in a fixed finite basis, expand the multilinear correlation as the finite sum over all basis choices, and use convergence of the finitely many coordinate products and the fixed-label point limits. For every sufficiently large replica displacement the limiting positions are generic, so all these fixed-label limits apply. Therefore the finite even-moment expression converges first along the regulator filter to its continuum replica expression. Sending the replica displacement to infinity gives the vacuum expectation of the corresponding even power of the ground compression, by the long-cluster moment formula already proved. If the finite paid-product estimate bounds the whole even-moment expression in norm by c2h, its real part is at most c2h. Both limits preserve that inequality, for every h. The ground-algebra even-moment norm argument then gives ∥ME∥≤c. Equivalently, one may apply the same moment inequality to each character and use the Gelfand norm. Here c is the product of the main and ex- terior insertion allowances; it is nonnegative because each paid allowance is nonnegative. Applying this construction to the exact register identity for the mismatch proves its norm bound. This proof uses finite-dimensional convergence of the moving label along the actual refinement filter; it requires no separate inner regulator limit for that label. The local grouping allowances just used are λi(H, l) = (δi/H)δi exp((ei + 1 2)l), W ′ γ = Y j∈γ λi(H, l)Wj, ei = supportExponenti . (OE9) Indeed each occurrence first receives δiWj, the allocation among at most H occurrences gives the factor δi/H, and moving its support to the chosen representative costs exp((ei + 1 2)l). The occurrence sets of distinct groups partition the old occurrences, so Q γ W ′ γ = λi(H, l)s Q j Wj exactly. This proves the group-product formula used in (OE8). To obtain the grouped local functions, apply the finite marker fusion estimate to those oc- currence sets, using the cover bound l, the occupancy bound, and the two smallness inequalities 349 ## Source page 352 A Wilson Yang–Mills construction J. Rodgers displayed above. The original insertion allowance Wj controls its actual local analytic norm; multi- plication by δi ≤1 preserves that bound. The positive background density permits normalization, and the representation of each lattice stage identifies this normalized density with the stage’s ac- tual background. Fusion produces one local function per group, together with dominated activity profiles, representing the exactly grouped marker polynomial. Their normalized remainder bound is 4/W ′ γ. Pruning each group allowance to W ′ γ/8 rewrites this as 4/W ′ γ = 1 2(W ′γ/8), the bound required to transport the group to the next scale. The leading counterterm of a newly grouped function is zero. The copied functions and profiles retain their common translations and reflections, since the same groups and representatives are used in each copy. For the terminal grouping there is just one group in each copy; no factor 1/8 is needed before normalization, and the common allowance is Q j λi(H, l)Wj. Its remainder bound is 4 divided by this allowance, exactly the input to the terminal step above. Finally, the paid even-moment bound can be seen directly from its lattice layout. The two groups of fields have separation scales ab and al. Choose buffer and insertion levels satisfying b ≤f ≤l≤n, f ≤Q, and the finite packing inequalities |I| 10af < ab, |J| 6an < al. The seating lemma places every original insertion with the required interior margins in the appro- priate translated reflection chamber. Translation copies are displaced by the prescribed replica itinerary. Within one replica the original spacing is unchanged; between different replicas all distances tend to infinity with the displacement. At sufficiently large displacement, and then suffi- ciently fine regulator, every required lattice separation exceeds its chamber scale. The same interior margins hold in all replicas because the itinerary preserves the chamber margins. At the 2h occurrences replace each field by its adjoint on the alternating even positions and keep it on the odd positions. Adjoining preserves its paid allowance. The mixed chessboard inequality on the two nested chambers now bounds the integral of the product of all these translated fields by the product of all their allowances, namely   Y i∈I Bi !  Y j∈J Dj     2h . The register translation identity and conjugation of finite products identify this integrand with the even-moment expression of the entire registered insertion Z. For a mismatch, Z is the mismatch times the exterior register, so the main group has exactly one occurrence per replica with allowance A. Nothing expands that occurrence into two independently estimated fields. This proves the finite bound required by the compressed-norm limit. We justify the nested-chamber inequality used in the last paragraph. A paid field has an interior support in its original central chamber and a bound A for that chamber’s root test. Any permitted diagonal reflection and conjugation preserves both properties and its price. If its placement has a margin of three original chamber scales inside a larger chamber, the translated field is a copy, in the appropriate cell of the larger tiling, of another interior function with the same bound A. This is the offset-and-reflection identity for the tiling copy map. 350 ## Source page 353 A Wilson Yang–Mills construction J. Rodgers First collect all finer fields assigned to one larger cell. Copy this collected product to every cell of the larger tiling. Within the finer tiling all individual copied fields occupy distinct cells: distinct larger cells cannot meet in a finer cell, and two original finer placements in one larger cell are farther apart than the finer cell’s extent. Reflection preserves that separation. The finer chessboard inequality bounds the full copied product by the product of the finer root tests. Taking the root of order equal to the number of larger cells gives the collected product a larger-chamber root test at most the product of its individual prices. This root extraction is legitimate because the prices and root tests are nonnegative and the number of cells is positive. Now the coarser main placements occupy distinct larger cells, and no one of them occupies a larger cell containing a finer exterior placement: otherwise the cell’s extent would contradict the assumed main-main or main-exterior separation. Put each main field in its own cell, and each collected exterior product in its occupied cell. One application of the larger chessboard inequality bounds the full mixed product by the product of their root tests. Multiplying the collected bounds gives exactly Q Bi Q Dj. Thus the different insertion scales are retained; replacing them by a single common scale is unnecessary. The common seating translation exists by two finite avoidance steps. First choose a translation placing the main positions at distance five buffer scales from the coarse cell boundaries; the total excluded intervals fit because |I|10af < ab. Then shift by at most one intermediate cell scale to place the exterior positions at distance three fine insertion scales from their cell boundaries, using |J|6an < al. The second shift consumes at most two of the five main buffer scales, leaving the required three. The nested cell periods divide the larger periods. Replica displacements are integer multiples of these periods, so they leave every boundary residue and hence all margins unchanged. Actual torus separations follow from the Euclidean ones for sufficiently large periods: the physical distance is at most twice the mesh times the wrapped lattice separation. Subsequent embedding multiplies lattice separations by the mesh ratio, proving the required inequalities at every finer cutoff. The one-step sweep transport uses allowances W ′ γ = Wγ/L4 i . Its local driving estimate multiplies the analytic remainder bound by 6Li; this fits since 6Li ≤L4 i for Li ≥2. Thus (6Li)/(2Wγ) ≤1/(2W ′ γ). All marker tolls are finite sums of nonnegative products of allowances. Decreasing the allowances therefore preserves every toll estimate. The separated representatives give the support substitution bound, the occupancy hypotheses hold before and after taking parents, and the prescribed transition density gives the exact descended polynomial. The leading counterterms retain their copied scalar transformations, and W ′ γ|cγ| ≤1 follows from W ′ γ ≤Wγ. At the initial fine scale, the solved activity representation gives remainder bound at most 4/Wγ; taking 8W ′ γ ≤Wγ converts this to 1/(2W ′ γ). These are precisely the numerical transport and initialization bounds used in the descending compensation induction. The marker fusion has the following quantitative construction. For a surjection of old occurrence labels onto new group labels, with a chosen representative of each group, extend every old support by the fixed connecting cover from its occurrence to its representative. If this cover costs at most l, the new allowance per old occurrence is f Wi = (η/H)Wie−αl, α = e + c. An old coefficient involving a set A of occurrences therefore gains (η/H)|A| in its weighted envelope, after paying all connecting covers. The group allowance is the product of these f Wi over its full fiber. Distinct original occurrences retain distinct markers, so whenever a product is eligible its 351 ## Source page 354 A Wilson Yang–Mills construction J. Rodgers marker sets are disjoint; the product of their tolls is the toll of their union. The latter is exactly the product of the full group allowances for the selected new markers. This is the algebraic reason fiber sizes can vary without changing the coefficient identity. The local coefficient inventory consists of two scalar-entry contributions and the activity-entry contributions. If the occurrence occupancy is at most s, the scalar coefficients have weighted bound a, the activity coefficients bound b, and Wi|ci| ≤T, extending their supports and charging representatives gives total sitewise envelope γ = 2(1 + s)η sTeα + a  + (1 + s)b. (OE10) For nonempty marker sets, (η/H)|A| ≤η/H. When a chosen representative is charged, there are at most H old siblings in its fiber; the factor H−1 cancels precisely that multiplicity. The representatives themselves satisfy the same occupancy bound, because the section is injective and their locations are old occurrence locations. These observations give the 1 + s incidence factor in each contribution. The scalar-entry inventory has two copies, producing the first factor 2; the activity inventory is charged once. Summing their nonnegative envelopes proves (OE10). Expand the grouped coefficients as finite eligible products of these inventory entries. Give a product the product of its entry envelopes. The disjoint-marker identity in the preceding paragraph proves that this is a valid weighted coefficient bound. The connected-set forest estimate, at excess support exponent c, applies provided (2B + 1)4γ ≤c. It bounds both the grouped ordinary coefficients and grouped activity coefficients by eγ −1 at support exponent e. This is the same connected-set estimate already proved for the general insertion calculus, now with the extended supports and group tolls. Selecting eγ −1 ≤Z gives the required two coefficient budgets Z. Read the singleton coefficient of the grouped ordinary polynomial as the new local function and the singleton coefficient of the grouped activity polynomial as its activity profile. The sin- gleton restoration identity and the normalized nonvanishing background give their exact restored polynomial and a local remainder bound 2Z/W ′ γ + 2Z/W ′ γ = 4Z/W ′ γ. The domination construction for the activity singleton retains the actual background activity coef- ficient at the empty marker. Hence it yields an admissible dominated activity profile, not only a numerical estimate on one coefficient. The higher coefficients give the simultaneous served-insertion representation. In the sweep specialization c = 1/2, Z = 1, these are exactly the bounds and hy- potheses used in (OE9). Copy covariance of these new singleton functions follows at the level of every grouped marker coefficient. A marker set focused in a given copy involves only inventory entries in that copy; its transformed set is focused in the transformed copy. The copy permutation is a bijection on these inventories, takes extended supports to their transformed supports, and conjugates each coefficient by the prescribed scalar transformation. Reindexing the finite sums and products proves covariance for the grouped ordinary and activity polynomials. Reading their singleton coefficients then proves covariance of the local functions and profiles. The same argument includes the empty marker and all interactions internal to a copy, so repeated regrouping preserves the hypotheses required at the next scale. We spell out the inventory underlying this restoration. Write ci for the scalar marker entry, AS for the ordinary polynomial on a support S, and KS,F for the activity polynomial on a support and 352 ## Source page 355 A Wilson Yang–Mills construction J. Rodgers its flagged subset. For every scalar ci, take a singleton-marker entry on its occurrence location; for every nonempty marker coefficient of AS, take that coefficient on S. Extend each of these supports by the representative locations of the groups mentioned by its markers. Split its scalar value into two entries: one equal to the value when this extended support is healthy and zero otherwise, and the other equal to the value when it is unhealthy and zero otherwise. Their sum is exactly the original value, at every configuration. Keep every allowed coefficient of KS,F , with the same extended support and its original flagged data. These are the two scalar-entry inventories and the single activity inventory in (OE10). They are finite on each lattice. Summing the two scalar entries recovers P i cizi + P S AS coefficient by coefficient. Collecting activity entries by their original support-and-flag label recovers KS,F , also coefficient by coefficient. Each nonzero coefficient has its marker locations inside its support by locality, so every term required for either recovery is present. The grouped construction forms eligible connected collections of these entries and assigns their union support, union markers, and flag intersection to the resulting coefficient. Expanding the marker exponential and activity product, then partitioning a finite collection into its connected components, gives R∆ " expz X i cizi + X S AS ! Gasz(K) # = expζ X S eAS ! Gasζ(f K). (OE11) Here R∆reads the old coefficient on the full inverse image of each new marker set, and the exponen- tials are the finite exponentials in the nilpotent marker algebras. The eligibility condition requires disjoint old markers and complete group fibers. It is exactly what identifies the left coefficient with the right one; the connected-component partition is unique. The healthy/unhealthy split routes a component to the ordinary polynomial or the flagged activity polynomial without changing its scalar value. This is a finite polynomial identity, so it holds regardless of zeros of any individual entry. The empty activity coefficient is preserved in (OE11). A connected eligible collection with no new marker can contain no scalar entry, since each scalar entry has a nonempty old marker. Starting from any one of its activity entries, a connection to a second entry would have to use a group marker or intersect a flag support. There is no group marker, and distinct old empty- marker activity entries have disjoint prescribed flagged supports. Thus the collection consists of a single old empty-marker activity entry. Conversely each such entry is eligible and contributes its original value and its original support-and-flag label. This bijection proves preservation of the empty coefficient before any envelope estimate. After restoring the nonzero background, (OE11) is the served-polynomial identity used to construct the new local functions. The sitewise scalar-entry estimate in (OE10) can also be read directly from this inventory. At a site there are at most s singleton scalar entries, each of weighted size at most T, and a singleton support has cover number at most one. Their contribution is at most sTeα. The nonempty coefficients of the ordinary polynomials contribute at most a. The two healthy/unhealthy copies are each bounded by this same envelope. Activity coefficients contribute at most b. Extending the supports and reallocating to representatives gives the 1 + s factors already described and the occurrence factor η, proving (OE10) with no omitted unit term. For precision, the eligibility and connectedness in (OE11) are as follows. An old marker set A is retained by R∆exactly when it is a union of complete fibers. Its new marker set is ∆(A). For any old monomial, R∆(zAa) = ( ζ∆(A)a, A = ∆−1(∆(A)), 0, otherwise. 353 ## Source page 356 A Wilson Yang–Mills construction J. Rodgers For a product of retained monomials the new marker sets are disjoint exactly when the old ones are disjoint: an overlap in a new group would force its entire nonempty fiber into both old sets. Thus reading the full-fiber coefficients commutes with such eligible products. Connect two inventory entries when their group rosters meet, or when their extended supports meet and both entries are of the type that must be collected into an activity. The latter types are the unhealthy scalar entries and all old activity entries. A grouped ordinary entry is a nonempty connected eligible collection containing no such type. A grouped activity entry is a connected eligible collection containing at least one old activity entry, with its total flagged intersection prescribed to be the union of the old flagged subsets. Its coefficient is the product of the inventory coefficients when this flag condition holds, and zero otherwise. The old activity supports inside a collection are required to be pairwise disjoint, as they were in the original gas expansion. Partition an eligible old collection into components for this relation. A component containing an unhealthy scalar entry but no activity entry has zero contribution whenever the old flag pattern is selected: the unhealthy entry requires a flag in its extended support, and that flag must belong to an old activity support, which would connect that activity to this same component. Thus every contributing component is either an ordinary entry or a grouped activity entry as defined above. Conversely, a selected collection of grouped entries has disjoint group rosters, and its activity supports obey the selected disjoint-support pattern. Its expanded components therefore have no additional connections to each other. Taking their union and then taking components are inverse operations. The union of the old flagged subsets equals the union of the grouped flagged subsets. Finally, under the selected pattern, a grouped activity’s own flag condition holds separately: a flag in its support assigned to another grouped activity would contradict disjointness of the grouped activity supports. The coefficient product is unchanged by this bijection. Summing proves (OE11). Locality and measurability also survive this construction. Every grouped support contains its mentioned representatives. Each entry depends only on the original variables and flag pattern in its extended support; the product depends only on their union. Its flag indicator is measurable and gauge invariant, and every underlying coefficient is measurable and gauge invariant. A nonzero ordinary grouped entry is healthy on its entire support, since all its scalar entries are healthy there. A nonzero activity entry has exactly its recorded flag intersection. Thus the new ordinary and activity polynomials satisfy the same support and flag conditions as the old ones, enabling the next grouping step. For the final identification of one-step transport, extract each singleton coefficient of the fine served polynomial. The singleton restoration law identifies it with the normalized local insertion, including its dominated activity correction. The local analytic transport estimate gives at the anchor one coarse local function and a dominated coarse activity profile, with the same leading counterterm and remainder multiplied by the local driving constant. Define the functions in every other copy by the prescribed reflection and translation of this anchor function; reindex the activ- ity labels by the actual activity-motion bijection. The conditional integration intertwines these transformations, so their normalized insertions agree with the corresponding singleton coefficients of the pushed polynomial. Coefficient extraction commutes with this integration, since the fine polynomial is integrable. The marked transport theorem then gives the full coarse activity rep- resentation at weights δI, not just singleton identities. Finally the support substitution estimate with target weights J, the occupancy bound, and the one-step duty inequalities restores the served representation at J. These are exactly the analytic transport, singleton cancellation and served restoration estimates developed above, with occurrence set V × I. No two-element property of the earlier occurrence set is used: all coefficient and support estimates are stated for an arbitrary finite marker set. This completes the transport identification required in the sweep. 354 ## Source page 357 A Wilson Yang–Mills construction J. Rodgers 13.2 Nested expansions Nested expansions require an order of scales. The inner cluster is replaced first with the common coefficient reader, and its remainder is estimated before the outer replacement is summed. The scale- ratio gain applies on the declared nested geometric domain. Terms with zero exponent are retained with their stated logarithmic or subpower qualification, rather than being assigned a nonexistent positive power gain. Nesting keeps one reader At a nested node, the output of the inner expansion is an input to the outer one. Both therefore use the same homogeneous channel labels and finite projections on the physical filtration. The lower-degree terms are retained before the next substitution; otherwise the outer reader would see a different coefficient. The estimates distinguish a small ratio of nested radii from a fixed geometric separation, and keep the zero-exponent cases in their stated scope. Each nested tree and degree horizon is finite. Let t be a finite rooted tree, with leaves carrying input fields and with every internal vertex carrying a center zv and radius sv > 0. A leaf also has its insertion position. Vertices may have zero or one child; all finite arities are allowed. At every internal vertex all positions in its subtree lie within Msv of its center, and positions in distinct immediate child subtrees are at distance at least ηsv. Radii of internal children are at most their parent’s radius. The root radius is r ↓0. There is no lower bound on any child-to-parent ratio other than positivity. These conditions may equivalently be obtained from bounds on immediate child centers after requiring a sufficiently small uniform upper bound on child-to-parent radii. To see this, induct over the finite tree. Let Ci be the already obtained displacement constants of the child subtrees and C′ = P i Ci. Choose 0 < β ≤1, β ≤η/[4(C′ + 1)] and below every child’s previously chosen ratio bound. The displacement of any descendant from its immediate child center is at most C′βsv. Two such deviations reduce the immediate sibling separation by at most 2C′βsv ≤ηsv/2. The displacement from the parent center is at most (M + C′)sv. This proves the stated full-subtree geometry recursively, with separation η/2. Write wlfor each input degree, Wv = P lbelow v wl, and qv = sv/sparent(v) for a nonroot internal vertex. The virtual parent radius of the root is 1, so qroot = r. For fixed output degree m, define the common price Tm,ε(s) = r m−P lwl−ε Y v internal v̸=root q−Wv−ε v . (NE1) Choose fixed truncation degrees Dv at the internal vertices, before taking r →0. The direct coefficient is the coefficient Cm,α of the entire leaf cluster at the root center. The iterated coefficient is the finite sum obtained by expanding each nonroot child cluster through its assigned degree, substituting its homogeneous basis fields at that child’s center, and repeating upward to the fixed root output m, α. Multiplication of the successive coefficients and summation over all intermediate degrees and channels is meant literally. These are the same C’s and homogeneous bases used in (OE1), with no new reader choice. Uniformly in the admissible positions and positive radii, for sufficiently small r, |Cdirect| ≤Tm,ε(s), |Cdirect −Citerated| ≤Tm,ε(s) X v internal v̸=root qDv+1 v . (NE2) 355 ## Source page 358 A Wilson Yang–Mills construction J. Rodgers If there are no nonroot internal vertices, the iterated and direct coefficients agree exactly; the sum on the right is zero. The degree assigned to the root does not enter that sum, since the root output has already been fixed. For exterior fields whose positions stay at least δ > 0 from all tree positions and from one another, the full leaf product minus its direct root expansion through degree m has ground-algebra norm at most Tm+1,ε(s). (NE3) The same estimate holds for the unselected law and every character, uniformly in their choice. The coefficients, the finite iterated sums and the tested expressions are smooth on the open set where all their finitely many reader configurations are admissible. For every fixed collection of requested degrees the geometry above eventually belongs to this open set. This follows recursively from admissibility of each child-center list and the direct leaf list; their diameters tend to zero and their required distinctness follows from the sibling separation. We explain the scale accounting used to prove these assertions. If sv has logarithmic rate −av, with nondecreasing av down the tree and virtual parent exponent zero, set ∆v = av −aparent(v) ≥0. Then the engineering cost of the original leaves is E = X v internal Wv∆v. (NE4) This is also the sum over leaves of their degree times their nearest internal parent’s exponent: each leaf’s path telescopes. Thus a direct output of degree m has rate E −maroot, and a tested remainder through degree m has rate E −(m+1)aroot. The price (NE1) has exactly this engineering rate plus ε P v ∆v. The latter is strictly positive after normalizing P v av = 1: at least one av > 0, and each av is at most the sum of the nonnegative advances along all vertices. Therefore it absorbs every fixed multiplicative constant, even when individual advances are zero. We prove the recursive rate assertions, keeping all truncations finite. A leaf has just one alternative: its original field at its position, with coefficient 1. An internal child’s alternatives through horizon D are the homogeneous fields of degrees 0, . . . , D, at its center, weighted by their direct coefficients. For any specified alternatives of the children, the parent coefficient is the product of their coefficients times the parent OPE coefficient for their resulting fields. A bud is never replaced by a degree-zero projection: its stage set is a singleton carrying its actual input degree. This convention includes arbitrary input labels correctly. The main inductive assertion has two parts. Every raw subtree can be represented, up to any prescribed tested error rate, by a finite family whose positions all have separation at its parent’s exponent and whose coefficient amplitudes pay its total engineering cost minus the total resulting field degree times that parent’s exponent. The same raw subtree has direct coefficient and full tested remainder bounds at its own exponent, with its engineering cost. For a leaf both statements are immediate. At an internal vertex, combine the child families by finite products. Their costs add exactly to (NE4). Sibling separation makes the combined positions admissible. If a child’s exponent equals its parent’s, retain the child’s comparable finite family obtained from its first inductive assertion. If it is strictly larger, use its second assertion to replace it through a sufficiently high auxiliary finite horizon H. Specifically, if its engineering cost is Ω, the parent exponent is k, its own exponent is a > k, and later exterior costs are bounded by Wb, the tested error rate is at most Ω−(H + 1)(a −k) + Wb. Given any requested rate ρ, choose H > (Ω+ Wb −ρ)/(a −k), 356 ## Source page 359 A Wilson Yang–Mills construction J. Rodgers and above any previously requested horizon. This makes the error smaller than requested. Perform the finitely many child replacements in order. At each step the preceding finite replacements and their field degrees have already been fixed, so choose the next auxiliary horizon to pay their actual exterior cost. The exact telescoping sum contains one replacement error at a time. This triangular choice avoids circular dependence among the horizons. Finite readouts of the tested errors then give the same comparison for coefficients. Applying the single-center estimate to the resulting comparable finite family establishes both assertions for the parent. To compare with prescribed, rather than auxiliary, truncations, fix a gain γ ≥0 such that γ ≤(Dv + 1)∆v for every nonroot internal vertex v. (NE5) Inductively, each recursively composed child coefficient has its direct coefficient’s engineering rate, and their difference improves that rate by γ. The product identity Y j xj − Y j yj = X j (xj −yj) Y ij xi therefore improves the product rate by γ, not by a multiple of γ. Multiplication by the parent coefficient preserves this improvement. For γ > 0, every nontrivial immediate child has a strictly positive exponent gap, by (NE5). Replace all such children through auxiliary horizons Hj ≥Dj with error of any desired rate, using the preceding triangular argument. Subtract the finite parent sum through the desired horizons. Its difference from the auxiliary sum is exactly the sum over omitted channel choices. In every omitted choice at least one child degree is at least Dj + 1. That child’s coefficient costs its degree at the finer exponent and the parent coefficient returns it at the parent exponent; hence the choice gains at least (Dj + 1)∆j ≥γ. All other degree-gap contributions are nonnegative. Thus the omitted finite sum gains γ. Replacing the remaining direct child coefficients by their recursive coefficients gains the same γ by the product identity. Adding these three differences proves that the direct-to-iterated coefficient error has rate at most E −maroot −γ. (NE6) If γ = 0, direct and iterated coefficients already have the common raw rate, so their difference has that rate by the triangle inequality; no high-accuracy replacement is needed at a zero exponent gap. If there are no nonroot internal vertices, there is one choice at every child, its coefficient is 1, and permutation invariance identifies the parent sum with the direct coefficient exactly. Take γ to be the minimum of the finitely many (Dv + 1)∆v. Choose a vertex attaining this minimum. Its ratio power qDv+1 v has rate −γ, while the positive slack in (NE1) absorbs the fixed constants in (NE6). Its single positive ratio term is bounded by the sum in (NE2), proving the claimed sum of gains. The direct coefficient and exterior remainder rates similarly give (NE2)’s first inequality and (NE3). Contractivity and the character supremum give the latter in the common ground-algebra norm. The finite-family calculus in that induction has explicit quantifiers. An alternative is a finite word at selected subtree positions, multiplied by a scalar amplitude of some finite upper rate. A family is eligible at exponent b when each alternative’s positions are separated at b. Two families agree to rate ρ, up to exterior weight W, when their complete correlations differ with rate at most ρ against every fixed exterior label list of total degree at most W, with arbitrary admissible position paths and arbitrary viewer paths. A supply at parent exponent k and cost M gives such an eligible family for every sufficiently large exterior exponent b; for every W, ρ it gives a comparable replacement agreeing in this sense. These choices concern exact continuum correlations throughout. 357 ## Source page 360 A Wilson Yang–Mills construction J. Rodgers To change one branch while retaining all other branches, fix an alternative of the other branches. Let its field degree sum be d and its amplitude have rate ζ. Include its fields in the exterior test of the changed branch. Request agreement through weight W + d and rate ρ −ζ. Multiplying by that alternative’s amplitude returns rate ρ. The finite sum over all other alternatives has the same rate. Since there are finitely many alternatives, one may take the maximum of their W + d and ζ first. Sibling separation and the old eligibility show that all these augmented exterior tests have the required separation. This proves the single-branch replacement formula. Induction over the finite branch set proves the triangular replacement assertion: at a given replacement only the previous choices and unchanged later branches enter its finite maxima. Transitivity follows from adding the complete correlation differences. A tested agreement transfers to coefficient agreement by the actual finite reader. For a requested horizon m, take W at least the maximum degree sum of every exterior word in all detector supports through that horizon. These supports form a finite union. The rotating detector words retain their degree sums and have bounded label coordinates; their positions stay a fixed positive distance from the center and from one another. A shrinking eligible family is therefore admissible against every such moving detector eventually. Write each rotated reader as its finite sum of these tested correlations times its fixed weights and rotated loaded vectors. Apply the agreement estimate and then average over the compact rotation group. The arbitrary-path quantifier makes the bound uniform in the rotating group element, as proved for (OE4). Apply the real symmetrization and Km, then a basis coordinate; these fixed finite-dimensional linear maps preserve its rate. This proves the coefficient transfer with the same ρ. There is a further debit when transferring a whole remainder, which is why agreement of prod- ucts alone would not suffice. Let the desired remainder rate be ρ0 = M −(m + 1)(k −u), before exterior cost, where the center-to-exterior exponent is 0 ≤u ≤k. Choose a comparable replacement agreeing at rate ρ0 −mu, with enough exterior degree to cover both the actual exterior fields and the finite detectors through m. Its full-product difference has at most this rate, hence at most ρ0. The coefficient differences through degree j ≤m have rate ρ0 −mu; their centered exterior correlations cost ju + |e|b. Each resulting term therefore has rate at most ρ0 + |e|b. Add the comparable replacement’s own complete remainder at that rate and subtract these coefficient differences. The exact resulting expression is the original family’s complete remainder. This proves the second inductive supply assertion with all coefficient subtractions retained. The comparable-family estimate used above includes the boundary exponent cases. An alter- native of field degree d has amplitude of rate M −dk, and its positions have size and separation exponent k. For k > 0, its degree-m coefficient costs (d −m)k by the homogeneous reader bound. Multiplication gives M −mk. If k = 0, test its amplitude times the word against the finite detector exteriors. The word has rate zero, the amplitude rate M; the readout comparison with the empty family gives rate M for its coefficient as well. Actual shrinking of its positions ensures eventual reader admissibility even though its exponent is zero. For the full remainder, when the exterior exponent u < k, multiply the single-center remainder rate by the amplitude to obtain M −(m + 1)(k −u) + |e|b. If u = k, estimate the whole correlation and each finite centered term separately; the engineering degrees cancel and give exactly M + |e|b, the same expression at zero gain. Summation over the finite comparable family proves the estimate in every case. 358 ## Source page 361 A Wilson Yang–Mills construction J. Rodgers We now remove the auxiliary assumption that radii already possess logarithmic rates. Along any approaching admissible family all internal radii tend to zero, since they are positive and bounded above by the root radius. Put T = X v internal log(1/sv). Eventually every summand is nonnegative, T > 0, and T →∞. The vector with coordinates log(1/sv)/T belongs to the finite compact cube [0, 1]Inner(t). On a nontrivial refining filter it con- verges to a; summing the coordinates proves P v av = 1. Hence each sv has rate −av on that filter. Monotonicity sv ≤sparent(v) implies av ≥aparent(v), since the opposite strict inequality would eventually force the smaller radius to exceed the larger one by the exponential-rate comparison. Take the virtual parent exponent to be zero. The geometric size estimate Msv gives the required envelope rate at vertex v, and also actual convergence of every descendant displacement to zero. The sibling lower bound ηsv gives separation at every exponent strictly larger than av. Induction over the tree thus gives all the regularity hypotheses of the preceding rate proof, including vertices with av = 0. Since P av = 1, the slack ε P ∆v is strictly positive, as required in (NE4). Applying the rate estimates gives (NE2) and (NE3) eventually on this refining filter. If a single common small-radius threshold for all admissible scales and positions failed, choose a violating scale and position array at arbitrarily small radii and restrict to those radii. The compact layer extraction just proved gives a nontrivial refining filter of this very array where the desired inequality holds eventually, contradicting its selected violations. The same argument handles the exterior geometry and all viewers; for viewers one first selects a violating viewer at each parameter. Thus the final thresholds depend only on the fixed tree, field labels, horizons, M, η, ε, and exterior clearance δ, with no lower bounds on the ratios. This proves the uniform quantifiers in (NE2)– (NE3). The bounds admit arbitrary joint weighted integration. Let (K, μ) be any measure space, u ∈L1(μ; C), and R |u| dμ ≤B. Let z : K →Plot(t) be almost everywhere measurable, with admissible geometry wherever u ̸= 0, at one fixed array of radii s. The direct coefficient and the finite iterated coefficient are smooth on their common open reader domain. Extending them by zero outside that domain gives measurable functions. If |g(z(i))| ≤C on the support of u, then |u(i)g(z(i))| ≤C|u(i)| almost everywhere, so this product is integrable and Z u(i)g(z(i)) dμ(i) ≤BC. (NE7) Apply this first to the direct coefficient with C = Tm,ε, then to its difference from the iterated coefficient with C = Tm,ε P v̸=root qDv+1 v . The iterated coefficient is integrable because it is the difference of those two integrable quantities. Subtracting their integrals proves the weighted version of (NE2). No product structure or absolute continuity assumption on μ is used. For jointly varying exterior positions y(i) and phase characters χ(i), the same conclusion applies to the complete tested remainder with C = Tm+1,ε. Its measurability follows on the common open guard domain from smoothness in positions and continuity of the character evaluation of each fixed ground-algebra element. The character space is compact metrizable, and these separate properties give joint measurability. Almost everywhere measurable z, y, χ may therefore be composed with it. The full raw correlation is integrable too: its bound is T0,ε, obtained by adding the finite degree-zero terms to the degree-zero remainder and applying the engineering rates. The truncated expression 359 ## Source page 362 A Wilson Yang–Mills construction J. Rodgers is then the difference of two integrable expressions. Consequently the finite sum may be integrated term by term and the norm of the full integrated discrepancy is at most BTm+1,ε. A probability mixture of characters is the special case u = 1, B = 1; its expansion uses the identical coefficients outside the phase integral. Finally, covariance concerns the full homogeneous vectors, rather than individual scalar basis channels. At a leaf the emitted vector is its actual label. At an internal vertex first form the emitted vectors of the children, include each in its fixed horizon space, and apply the multilinear hj to their centers for each j through the parent’s horizon. The finite sum of these included outputs is the emitted vector of that subtree. Expanding multilinearity shows that its coordinates are exactly the finite iterated channel formula. Translation invariance, conjugation compatibility and orthogonal covariance now follow by induction from the corresponding laws of hj and the inclusions. Applying just hm at the root proves the same laws for the degree-m iterated output. The basis can transform nontrivially within Hm; no assertion that its vectors are individually rotation invariant is needed. The full tested discrepancy inherits these laws from the hierarchy and these homogeneous outputs. The separated and nested expansions, their compatible coefficients, all-viewer estimates, joint weighted versions and covariance have now been translated. 14 Extension across collision sets Extending a separated correlation across collisions requires a choice of subtraction device. Tay- lor jets normal to a collision set are subtracted on dyadic shells; the cancellation-bearing shell expressions then have an absolutely summable L1 bound. Forests organize overlapping collisions with finitely many branches at a fixed degree and arity. The construction proves compatibility for the chosen devices and finite cutoffs; independence of arbitrary subtraction prescriptions is not an additional conclusion. The additional contact conclusion concerns Euclidean distributions, constructed with a speci- fied subtraction system using the same reader prescription. Its scope is the following. Restrict the k-point configuration space by a real linear isometry P : E →(R4)k, where E is finite-dimensional and carries its Haar volume. A collision group specifies a subset of the k labels and its collapsed configurations. The permitted forest consists of compatible collision groups together with local continuation charts, support windows, finite Taylor orders and projection choices. On every admis- sible such forest, the construction supplies an integrable Taylor-subtracted pairing for a compactly supported smooth test function in the chart. Its expansion into the prescribed annular pieces is absolutely summable in integral norm and sums to that pairing. Iterated expansions over any finite ordered list of centered collision groups have the same property. On annuli away from total collapse the result agrees with the punctured correlation. The public endpoint also bounds the resulting distribution on every permitted isometric collision plane containing the collision directions, at the stated finite total test order. The subtraction devices carry the actual rotation action of the reconstructed hierarchy, and their finite projection basis is exactly the reader prescription used for the OPE. A real system basis and compatible positive threshold choices exist simultaneously. These contact assertions must therefore be proved with these specified devices and admissible forest conditions; they are not a claim of arbitrary subtraction-independent contact values. For a nonempty label set g, its normal displacement space in the full configuration space is Nfull g = {x : xj = 0 (j /∈g), X j∈g xj = 0}. 360 ## Source page 363 A Wilson Yang–Mills construction J. Rodgers Its orthogonal complement contains the collapsed configurations of the group, including its centroid. In E use Ng = P −1Nfull g . The admissibility assumption is Nfull g ⊂ran P; hence P identifies these normal spaces isometrically. Omitting any one member of g identifies Nfull g linearly with (R4)|g|−1: the omitted coordinate is minus the sum of the others. In particular dim Ng = 4(|g| −1) =: νg. (CE1) For disjoint groups the normal spaces are orthogonal. Centroids of two disjoint nonempty proper groups are different almost everywhere in these normal variables: their equality is the kernel of a nonzero linear map. It is nonzero by placing a vector at one member of each group with oppo- site signs. Its proper linear kernel has Haar measure zero. This permits the subsequent almost everywhere shell identities without assigning a correlation directly on a coincidence plane. Write πg for orthogonal projection onto Ng. The degree-j normal Taylor term of a smooth test f : E →C is Tg,jf(x) = 1 j!Djf(x −πgx)[πgx, . . . , πgx]. For a fixed positive subtraction length s, choose the prescribed smooth radial cutoffχs(πgx), equal to one near zero and zero for ∥πgx∥≥2s. The subtraction operator through order m is Tg,s,mf = χs(πgx) m X j=0 Tg,jf, Rg,s,m = 1 −Tg,s,m. (CE2) Tangential coordinates remain parameters in this Taylor expansion. The subtraction depends only on the normal jets through m along N⊥ g , and vanishes on a test supported away from that subspace. Its support is contained in {x : x −πgx ∈supp f, ∥πgx∥≤2s}. These facts follow directly from the displayed derivative formula and the cutoff. Orthogonal groups have commuting Taylor operators: their normal projections annihilate one another, their normal derivatives commute, and their cutofffactors depend on different normal variables. Thus their remainders commute as well. The finite field projection is combined with (CE2), rather than subtracting every component equally. For input degrees dj, set ωg = X j∈g dj, Lg = max(0, ωg + 1 −νg). Resolve the group’s field expression into projected components of degrees h = 0, . . . , Lg −1 and its remaining component. The remaining component is paired with the original test; the degree-h projected component is paired with the normal Taylor remainder through order Lg −1−h. Denote these finitely many choices by v, with gain Gv = Lg for the remaining component and Gv = h for a projected component. After normal integration the effective scale exponent is ev = νg + Gv −ωg. (CE3) For the remaining component ev ≥1, including when Lg = 0. For a projected component, h < Lg implies ev + (Lg −1 −h) = 0. At a relative shell radius a ≤1, Taylor’s remainder contributes aLg−h, leaving one full power: aevaLg−h = a. (CE4) 361 ## Source page 364 A Wilson Yang–Mills construction J. Rodgers For a > 1 the cutoffestimate needs only aLg−1−h, so the projected product is exactly 1. The remaining component can meet such a shell only up to the fixed geometric reserve, where aev is bounded. This near/far distinction is the central integrability gain. To make this argument uniform under further collisions, a local chart records a partition Q of the current label group into disjoint nonempty parts, a positive length s, a collapsed offset, a linear resizing frame A, and a closed stage inside a fixed bounded set in resized coordinates. Each part has an assigned radius Ri ∈(0, 1]; the frame rescales its normal directions by that radius. The whole normal extent and its subtraction halo fit a common detector collar B ∈(0, 1], with 2s ≤B. For every nonsingleton part, Ri is at most a fixed multiple of s. The detector requirement is B ≤clearance  X j dj  . The clearance is chosen decreasing with degree, so it also covers every projection degree used below. Other groups, their centers and their field expressions are read by a continuation constant along each current part’s normal directions; this permits use of ancestor-subtracted expressions as the exterior continuation, with their cancellations retained. Refine one nonsingleton part i into a proper partition U on an angular annular patch. Assign radius a to the new parts and leave the others unchanged. Replace the frame by its composition with the map multiplying Ni by a/Ri and fixing N⊥ i . The determinant changes by | det A′| = | det A|(a/Ri)νi, (CE5) and the engineering field factor changes by (a/Ri)−ωi. The gain in the continuation contributes (a/Ri)Gv, so their product is precisely (a/Ri)ev. Supports in the Taylor branch may enter its sub- traction halo; this is why the refined route retains both the old stage and its normally collapsed, cutoff-bounded part. Orthogonality keeps the other groups’ normal data unchanged. Their contin- uation remains locally continuous because the Taylor operation is applied within its detector collar and the refined angular patch keeps the distinct part palettes disjoint. Here is the uniform integrated estimate. Let τQ = 3 |g|−|Q|, εQ = τQγ, γ > 0, (τQ + 1)γ ≤1. A proper refinement increases the number of parts, so 2εQ′ + γ ≤εQ. (CE6) The local base budget for degree list d, continuation bound Bw, frame Aw, and radii Rj(w) is Bw(ε) = | det Aw| Bw Y i∈Q Ri(w)−ωi Y j Rj(w)−ε. (CE7) The last product includes all recorded radius registers, including unchanged ones. If selected parts are restricted to dyadic shell numbers ni, multiply this budget by Q i∈Q, |i|>1 2−niγ; an unrestricted part contributes 1. The number of required test derivatives is NQ = (2 + X j dj)(|g| −|Q|). For every permitted choice of projected or remaining components and shell restrictions, the complete resulting integrand F is integrable and Z E ∥F(x)∥dx ≤C ∥f ◦Aw∥CNQ Bw(εQ) Y i∈Q |i|>1 2−niγ. (CE8) 362 ## Source page 365 A Wilson Yang–Mills construction J. Rodgers The constant is uniform in the permitted charts and parameters, after fixing the geometric reserves, degree bounds, and continuation estimates. In (CE8) the absolute value is taken after the finite subtraction expressions have been combined. The induction proving (CE8) uses (CE3)–(CE6). Under refinement the child’s budget is the parent’s budget at εQ′, times (a/Ri)eva−εQ′. The Taylor estimate supplies the extra factor from (CE4). For every 0 ≤t ≤1 their product is at most a fixed reserve constant times (a/Ri)t: use (CE4) when a/Ri ≤1, and the bounded far-shell estimate otherwise. Choose t = γ +εQ′ ≤1. Then (a/Ri)γ+εQ′a−εQ′ = aγR −(γ+εQ′) i . The last Ri loss is absorbed by the parent’s radius surcharge using (CE6). The other shell factors are unchanged. Thus each new dyadic annulus supplies aγ, and summation over its annulus index converges. A proper partition strictly decreases |g| −|Q|; this supplies the finite induction and the stated derivative budget. Here are the shell and angular constructions used in the induction. Put ψg,a = χa(πgx) −χa/2(πgx), an = 2−n. The support of ψg,a lies in a/2 ≤∥πgx∥≤2a, and X n 0: at radius one take the positive minimum of their continuous distances on the compact support, then rescale. This proves all the reader separations needed after the refinement, even for subgroups used later. The normal Taylor bound in these coordinates is uniform in the subtraction length. On a fixed compact unit annulus, if f has bounded derivatives through n + m + 1, normal rescaling by a gives ∥Rg,s,mf∥Cn after rescaling ≤C∥f∥Cn+m+1 ( am+1, 0 < a ≤1, am, a > 1. (CE10) Here the parent frame has normal scale R, and the allowed parameters satisfy R ≤Bs, Ra ≤Bs, with the fixed normalized support bound. To prove (CE10), bound the family of cutofffunctions χ1(uπgx), 0 ≤u ≤B, and all derivatives through n on the compact annulus; smoothness on the compact product gives one constant. The normal Taylor polynomial after rescaling is bounded by a constant times max(1, a)m. The rescaled original function has a uniform derivative bound when it meets its compact support, and is zero as a germ when a is beyond that support. Leibniz’s rule gives the am bound for a > 1. For a ≤1, if 3Ra/s ≤1, the cutoffis identically one near the annulus and the integral Taylor remainder gives the factor am+1. Otherwise 1 ≤3Ba; multiplying the rough 363 ## Source page 366 A Wilson Yang–Mills construction J. Rodgers uniform bound by (3Ba)m+1 gives the same am+1 estimate with a larger constant. The angular masks have fixed normalized derivative bounds, so their product costs another fixed constant. This proves (CE10). We may now make the induction in (CE8) completely explicit. If every current group is a singleton there is no unresolved normal direction; use the permitted continuation bound, its local continuity and the bounded support in resized coordinates. Otherwise choose a nonsingleton part i, and decompose its election into the finite choices of remaining component and projected degrees. For a fixed such choice v, apply (CE9) to its test after the normal Taylor operation and then the finite angular partition. The resulting annular term equals the refined forest expression, read with the continuation that first regroups the children into i and applies its fixed component projection or remainder. This is an identity of the underlying linear expression before evaluating any hierarchy or integrating. The refined partition Q′ has more parts, so its value of |g| −|Q′| is strictly smaller. Its con- tinuation retains the original offsets, length and normal constancy. Its component fields have the prescribed palette of subcentroids; the angular separation and common detector collar ensure the projection is admissible. The inductive bound applies. Formula (CE10) pays its test derivatives, while (CE3)–(CE7) give precisely the budget Cv∥f ◦Aw∥CNQBw(εQ)aγ n Y j∈Q j̸=i 2−njγ. (CE11) The number of angular sectors and component choices is finite; summing their constants preserves uniformity. If the specified annular stage is empty, the term is zero and the same estimate holds. Otherwise its geometry is exactly the refined route already described. Since P n aγ n = (1 −2−γ)−1 < ∞, (CE11) proves summability of the integral norms of the complete annular terms. The eventual germ equality from (CE9), together with the proper-normal exceptional sets of Haar measure zero, identifies the pointwise sum almost everywhere with the unsliced component expression. To pass to its integral without any unproved domination, regard each annular term as an element of the Banach space L1(E; X). The sum of their L1 norms is finite, so they sum in L1 to an integrable function. Absolute integral-norm summability also gives their almost everywhere pointwise sum; uniqueness and the eventual identity identify that function with the prescribed unsliced expression. Continuous integration then gives Z Fv = ∞ X n=0 Z Fv,n, X n Z ∥Fv,n∥< ∞. (CE12) This proves the bound for the unsliced choice. For a specified shell index use (CE11) directly. Summing the finite component choices establishes (CE8), completing the finite induction. If the election at i is the whole expression, all its choices occur, so (CE12) yields the exact root formula Z F = X v ∞ X n=0 Z Fv,n. (CE13) For any ordered list of distinct nonsingleton current groups with no preassigned shells, one can resolve their shells successively. The common bound becomes a constant times Q j 2−njγ. This is summable over the Cartesian product of all shell indices. Repeated use of (CE12) and absolute Fubini therefore gives the full iterated sum, equal to the original integral. In particular every permitted resolution order gives that same pairing. This does not assert absolute convergence of 364 ## Source page 367 A Wilson Yang–Mills construction J. Rodgers the individual uncombined Taylor monomials; the terms in (CE12) and (CE13) are the already combined expressions. For any requested loss ε > 0, choose γ = min(1, ε)/(τQ + 1). Then the economy condition holds and εQ ≤ε. On unrestricted shells the shell factor is one, and monotonicity of R−ε for 0 < R ≤1 upgrades (CE8) to the requested arbitrary positive loss. Thus all shell identities and bounds apply with the same finite projection choices and devices. We verify the continuation inheritance required in that induction. Its hypothesis is a tested rate inequality on every logarithmic path through the allowed chart: if the scalar continuation price has rate b, and each current group expression is affordable at its assigned radius with rate ci, the reading of their tagged product has rate at most b + P i ci. The continuation is also locally continuous on the specified palettes and constant in the normal directions already assigned to each group. These properties hold for the actual hierarchy with separated exterior fields by the signed expression estimates described below. For a refinement of i at radius a, suppose a has rate −s and its parent’s recorded radius has rate −t. The angular separation gives positive lower bounds of rate −s between all palettes of different child parts. The finite product rule for affordable expressions therefore gives the product of the children rate P cj on the parent palette at exponent s. Applying the remaining component or degree-h projection at the parent’s center changes this to X j cj + (t −s)Gv. (CE14) For the remaining component the chart’s support forces t ≤s; use the full finite projection remain- der estimate through Li−1, giving Gv = Li. If Li = 0, there is no projection and the same estimate is the coarsening inequality with zero gain. For a projected component, the degree-h readout esti- mate supplies Gv = h. The common detector collar justifies these operations on the actual moving palettes. The new scalar price is exactly B′ w = Bw(a/Ri)Gv. If Bw has rate b, this has rate b + (t −s)Gv, matching (CE14). On the subtraction halo, choose a point of the original stage differing from the present point only in Ni. Such a point exists by the definition of the halo; for the remaining branch it is the point itself. Constancy of the original continuation in Ni permits evaluating there. Other current parts are orthogonal to Ni, so their palettes and expressions are unaffected. Substituting the child product with its fixed component treatment into the old tagged product is exactly the refined reading. The original path inequality now yields b + (t −s)Gv + X refined parts cj, the required refined continuation bound. This proves the inheritance used in (CE11). The local continuity assertion follows by the same substitution on two closed pieces, the original-stage piece and the normally collapsed-stage piece of the halo, and gluing their agreeing restrictions. Thus the auxiliary continuation is justified at each step, including expressions retaining earlier Taylor cancellations. We next obtain a distribution on the whole allowed position space. At each x ∈E, partition the labels by equality of the actual positions Px+o. Each part is collapsed at x; its normal displacement is zero. All subcentroids of different parts are distinct. These conditions persist on a sufficiently small closed ball about x: all part normal displacements remain inside the common collar B, and all distinct-part subcentroid distances have a positive lower bound. Crucially the same positive 365 ## Source page 368 A Wilson Yang–Mills construction J. Rodgers subtraction length s, with 2s ≤B, works on each of these balls. Applying the local forest estimate to the equality partition gives its integrable test pairing. The exact expression overlap identities identify that pairing with the single root prescription on tests supported in the ball, so no new extension values are assigned on chart overlaps. For a compact test support K, choose finitely many such balls covering it and a smooth partition of unity {θj} subordinate to their interiors. On every chart the test estimate uses at most N = (2 + X i di)|g| derivatives, since NQ ≤N. Leibniz’s rule gives ∥θjf∥CN ≤CK,j∥f∥CN . The root prescription is linear in its test, and P j θjf = f. Add the finitely many local integral-norm bounds to obtain Z E |Ks(f)(x)| dx ≤CK∥f∥CN (supp f ⊂K). (CE15) The linear functional Ss(f) = Z E Ks(f)(x) dx is therefore continuous on each fixed-compact-support smooth test space, and hence on the usual compactly supported test-function space. This is the asserted Euclidean distribution of finite order N. Its restriction to a local forest is the local pairing, because both are readings of the same test expression. The construction works for every unselected or character reader. The extension and its bounds retain the chosen subtraction length, finite readouts and normal Taylor convention. We specify the finite expression prescription, including its recursion. A score is a finite sum of field expressions tensored with linear operations on smooth tests; its action at x multiplies the field expression at x by the operated test at x. Products compose test instructions in their prescribed order. For disjoint label groups those instructions commute, by orthogonality of their normal spaces. Let Cg denote the recursively constructed local correction, and let Pg be the punctured score formed from proper partitions: Pg = X Q partition of g Q̸={g} Y i∈Q Ci. For a singleton, C{j} is its actual field expression tensored with the identity test operator. For |g| ≥2, set Cg = − Lg−1 X h=0 reshape(readerg,h)Pg Tg,s,Lg−1−h. (CE16) The full score is the sum over all partitions, or equivalently Sg = Pg + Cg. Every group appearing in a proper partition has strictly smaller cardinality, so (CE16) defines a finite recursion. The reader in this formula is the fixed physical projection at the group centroid; it acts on the field expression, while the right Taylor factor acts on the test. For Lg = 0 the sum is empty and there is no local correction. Formula (CE16) is the exact prescription whose finite rearrangement gave the component decomposition preceding (CE3). Every Cg depends on the test only through its germ at the normally collapsed point x −πgx. This follows from (CE2) in each summand of (CE16); for a singleton its normal space is zero. Now compare two partitions Q, R which are both separated on a test support. Expanding Q i∈Q Si gives precisely the terms Q j∈T Cj over all partitions T refining Q: choosing a partition separately inside every part of Q is bijective with choosing a refinement. The same holds for R. A term refining Q 366 ## Source page 369 A Wilson Yang–Mills construction J. Rodgers but not R has a group j containing labels in two different parts of R. At its normal collapse those labels coincide, whereas they are separated everywhere on the test support. Thus the test vanishes in a neighborhood of this collapsed locus, and its germ-dependent factor Cj annihilates it. All such crossing terms vanish as scores after multiplication by the test mask. Both expansions therefore reduce to the common refinements of Q and R. Acting on the test proves their equality before hierarchy evaluation and integration. This proves the overlap identity used in (CE15), including partitions not subordinate to one another. The signed expression bounds supplying the initial continuation estimates follow a finite proof tree. Its leaves are bounded labels of degree d at one position, with price R−d, and the unit with price 1. One may multiply two already priced expressions when their position palettes are separated by a positive multiple of their common radius; their prices multiply. One may carry an expression from radius R to R′ when its palette has diameter O(R) about its chosen center and R = O(R′). Subtracting its full projection through degree h multiplies its price by (R/R′)h+1, with the same size condition and the fixed reader clearance. Projecting onto just degree h multiplies its price by (R/R′)h and replaces its palette by the center alone; no ratio bound is required for this projected operation. Palette enlargement is also permitted. Along any logarithmic path where the finitely many radii have rates, induction on this tree proves the corresponding affordable-expression rate. The leaf uses the full field bound, multiplica- tion uses separated products, carrying uses coarsening, and the two projection steps use the full tested remainder and output estimates with the displayed gains. All use the same hierarchy and reader, and the remaining component is kept as its signed difference. To obtain a uniform bound in the radii, normalize their logarithms as in the nested expansion proof and pass to a compact refining limit. The product surcharge Q R−ε j supplies a strict positive rate margin against any proposed violation. Hence an expression of price p satisfies |⟨E⟩χ| ≤Cεp Y j R−ε j for all admissible geometries and all viewers. Testing its output at degree h gives the additional factor Rh. This proves the signed moment and projection bounds for every finite expression tree. It also supplies the initial continuation bound with separated exterior words: their palettes have fixed positive clearance, their label norms are bounded, and their radius can be set to one. Their full expectations and local continuity are those of the previously constructed hierarchy. The same construction permits an extra projection or remainder at the root whenever its full palette lies in the required root detector collar. Its scalar continuation price is Rh for degree h output and Rh+1 for the remainder through degree h. These operations are applied after the root’s field expression has been formed, and before the lower forest is tested; (CE14) shows that the subsequent shell induction keeps this root price intact. Consider now normalized normal tests fR(x) = R−νgf x −πgx + R−1πgx . (CE17) The normal frame is multiplication by R on Ng and identity on its complement. Its determinant is Rνg; composing fR with that frame gives R−νgf, so the determinant and test normalization cancel exactly. In a one-root chart, (CE7) consequently becomes Ru−ωg−ε, where u = 0 for the full reading, u = h for degree-h root output, and u = h + 1 for its root remainder. Thus each complete contact pairing obeys Z ∥Ks(fR)(x)∥dx ≤C∥f∥CN{g}Ru−ωg−ε (CE18) 367 ## Source page 370 A Wilson Yang–Mills construction J. Rodgers on the permitted normalized supports, uniformly over the chart parameters. The subtraction length may be fixed or varying, subject to the same support, halo and clearance conditions; it need not equal R. If f vanishes near the total diagonal N⊥ g , every term Cg in (CE16) annihilates it, by the normal jet formula. Therefore the full score agrees with the punctured score on such tests, even when they meet proper-subcluster collision sets. A test supported in any positive normal annulus has this property. Normal rescaling preserves it, so (CE18) holds for the punctured pairing with the same constants. No uniform positive lower bound on the annulus hole is required. These are the full, output and remainder scaling assertions and their punctured counterparts. A real basis can be chosen in each homogeneous contact grade simultaneously. This is a coor- dinate basis for the strata of the already fixed splitting; it does not replace the OPE prescription or its projection functionals. The splitting’s stratum is invariant under conjugation and rota- tions, because its lower-grade map intertwines them. Every vector is a complex combination of conjugation-fixed vectors by the real-and-imaginary decomposition already used for Hm. Extend a linearly independent subset of these real vectors to a maximal such subset; it spans the complex stratum, so finite-dimensionality makes it a finite real basis over C. At degree zero start with the nonzero unit, which is fixed by conjugation, and extend it. This gives a homogeneous system basis containing the unit. Expand a general input field into its finitely many homogeneous parts and then into this basis. Multilinearity of the score and reader gives the general contact expression as the finite sum of its basis expressions. Each term retains its actual assigned homogeneous degree in (CE16). Raising the ambient filtration horizon changes none of these parts, so the resulting linear extension is well defined. The compatible radius-dependent prescription is also explicit. Given any positive requested bounds fn, choose gn = min  fn, 2−n, clearance(n) 2(n + 1)  , t0 = g0, tn+1 = min(tn/8, gn+1). Then tn > 0, 4tn+1 < tn, tn →0, and 2(n + 1)tn ≤clearance(n). Let b0(s) = 1 and, for n > 0, let bn(s) be the prescribed smooth cutoffat tn. It is one for 0 ≤s ≤tn and zero for s ≥2tn. For every s > 0 these cutoffs are locally finite in n. Moreover, a nonzero bk(s), k > n, implies bn(s) = 1, by the strict threshold spacing. Let N(s) be the last nonzero cutoffindex. It exists, and N(s) →∞as s ↓0. For every active k > 0, ks ≤clearance(k). Indeed s < 2tk and the displayed threshold bound is stronger. Consequently for any fixed C, eventually Cs ≤clearance(N(s)). Define the finite smooth projection Ds(x)E = N(s) X j=0 bj(s) insertRowx,j E. (CE19) Local finiteness makes this smooth for positive s, even where the last active index changes. Biorthog- onality of the homogeneous rows gives exactly rowx,n Ds(x)E = bn(s) rowx,n E. For s ≤tn, all rows through n equal one, and projecting (CE19) through degree n is exactly the original degree-≤n projection. Thus no limiting approximation is substituted for a finite reader in 368 ## Source page 371 A Wilson Yang–Mills construction J. Rodgers the contact estimates. The construction may be slowed to meet all additional positive fn bounds while preserving these identities. It asserts no convergence as the growing cutoffhorizon itself varies. This completes the contact endpoint’s real basis and threshold existence clauses. The affordable- expression rules, the uniform logarithmic compactness argument, and the mixed-label inclusion identities are proved in (UC1)–(UC3) below. 14.1 Uniform subpower bounds on contact layers The layer estimates used in the contact construction are uniform in all radii. Their deduction from logarithmic-path estimates is the following compactness argument. Let R1, . . . , Rm be radii in (0, 1] on a configuration set U, let q > 0 be the bill without the small losses, and let F be the normed quantity to estimate. Suppose that along every nontrivial filtered sequence with an increasing clock T →∞and log Ra T −→−sa, 0 ≤sa ≤1, the bill has some exact exponential rate c, and F has upper rate at most c. Then for every ε > 0 there is a single finite C > 0 such that ∥F(w)∥≤Cq(w) m Y a=1 Ra(w)−ε (w ∈U). (UC1) If this failed, choose wn violating the inequality with C = en. Put pn,a = −log Ra(wn) ≥0, Sn = X a pn,a, Tn = n + Sn. Take an ultrafilter refining the sequence tail. Compactness of [0, 1], for finitely many coordinates, gives limits pn,a/Tn →sa ∈[0, 1] and n/Tn →k ∈[0, 1], with k + P a sa = 1. The path hypothesis applies. The supposed upper bound for F has rate c, while the violating comparison quantity enq(wn) Y a Ra(wn)−ε has exact rate c + k + ε P a sa > c. Positivity of the increment follows because k and all sa are nonnegative and their sum is one. A quantity of strictly larger exact rate eventually exceeds one of smaller upper rate, contradicting every chosen violation. The added term n in Tn handles sequences whose radii remain away from zero. Every bill used here has the required exact rate: constants, positive radii, products and fixed real powers have exact rates, and these operations generate the finite radius-power bills. Thus the argument does not require a common OPE threshold for all multiparameter paths or a common intermediate output horizon. It supplies the ordinary uniform layer inequality actually used in the contact integral. 14.2 Extending contact expressions from homogeneous to arbitrary labels Let M(d) a : E≤d →E≤a, 0 ≤a ≤d, be the fixed homogeneous splitting maps, with their inclusion maps between grade caps. The contact route construction for homogeneous labels is multilinear in each input. To see this before integration, fix all other inputs; its grade-transport identity expresses the remaining dependence as a linear map. Hence addition and scalar multiplication hold in that slot for the full expression-valued score. 369 ## Source page 372 A Wilson Yang–Mills construction J. Rodgers For labels fi ∈E≤di, define the mixed score by the finite sum Bd(f1, . . . , fk) = X 0≤ai≤di A(ai)(M(d1) a1 f1, . . . , M(dk) ak fk), (UC2) where A is the homogeneous score for the fixed chart, partition and route parameters. If all inputs are homogeneous of their stated grades, every summand except ai = di for all i vanishes, so (UC2) is the original homogeneous score. Raising the grade caps leaves (UC2) unchanged. Embed every original tuple (ai) in the enlarged finite index set. The splitting maps commute with inclusion, so those summands agree exactly. Any additional tuple has some ai > di; the corresponding split component of the included input is zero, and multilinearity makes that entire summand zero. This proves cap independence at the expression level. On a test support containing only distinct configurations, compatibility of contact charts iden- tifies each homogeneous local image with the ordinary field product times the test function. At each insertion point the splitting identity and coherent inclusion give d X a=0 inserta(M(d) a f) = insertd(f). Expanding the finite product of these sums recovers the ordinary mixed-label field word. Evalua- tion by any allowed spectator then yields exactly the original separated correlation times the test function. Every sum is finite before integration; the previously proved per-grade integral bounds justify taking its integrals term by term. Thus the mixed extension has the prescribed separated restriction, cap independence and multilinearity on the full field space. The projection gain underlying (UC1) can now be made precise. Along a logarithmic clock T, an upper rate c means a bound by e(c+δ)T eventually for every δ > 0. A finite expression is affordable at layer s and cost c when it is a finite sum of separated field words whose coefficient for a word of total field weight D has rate at most c−sD. This class is closed under finite sums, scalar coefficients with their added rate, and eventual equality. Its supports follow its finite placement palette. A tested error of precision q at a palette means that after multiplication by any separated affordable spectator word, at terminal layer one and of cost b, its expectation has upper rate q + b. This quantifies over the unselected hierarchy and every character, including varying choices of character. An expression has funds at layer s, cost c, if it has some finite terminal affordability bound and, for every precision q, differs by such a tested error from an affordable expression at layer s, cost c. When a palette lies at radius rate e−sT about a center and 0 ≤u < s ≤1, the one-scale OPE replaces an affordable word by a finite sum of fields at that center to arbitrary tested precision at layer u. Indeed choose the output horizon n so large that the gain n(s−u) dominates the prescribed precision deficit. The coefficient estimate assigns every retained output its rate c−ds, where d is its output weight. The tested remainder estimate applies to each allowed spectator geometry on this path. The choice of horizon is allowed to depend on that path and precision; it is not asserted to be uniform before (UC1). Finite sums and scalar coefficients give the same replacement for every affordable expression, and adding its existing tested error gives the same replacement for every expression with funds. For a centered finite sum of fields with coefficient bounds c −ds, the retained coefficient row of weight w has rate at most c −ws. A field of weight below w gives zero, while splitting a field of weight at least w applies a fixed linear map and cannot worsen its coefficient rate. To transfer this 370 ## Source page 373 A Wilson Yang–Mills construction J. Rodgers statement from a finite approximation to the original expression, test the error against the finite detector family for that row, including its rotations and conjugates. All these tests lie outside the fixed row aperture. Their placement distances remain bounded below independently of T, so their mutual separation has layer zero. If probeFee(w) is the sum of their fixed weights, a precision-q error gives row rate at most q +probeFee(w). Choose q sufficiently small to absorb this cost. Hence an expression with funds (s, c) has row rate c −ws and its projection through any finite horizon still has funds (s, c). This row estimate includes the full rotation average. For any varying rotation, finite-dimensional covariance and compactness of the orthogonal group preserve the rate of a coefficient vector. The rate bound for every varying choice is uniform over rotations eventually: otherwise selecting a vi- olating rotation at each bad index contradicts that bound. The normalized Haar integral is then bounded by the same exponential bound. Conjugation preserves norms, and the fixed finite decod- ing maps preserve rates. This proves the detector-row estimate without exchanging a nonuniform family of limits and an integral. Let Π≤w be the prescribed finite projection. Subtracting it and passing to a coarser layer u ≤s gives Funds(s, c; e) =⇒Funds u, c −(w + 1)(s −u); e −Π≤we . (UC3) For a centered finite field sum this is immediate: all outputs of weight at most w cancel, and any surviving weight d ≥w + 1 satisfies c −ds ≤c −(w + 1)(s −u) −du. For a general expression choose its centered approximation at precision min(q, q−maxFee(w)−wu). Projection of its error costs at most maxFee(w)+wu, where maxFee(w) = Pw j=0 probeFee(j). Both the original error and its projection consequently fit precision q. Subtract them and use the finite- sum result for the approximation. Since q was arbitrary, (UC3) follows. At u = s, simply subtract the projected affordable expression, with no gain. Repetition of (UC3) supplies the full w + 1 gain at every signed ancestor move of the contact forest. Funds multiply across separated palettes with addition of costs. Suppose e has funds (s, c), f has funds (s, d), their palettes are separated at layer s, and s ≤1. Fix terminal affordability costs a, b for e, f. For any desired tested precision q, first approximate e by an affordable g with error precision q −b. After choosing g, enlarge its layer to one and record its finite terminal cost c′. Then approximate f by an affordable h with error precision q −c′. The identity ef −gh = (e −g)f + (f −h)g places both terms at precision q, by the definition of tested precision and the two recorded terminal costs. The product gh is affordable at layer s, cost c + d: concatenate its separated words and multiply their coefficients. This proves the multiplication rule for funds. Induction proves it for any finite family of pairwise separated palettes. The order of these approximation choices matters. Although g has a controlled outgoing layer-s cost, its terminal cost at layer one can be much larger. Choosing the second approximation only after recording that finite cost pays for it without imposing a false uniform bound on all OPE horizons. The arbitrary-precision definition is exactly what permits this successive choice. Funds also imply the stated expectation bounds. To test e against an affordable spectator g of layer v, cost q, with s ≤v ≤1, first enlarge g to terminal layer one and record its finite cost b. Choose an affordable layer-s approximation to e whose error precision is c + q −b. The direct separated correlation estimate bounds the approximation times g at rate c + q, and the definition 371 ## Source page 374 A Wilson Yang–Mills construction J. Rodgers of precision bounds its error times g at that same rate. Addition proves the desired tested bound. Taking the spectator to be the empty word gives expectation upper rate c, uniformly over all character choices. This closes the expectation and multiplication rules used with (UC3). The avoidance geometry used for the equation lift can also be given explicitly. A lattice step that would traverse the forbidden positive edge in either direction can be replaced by three steps: move one unit in a different coordinate, make the intended step, and move back in that coordinate. The replacement has the same endpoints and avoids the forbidden stored edge. Replacing each offending step of any fixed coordinate path constructs a path between any two cover sites avoiding that edge. For a plaquette of axes i < j, its boundary can use a fixed forbidden edge based at b only if its own base belongs to {b, b −ei, b −ej}. This follows by listing its four stored positive edges, including the bases of its two reversed steps. Outside this finite set, take an avoiding transport path to the plaquette base as above; both the transport and the boundary then avoid the edge. These are the safe sampling probes used in the finite-dimensional span argument. Avoidance on the integer cover yields exact independence on all sufficiently large tori, uniformly over translations of the stencil. For each of the finitely many used cover edges choose the side larger than twice the largest absolute coordinate of its base and of the forbidden base, plus one. Their centered wrapped lifts then recover those bases exactly. Distinct stored cover edges stay distinct after wrapping. The link variation therefore changes none of the path factors, so their holonomies, curvatures, adjoint transports and all finite products are unchanged. Taking the maximum over the finite used-edge set gives the common side threshold in the marking identity. 15 The Gaussian short-distance comparison The short-distance comparison has two normalizations. A weak scaling sequence compares the actual hierarchy with the free Gaussian hierarchy. A finite determining collection of position tests then adjusts the coefficient frame by a small invertible map. Those finitely prescribed columns are matched exactly. General interacting moments still satisfy the purchased error bounds; they are not all equal to their Gaussian counterparts at a finite station. 15.1 Leading perturbative comparison on a common weak scaling sequence There is a further conclusion about the mean Euclidean hierarchy. Its scale parameter is the same weak sequence rk →0, with ultraviolet inverse scale r−1 k ; it does not assert interpolation to every small positive real scale. Let E0 d be the free Maxwell field spaces and Uk,d : E0 d →Ed the simultaneous scaling frames. Their unit, conjugation, signed rotation and filtration compatibilities hold eventually in each fixed degree. The transported first derivatives satisfy rkU−1 k,d+1∂aUk,d −→∂0 a, (PT1) and the transported infinitesimal rotations converge to the free ones. The real conserved stress space has dimension one. The free stress comparator is translation-normalized: it is the negative of the source’s original Wick Maxwell tensor. This sign is fixed by a Ward identity, rather than by conservation alone. If a nonzero first-derivative family D and one tensor T satisfy the translation shell identities both with cD and with c′D, subtract those identities, test against arbitrary separated exterior fields and use 372 ## Source page 375 A Wilson Yang–Mills construction J. Rodgers field separation. The resulting equality (c −c′)D = 0 gives c = c′. Apply this to the constructed free tensor and the explicit Maxwell tensor, whose translation convention is −D0. It forces their comparison constant to be −1. The explicit curvature-label calculation identifies the underlying tensor with the original Maxwell tensor, giving the claimed negative comparator, which we denote T 0,tr. Fix any finite list of free homogeneous labels qi, any subset of its slots to be replaced by stress components, and a pair of tensor indices in each such slot. In a regular slot use Uk,diqi; in a stress slot use r4 kTaibi. Then, for every compact set K of jointly distinct positions and every position- derivative order j, their mean moments at rkx converge uniformly on K, with those derivatives, to the free Wick moments with stress slots T 0,tr aibi. All arities, finite label choices and derivative orders use the same sequence and frames. To prove this passage, pull every slot back by U−1 k,di. Regular slots become constant labels; stress slots converge to T 0,tr aibi by the physical stress comparison and its Ward normalization. Expand the resulting finite-dimensional labels in a fixed basis. The common compact jet comparison handles the finitely many basis moments, while coordinate convergence and compact boundedness of the free derivatives handle the varying coefficients. This proves the mixed statement with any number of stress slots. The construction of the sequence supplies the mean hierarchy, derivatives, rotations and physical stress on the same table as the selected and reconstructed theory; no change of continuum witness is made here. The position normalization through degree four is a second finite-dimensional construction. At a general finite degree horizon d, let Bd be the real vector space of allowable small endomorphisms b of the free field space E0 d. They annihilate the unit, preserve every lower filtration subspace, commute with conjugation and signed symmetries, and commute with multiplication by i. The last property makes I + b complex linear. Put Mb = I + b. (PT2) For b near zero it is invertible; the inverse follows from a geometric series in the endomorphism norm. It fixes the unit and preserves the listed structures. The determining finite collection of position tests will choose b = bk →0, and the adjusted frame is Uk,dMbk. At horizon four the adjustment preserves all lower fields exactly. The free degree-d field space for d < 4 consists of unit multiples for a simple gauge algebra. The degree-raising maps take that unit to the unit. Since b annihilates the latter, Mb fixes every raised lower field. Thus the degree-four adjustment is compatible with the already chosen frames in degrees zero through three. A determining position test consists of a finite arity ni, labels vi1, . . . , vini ∈E0 d, and a smooth compactly supported complex profile ψi whose support lies in the open set of distinct configurations. A finite real linear map Λ combines the real parts of these readings. Choose a complex basis ea of E0 d. A finite array w records the profile-integrated basis moments, for each test and every basis tuple. Multilinearity reconstructs the reading of any tuple: Di(w; v1, . . . , vni) = X a1,...,ani Y j (vj)aj wi,a1,...,ani. Define the finite position map CQ(w, b) = Λ Re Di(w; Mbvi1, . . . , Mbvini)  i. (PT3) It is a polynomial in its finite real coordinates. Let w0 Q be the array obtained by integrating the free Wick moments against the profiles. A square determining collection means exactly that ∂bCQ(w0 Q, 0) : Bd −→Rdim Bd is invertible. (PT4) 373 ## Source page 376 A Wilson Yang–Mills construction J. Rodgers This condition is proved for a suitable collection at degree four. For this existence proof, a single test’s derivative in direction b is the real part of the integral of Jb(x; v1, . . . , vn) = X j ⟨v1(x1) · · · (bvj)(xj) · · · vn(xn)⟩0. (PT5) If all these real integrated derivatives vanish, replace each profile by iψ to see that their imaginary parts vanish as well. Continuity on distinct configurations and smooth compact test separation then show that (PT5) vanishes pointwise there. The degree-four rigidity calculation below proves that this forces b = 0. The resulting real linear functionals therefore separate B4. In a finite- dimensional space finitely many suffice: choose a finite spanning subset of their span in the dual. Their combined map has zero kernel, hence a linear left inverse. Compose that left inverse with coordinates on B4 to choose Λ. By differentiation of (PT3), its b-derivative is then precisely that coordinate isomorphism. This proves (PT4), with profiles supported away from every collision. The finite-dimensional implicit-function theorem now provides a function φ on a neighborhood of w0 Q, with φ(w0 Q) = 0, CQ(w, φ(w)) = CQ(w0 Q, 0), (PT6) and local uniqueness: for (w, b) near (w0 Q, 0), the equality in (PT6) holds if and only if b = φ(w). The map is smooth to every finite order at the base point. The proof uses only the polynomial map and the invertible derivative (PT4). Let wQ(k) be the same array measured in the actual mean hierarchy with labels Uk,dea at positions rkx. Compact separated convergence implies wQ(k) →w0 Q. To justify the integrals, the smooth profile has a fixed compact separated support, so the uniform moment error is bounded by its supremum there times ∥ψi∥L1. Set bk = φ(wQ(k)). (PT7) Then bk →0, Mbk is invertible eventually, and (PT6) gives the exact determining equations in the interacting mean hierarchy. Equality of (PT3) with those measured equations follows by expanding the inserted labels in the finite basis and exchanging a finite sum with the integrable profile. For any other finite collection R of such separated profiles, continuity of CR and convergence (wR(k), bk) → (w0 R, 0) give CR(wR(k), bk) −→CR(w0 R, 0). (PT8) Thus the exact normalization does not spoil any other leading position test. The endpoint also compares this construction with numerical jets of finite Wilson data. For a fixed finite torus with side greater than one, let wQ,l(m) denote the coefficient of order lof its regulated profile series at mesh parameter m. Its order-zero coefficients tend to w0 Q as m →∞. For a fixed truncation order p, form the finite polynomial curve cQ,p,m(t) = Pp l=0 tlwQ,l(m). This curve is used only to compute numerical derivatives. It requires no convergence of the full formal series. Equation (PT6) holds along the curve near t = 0, for all sufficiently large m, because its base point tends to w0 Q. Therefore every positive derivative at zero of CQ(cQ,p,m(t), φ(cQ,p,m(t))) vanishes exactly, and its zeroth derivative equals the prescribed constant. For another query R, the order-zero numerical reading is CR(wR,0(m), φ(wQ,0(m))) −→CR(w0 R, 0). (PT9) Subtract (PT9), at m = k, from (PT8). Their difference tends to zero on the same actual-theory sequence. The order-zero numerical reading is independent of the chosen polynomial truncation 374 ## Source page 377 A Wilson Yang–Mills construction J. Rodgers p. These are exactly the numerical-jet calibration and leading-comparison assertions: calibration of the determining equations at each finite formal horizon, and order-zero comparison for other profiles. They do not assert comparison of every higher-order coefficient with the actual interacting hierarchy. Any other small correction satisfying the same exact determining equations equals (PT7) even- tually, by the local uniqueness in (PT6). This identifies the correction chosen by the local map with the correction appearing in the endpoint. Combined with (PT2), the unit and symmetry identities and the exact preservation of raised fields of degree below four, it proves all the frame properties claimed for the degree-four position normalization. Here is the degree-four rigidity used in (PT5). Its free field basis consists of the unit and the color-contracted curvature products ec = X α F α abF α uv, (PT10) indexed by unordered pairs c of oriented coordinate planes, each plane put in its increasing-axis orientation. Reversing a plane changes the sign and interchanging the planes preserves the product. The free quotient calculation proves that these labels, together with the unit, form a basis; its dual coordinates will be denoted l∗and lc. At the separation (e0, 0), the free curvature two-point form is diagonal on the six coordinate planes, with nonzero coefficients γp. Wick contraction therefore gives ⟨v(e0)ec(0)⟩0 = aclc(v), ac ̸= 0. (PT11) For a repeated plane the contraction has its two equal pairings; for two different planes it has the single matching pairing. Accordingly ac is the product of the two nonzero curvature coefficients and dim g, multiplied by the corresponding nonzero Wick multiplicity. The dual-coordinate nor- malization absorbs that multiplicity. The unit coordinate is expectation. If l∗(v) = 0, the reversed pairing ⟨ec(e0)v(0)⟩0 has the same formula. These facts follow by expanding v in the basis and applying the diagonal contractions, so no positive-definiteness of this particular bilinear form is required. For each plane pair c = (p, q), let its parity set be the symmetric difference p△q of their axis sets. Reflection in axis a multiplies ec by −1 exactly when a belongs to this parity set. If b commutes with all such reflections, its matrix coefficient bcd = lc(bed) vanishes whenever c, d have different parity sets. For equal parity sets there is an involutive permutation of the four axes exchanging the two plane pairs, up to reversing individual plane orientations and swapping the planes. This finite assertion has three cases. If the parity set is empty, both pairs repeat a plane, and a transposition or a product of two disjoint transpositions exchanges the two planes. If it has two elements, each pair consists of two planes sharing one of the other two axes; exchange those common axes when they differ. If it has four elements, each pair is a partition into two complementary planes; a transposition exchanges any two different such partitions. The identical-pair case uses the identity. These permutations belong to the list consisting of the identity, six transpositions and three products of disjoint transpositions. The induced action sends ec to εed and ed to εec, with the same sign ε ∈{1, −1}: applying the involution twice gives product of signs one and the basis vectors are nonzero. Commutation of b with this action gives bcd = bdc. Moreover ac = ad for equal parity sets. Their Wick multiplicities agree, and the product of their two curvature signs at the axis-zero separation depends only on whether zero belongs to the parity set; their absolute curvature factors are the same. This proves the equality of the coefficients in (PT11). 375 ## Source page 378 A Wilson Yang–Mills construction J. Rodgers Now suppose (PT5) vanishes for every separated configuration and tuple. Arity one gives l∗(bv) = 0 for every v. Arity two at (e0, 0) gives ⟨(bv)(e0)w(0)⟩0 + ⟨v(e0)(bw)(0)⟩0 = 0. (PT12) Take v = ec, w = ed. Equation (PT11) and the zero unit coordinate give adbdc + acbcd = 0. For different parity sets both coefficients already vanish. For equal parity sets, symmetry and ac = ad ̸= 0 give 2acbcd = 0, hence again both vanish. The unit column vanishes because b1 = 0; the unit row vanishes by arity one. Every coefficient of b is zero, proving the rigidity assertion. This closes the existence proof of the square position tests through degree four. The physical stress comparison used before (PT2) can be proved directly on this weak sequence. Let Jraw ̸= 0 be its constructed raw conserved centered tensor, and let ξk ̸= 0 be the scale used in its preliminary stress comparison. The one-dimensional divergence kernel implies Jraw = c center(T), c ̸= 0. The centered tensor is nonzero because its translation Ward charge differentiates a nonzero field. Compare its charges with those of Jraw. Field separation applied to every translation shell identity gives equality of their derivative families, Draw = cD. The weak derivative comparison then yields c/ξk →κ0, where κ0 ̸= 0 is the free stress Ward amplitude. Multiplying the preliminary raw stress limit by (c/ξk)−1 gives U−1 k,4(r4 k center(T)) −→κ−1 0 T 0. (PT13) The difference from T is a fixed constant unit tensor. Its inverse-scaled contribution is r4 k times that unit, tending to zero; the unit identifications are exact eventually. The Ward argument preceding (PT2) gives κ0 = −1, proving the stress-slot limit used there. Similarly, the preliminary angular- generator limit is normalized by the factor (c/ξk)−1, giving the transported rotation limit in the final physical normalization. The compact-profile convergence needed in (PT7) requires no distribution extension through collisions. If ψ is one of the allowed compact separated profiles and fk →f0 uniformly on its support, take δ = ε/(1 + ∥ψ∥L1). Eventually Z ψ(fk −f0) ≤∥ψ∥L1 sup supp ψ |fk −f0| ≤∥ψ∥L1δ < ε. Smoothness of the separated moments makes both integrals ordinary Bochner integrals. This supplies every component of wQ(k) →w0 Q. For the regulator arrays in (PT9), the order-zero Wilson coefficient is the Gaussian Wick con- traction of the reduced local field polynomials. The common field-address identity identifies these polynomials with the free field labels. At fixed lattice refinement base L > 1, the denominator is Lm, the mesh spacing is L−m, and the chosen period tends to infinity fast enough for every fixed polynomial margin. The Gaussian row limit holds for every refining subsequence and every sequence of grid tuples tending to a distinct continuum tuple. The grid-sum limit for a compact separated profile then gives wQ,0(m) −→w0 Q. (PT14) The Gaussian kernel, Wick row and grid-sum arguments underlying this assertion are developed next, including the finite-regulator subtraction and its limiting cancellation. 376 ## Source page 379 A Wilson Yang–Mills construction J. Rodgers 15.1.1 From moving Gaussian rows to profile integrals Here is the grid-sum passage in (PT14). Let hm > 0 tend to zero, hm = D−1 m with integer Dm, and let the torus side Nm satisfy hmNm →∞. For an arity n, round every coordinate of a continuum configuration x to the grid. Denote the rounded point by ρmx, so |ρmx −x| ≤4nhm. (PT15) Every fixed bounded set eventually embeds in the torus without wrapping, because hmNm →∞. Suppose lattice kernels Km have the following joint moving limit: for every subsequence mj → ∞, every x in the allowed open region, and every grid tuple oj with embedded positions tending to x, one has Kmj(oj) →K0(x). This is stronger than convergence only at a fixed rounded point and is exactly what the Gaussian row calculation supplies. It implies a uniform eventual bound for Km on rounded points in any fixed compact allowed set. Otherwise choose mj ≥j and points xj in that compact set with |Kmj(ρmjxj)| > j. A convergent subsequence xj →x, (PT15) and eventual nonwrapping give a sequence of embedded grid tuples tending to x, contradicting the moving limit. For a continuous compactly supported profile ψ with support in the allowed region, define the step function fm(x) = ψ(ρmx)Km(ρmx). It converges pointwise to ψ(x)K0(x). On the support use the moving limit and continuity of ψ; outside its closed support the rounded points are eventually outside it, so the expression vanishes. If ψ is supported in the radius-R ball, the step functions, for hm ≤1, are supported in the radius- R + 4n ball and are bounded by a common constant there. They are integrable step functions. Dominated convergence gives convergence of their integrals. Each grid cell has exactly volume h4n m . Summing the cells, with no wrapping on the fixed support, identifies the integral exactly with h4n m X o∈(Z/NmZ)4n ψ(hmo)Km(o). (PT16) The cell partition and its volume hold also at arity zero, when both the configuration space and the grid consist of one point and h0 m = 1. This proves convergence of (PT16) to the desired continuum integral with its correct normalization. The Gaussian row limit is reduced to a two-point covariance limit by an exact finite recursion. Give every occurrence of every generator in the finitely many local polynomial labels a distinguish- able index a. Let τ(a) be its insertion slot and let ζa be its field-generator label. At order zero, the row is the Gaussian moment Mm(A) of the product of the corresponding centered linear variables, for each finite set A of occurrences. Gaussian integration by parts gives |A|Mm(A) = X a∈A X b∈A\{a} Cm(a, b) Mm(A \ {a, b}). (PT17) For A = ∅, Mm(A) = 1 because the Gaussian law is a probability measure. The continuum Wick moments obey the same recursion with covariance C0(a, b). Induct on |A|: the covariance limit and the finitely many already proved smaller-product limits give convergence of the right side, and division by the positive integer |A| gives convergence of Mm(A). This occurrence indexing accounts for repeated powers without identifying identical generators prematurely. An arbitrary monomial is the product over its occurrence set; arbitrary finite poly- nomial labels are finite sums of such monomials with their coefficients. Multilinearity and finite summation extend the preceding limit to every tuple of the allowed local polynomial sections. The 377 ## Source page 380 A Wilson Yang–Mills construction J. Rodgers field-address identity sends the reduced polynomial section of a free label to that same label in the free Wick hierarchy. Therefore the limiting row is precisely the continuum free moment, not an independently fitted polynomial. Applying the result along every refining subsequence supplies the moving-limit hypothesis used in (PT16). This proves (PT14), once the corrected covariance limit used in (PT17) is established. 15.1.2 Private Gaussian marks and the subtraction covariance Private Gaussian marks distinguish contraction occurrences without changing the physical co- variance. Their subtraction covariance must be computed with the stated Fourier and opposite- momentum convention. Only after the corrected bilinear covariance is identified can the occurrence recursion and scalar return estimate be used for higher products. The covariance used in (PT17) has a finite-regulator subtraction that must be retained before taking the limit. Fix the largest jet grade d occurring in a polynomial row. Its finitely many linear coordinates are indexed by keys a. Let Z be the centered real Gaussian coordinate vector of the regulated potential, with precision matrix P and covariance P −1. Write kh,x,a for the coefficient vector of the linearized, symmetrically averaged jet coordinate at x. Thus its value on Z is kT h,x,aZ. These coordinates use the same Gaussian density as the regulated perturbative integral. The subtraction matrix is defined on the infinite lattice. If c ranges over sine and cosine waves, vector axes and color indices, and lh,a,c(p) is the jet coordinate evaluated on that wave, set Sh(a, b) = (2π)−4 Z [−π,π]4 δ(p)−1 X c lh,a,c(p)lh,b,c(p) dp. (PT18) Here δ is the lattice Laplacian symbol. The wave responses are continuous on the cube, hence bounded; δ−1 is integrable in four dimensions, so every entry exists. Symmetry is immediate. For any real vector v, vT Shv = (2π)−4 Z [−π,π]4 δ(p)−1 X c  X a valh,a,c(p) 2 dp ≥0. Consequently Sh is the covariance of a possibly degenerate centered real Gaussian vector. No inverse of this matrix is needed. For insertion indices i = 1, . . . , n, take independent copies Yi of this Gaussian vector, indepen- dent also of Z, and put Xi,a = kT h,xi,aZ + i Yi,a. All these variables are complex linear functions of one finite-dimensional real Gaussian vector. Their means vanish and their bilinear covariance is Ch((i, a), (j, b)) = kT h,xi,aP −1kh,xj,b −1i=jSh(a, b). (PT19) Indeed the mixed terms vanish by independence and centering; the two private marks have covari- ance zero for different insertions, and multiplication of their imaginary coefficients gives the minus sign for equal insertions. This is a bilinear covariance, with no complex conjugation. At finite regulator its same-insertion entries need not vanish. The constant term of the normalized perturbative integral is exactly the expectation of the product of the reduced insertion polynomials evaluated at these Xi,a. To see this directly, the weight and vacuum denominator both have constant coefficient one. The constant coefficient of each linear insertion generator is its physical Gaussian coordinate plus its imaginary private mark. 378 ## Source page 381 A Wilson Yang–Mills construction J. Rodgers Expanding a polynomial is a finite operation, so taking constant coefficients commutes with that expansion. Integration over each private mark produces the prescribed subtraction within its own insertion. The product law then combines these independent integrals with the physical Gaussian integral. Every integrand is a polynomial in a finite Gaussian vector and is integrable: a polynomial is bounded by C(1 + ∥z∥)q, and all Gaussian moments are finite. This also justifies each finite expansion and use of Fubini. For completeness the moment recursion does not require a nonsingular Gaussian law. A cen- tered real linear Gaussian variable L, of variance v, has moment generating function exp(vt2/2). Differentiating at zero gives ELr+1 = rv ELr−1. Apply this identity to arbitrary real linear combinations of a finite family of centered real linear probes and compare coefficients. Polarization gives |A| E Y α∈A Xα = X α∈A X β∈A\{α} E(XαXβ) E Y γ∈A\{α,β} Xγ. (PT20) Initially the probes can be taken real. Expanding each complex probe into its real and imaginary parts extends the identity complex multilinearly. Occurrences, including repeated occurrences of the same generator, remain distinct elements of A. The empty product has expectation one. These facts prove the finite Gaussian recursion used above. 15.1.3 The limit of the corrected covariance Same-insertion subtraction The corrected covariance removes contractions within one pre- scribed insertion while retaining contractions between distinct insertion positions. This is an occurrence-level subtraction: repeated generator names do not merge their occurrence indices. Its two limits have different purposes. At different positions it recovers the Maxwell two-point kernel; at the same insertion it tends to zero. The latter is a regulator-removal conclusion, not an identity imposed on every finite torus covariance. The subtraction in (PT19) has the following two different limiting roles. For distinct physical limiting insertion positions zi ̸= zj, Ch((i, a), (j, b)) −→K(ka, kb; zi −zj), where ka is the corresponding coordinate functional on free solution jets and K is their Maxwell two-point kernel. For i = j, Ch((i, a), (i, b)) −→0. (PT21) Thus zero same-insertion covariance is a conclusion after removing the regulator. It is not an identity for a finite torus. Here is the averaging step that identifies both limits. Let S be the finite group of signed coordinate transformations and cS = |S|−1. For a jet functional A, its averaged lattice stencil has the exact form A(wideLinearh,x) = cS X s∈S A ◦turns ◦radialMap(charts,h,x). (PT22) Denote the functional inside this sum by As. On free equation solution jets the radial reconstruction is an inverse to sampling; equivariance therefore gives plains(As) = A ◦gaugeJet . 379 ## Source page 382 A Wilson Yang–Mills construction J. Rodgers In particular pairing independently transformed stencils with their inverse turns gives the same con- tinuum two-point form for every pair s, t. The relative endpoint positions are kept fixed. Bilinearity expresses the finite-torus covariance as c2 S X s,t ordinarys,t(h; x, y; As, Bt). Similarly expanding the two factors in the wave integral (PT18), then interchanging its finite sums and integral, expresses Sh as the same double average of the infinite-lattice shell forms. The integrability already established justifies this interchange. The separated chart limit gives the same free pairing for every summand when x −y tends to a nonzero physical vector. The return chart limit says that, at equal endpoints, each ordinary form minus its shell form tends to zero. Taking the finite double average proves (PT21) and the separated limit. The reporting schedule meets both chart-limit hypotheses. With an integer side L > 1, its spacing and period are hm = L−m, Nm = 4Lm2+m+1, hmNm = 4Lm2+1. (PT23) In particular Nmhq m →∞for every fixed integer q, and 4 | Nm. Center the observed positions at the quarter-period point. If their centered physical coordinates converge, their centered integer coordinates are eventually less than Nm/4 in absolute value, since hmNm →∞. Hence every fixed stencil shift of their difference is represented without wrapping and has the required physical limit. A sum of the two positions lies at twice the quarter-period center plus a bounded physical displacement. Its lift has at least one coordinate of absolute value at least Nm/4. Its physical distance is consequently at least hmNm/4 →∞. This puts the inversion image outside every bounded physical region, including when the two observed positions coincide. It supplies the return limit used for the subtraction. Finally apply (PT20) by induction on the number of occurrences. Covariance convergence (PT21) gives convergence of each moment to the recursion with zero within-insertion pairings and Maxwell pairings between distinct insertions. This is precisely the free Wick rule for reduced in- sertion polynomials. Finite polynomial expansion proves convergence of every moving row. The compactness and cell-integration argument in (PT15)–(PT16) then proves convergence of their profile integrals. This completes the Gaussian constant-coefficient step needed in the leading per- turbative comparison. The reduction of the chart forms to scalar lattice kernels is also explicit. An elementary stencil a is a finite signed sum of shifted edge readings, with coefficients at, offsets ut, a vector axis, and an ordered derivative list Da. Its Fourier polynomial factors as a sign, a translation character, and the product of the difference multipliers in Da. Thus, for two such stencils, X t,s atbsχp(z + ut ± vs) = σa,b,±χp(z + ua,± + ub,±) Y ν∈DaDb qν(p). (PT24) This follows by multiplying their two finite Fourier polynomials and using χp(x + y) = χp(x)χp(y). The prescribed offsets and signs are precisely those in the elementary stencil factorization, so no limiting replacement occurs in this identity. Multiplying by the torus inverse-Laplacian Fourier weight and summing over p identifies their covariance with the corresponding differentiated scalar torus kernel, multiplied by h−|Da|−|Db|−2 and the same sign. For the infinite-lattice expression, sum the sine and cosine channels first. The elementary identity cos(p · u) cos(p · v) + sin(p · u) sin(p · v) = cos(p · (u −v)) 380 ## Source page 383 A Wilson Yang–Mills construction J. Rodgers shows that the sum over channels is zero unless both vector axes and colors match; when they match it is the cosine of the offset difference. Integrating against (2π)−4δ(p)−1 gives exactly the infinite-cover scalar Green function at that difference. The bound | cos | ≤1 and integrability of δ−1 justify every such integral. A curvature coordinate is a difference of two elementary edge stencils. Its two-point form consequently has four terms, with signs +, −, −, +. Both its finite-torus covariance and its infinite- cover subtraction have exactly this four-term expansion. At separated positions, (PT24) reduces each torus term to the scalar differentiated-kernel limit. At inversion-image positions it reduces each term to the scalar far-distance limit, which is zero. At coincident positions subtract the infinite-cover value before rescaling. The torus-to-cover return estimate makes this difference tend to zero even after multiplication by any fixed inverse power of h, because Nmhq m →∞for every fixed q. These are respectively the separated, far-image, and return scalar estimates needed by the chart argument. Finally elementary coordinate readings span the finite-dimensional dual of every raw jet-response space: these responses are arrays of color components, vector-index pairs and derivative-index words, and coordinate projections span the dual of a finite array space. Expand each of two arbitrary dual vectors as a finite linear combination of these readings. Bilinearity reduces their covariance convergence to the elementary curvature calculations. This establishes the passage from elementary stencils to arbitrary chart functionals used in (PT22). 15.1.4 Scalar return estimate with its rescaling rate The scalar subtraction at a fixed integer displacement has a useful quantitative proof. Rescale the Fourier cube to Q = [−1 2, 1 2]4, put d(x) = δ(2πx), and let fa(x) = d(x)−1 cos(2πa · x), fa(0) = 0. The value assigned at zero agrees with the zero-mode convention for the inverse Laplacian and does not affect its integral. The elementary lower bound for the lattice symbol implies |fa(x)| ≤∥x∥−2 (x ∈Q). Let vN(x) be the cell representative in the torus Fourier quadrature. Outside cell boundaries, which have measure zero, vN(x) ∈Q, ∥vN(x) −x∥≤4/N, and, when vN(x) ̸= 0, ∥vN(x)∥≥1/N. These imply ∥x∥≤5∥vN(x)∥, hence |fa(vN(x)) −fa(x)| ≤26∥x∥−2. If 0 < r ≤1, 8/N ≤r, and ∥x∥> r, then ∥vN(x)∥≥r/2. The numerator cosine and the symbol d have bounded derivatives on a ball containing Q. Subtracting two quotients therefore gives |fa(y) −fa(x)| ≤Car−4∥y −x∥ (∥x∥, ∥y∥≥r), and with r/2 in place of r, |fa(vN(x)) −fa(x)| ≤AaN−1r−4 (∥x∥> r). For example the quotient subtraction is (cosa(y) −cosa(x))d(y)−1 + (d(x) −d(y))d(y)−1d(x)−1 cosa(x), 381 ## Source page 384 A Wilson Yang–Mills construction J. Rodgers which proves the asserted bound directly from the Lipschitz bounds and d−1 ≤r−2. The singular part is integrable. In four dimensions the change of variables x = ru gives Z ∥x∥<2r ∥x∥−2 dx = r2 Z ∥u∥<2 ∥u∥−2 du. Split the quadrature error at radius r, use the bound 26∥x∥−2 inside and the uniform estimate outside, and obtain |Re GN(a) −G∞(a)| ≤Car2 + AaN−1r−4. (PT25) Here the torus Fourier sum is exactly the integral of fa ◦vN over Q: every representative cell has volume N−4. The infinite-cover value is exactly the integral of fa over Q, by p = 2πx. Even N ensures the chosen symmetric representatives tile this same cube. For any fixed integer d ≥0, choose r = hd+1. Along the schedule, eventually r ≤1 and 8/N ≤r. Multiplying (PT25) by h−d gives h−d |Re GN(a) −G∞(a)| ≤Cahd+2 + Aa(Nh5d+4)−1 −→0. (PT26) This proves the return estimate with every fixed rescaling power that occurs in a finite jet calcula- tion. 15.1.5 Recovering the whole separated scalar kernel from annuli Annular convergence alone determines differences of cutoffs and can leave an additive ambiguity. The small-cutoffbound and the large-cutofftail bound below fix that ambiguity. The comparison chooses the small cutofffirst, then a finite number of dyadic annuli, and only then the lattice index. No uniform convergence in a moving cutoffis assumed. The separated Fourier limit uses a cutoffcomparison that also fixes the otherwise undetermined constant between annular limits. Here is the complete comparison principle. Suppose Fm ∈C, cutofffunctions Am(s), a continuum cutoffA(s), and a candidate limit F satisfy, for every s > 0, Am(2s) −Am(s) −→A(2s) −A(s). Suppose there are constants C, B, E, D ≥0, independent of s, such that eventually in m, for each fixed s, |Am(s)| ≤Csp, |Fm −Am(s)| ≤B/s, |A(s)| ≤Esp, |F −A(s)| ≤D/s, (PT27) where p ≥1. Then Fm →F. Indeed choose 0 < s ≤1 so that (C +E)sp < ε/3, and next an integer q so that (B+D)/(2qs) < ε/3. Finite telescoping of the annular convergence gives Am(2qs) −Am(s) −→A(2qs) −A(s). For sufficiently large m its error is less than ε/3. Adding the two large-cutofferrors, this difference error, and the two small-cutoffsizes bounds |Fm −F| by ε. The order of choices is s, then q, then m; no uniform convergence in the cutoffparameter is assumed. Apply this principle to the differentiated torus Green kernel at a nonzero limiting physical displacement, with Am(s) its smooth Fourier cutoff, A(s) the corresponding continuum cutoff, and p = |D|+2. The cutoffkernel is bounded by Cs|D|+2. Its complement is bounded by B/s, uniformly 382 ## Source page 385 A Wilson Yang–Mills construction J. Rodgers when the physical displacement is bounded away from zero. The Fourier mesh condition required for this estimate, (2π/N)(|D| + 4) ≤hs, hs ≤1 2, holds eventually for each fixed s, since Nh →∞and h →0. The annular Riemann sum converges to the continuum annular integral. Thus (PT27) proves the separated scalar limit used in (PT24). The continuum candidate in this argument is the ordinary derivative ∂DG(z), with G(z) = (4π2∥z∥2)−1. One can identify it while retaining an integrable heat regulator b > 0: inverse Fourier transformation of the regulated differentiated inverse-square symbol gives ∂DG(z) −∂D  G(z)e−π2∥z∥2/b (z ̸= 0). The Fourier convention in this formula has character e2πik·z. To remove heat take b = v−2, v →∞. The second term equals v|D|+2∂D  Ge−π2∥·∥2 (vz). Every derivative of the fixed function on the right decays faster than any inverse power at infinity, so this term tends to zero for z ̸= 0. At a fixed Fourier cutoff, dominated convergence also removes heat under the integral. The annular tail bound survives this limit. Summing its dyadic estimates C/(2js) yields at most 2C/s, supplying the continuum half of (PT27). For a physical displacement whose norm tends to infinity, the same comparison principle applies with F = 0, A(s) = 0. The annular summation-by-parts estimate is | annulush,s,D(x)| ≤CD s  2π ∥h lift(x)∥ |D|+3 . For each fixed s this tends to zero. The small-cutoffsize and B/s tail estimates remain valid once the physical distance is at least one. This proves the far-image scalar limit. The uniform low-frequency bound in (PT27) can be proved without oscillation. In normalized momentum coordinates q = p/(hs), put t = hs. The lattice denominator is dt(q) = 3 X ν=0 q2 ν sinc(tqν/2)2. On ∥q∥≤1, 0 ≤t ≤2, the inequality sinc(u) ≥2/π for |u| ≤1 gives dt(q) ≥(2/π)2∥q∥2. Each normalized difference numerator has modulus at most |qν| ≤1, because |eiu −1| ≤|u|. The normalized cutoffsymbol for any derivative list is therefore bounded by (π/2)2∥q∥−2 on its unit-ball support, independently of the length of that list. For centered integer momentum lifts, let Bm contain those whose largest coordinate in absolute value is at most m. Injectivity of the lift gives |Bm| ≤(2m + 1)4. On Bm \ B⌊m/2⌋, the inverse squared Euclidean norm is at most 4/m2. Strong induction now gives X p∈Bm ∥lift(p)∥−2 ≤432m2, where the zero summand is assigned zero. Indeed the inner contribution is at most 432(m/2)2 = 108m2, and the outer contribution is at most (3m)4(4/m2) = 324m2; the case m = 0 is zero. 383 ## Source page 386 A Wilson Yang–Mills construction J. Rodgers Write η = 2π/N, T = η/t. Rescaling this last inequality bounds the inverse-square sum on ∥q∥≤1 by 432T −4. The physical normalization of the cutoffsum is (Nh)−4s|D|−2. Since Ts = 2π/(Nh), multiplication gives | pieceh,s,D(x)| ≤432(π/2)2 (2π)4 s|D|+2. (PT28) This is the first estimate in (PT27), uniformly also at coincident positions. For the oscillatory estimate, repeated discrete integration by parts is exact on the finite torus. If ∆−eν denotes a momentum difference, translation of the summation index gives X p χp(x)∆v −eνf(p) = (χeν(x) −1)v X p χp(x)f(p). (PT29) The normalized annular symbol is smooth, supported where ∥q∥≤2, and all its derivatives through any prescribed finite order are bounded uniformly for 0 < t ≤1. This follows from its cutoffaway from the denominator’s zero and the smooth sinc extension at t = 0. A momentum difference has step size T in q. Repeated use of the fundamental theorem of calculus bounds its v-th difference by CD,vT v. If vT ≤1, the difference vanishes for ∥q∥> 3. There are at most 84T −4 momentum representatives in this enlarged support: each integer coordinate has absolute value at most ⌊3/T⌋, and T ≤1. These statements remain valid across a representative boundary. The periodic uncut symbol agrees on the two lifts; the annular cutoffis zero on both whenever a shift crosses that boundary, provided ηv ≤1/2. For the complementary high-frequency symbol, which has no compact support in q, one instead uses its uniform derivative bound on the rescaled Fourier box when t ≥1/2. All N4 representatives then contribute, but N ≤4π/T, so the same CT v−4 bound results. Combining these facts with (PT29) gives |χeν(x) −1|v X p χp(x)f(p) ≤CD,vT v−4. Choose a coordinate whose physical displacement has absolute value at least half its Euclidean norm. For centered lifts, the chord inequality for the circle implies |χeν(x) −1| ≥∥h lift(x)∥/(Nh). Set v = |D| + 3 and restore the physical normalization. Using Ts = 2π/(Nh) gives | annulush,s,D(x)| ≤CD,v (2π)4s  2π ∥h lift(x)∥ v . (PT30) The same bound holds for the high-frequency complement when 1/2 ≤hs ≤1. To bound the entire tail choose the first integer q for which h2qs > 1/2. Then h2qs ≤1. Decompose the tail into the annuli at s, 2s, . . . , 2q−1s and the final high-frequency complement. At a physical distance at least R > 0, (PT30) bounds the annular sum by 2CD,R/s and the complement by CD,R/s. Hence | jointh,D(x) −pieceh,s,D(x)| ≤3CD,R/s. 384 ## Source page 387 A Wilson Yang–Mills construction J. Rodgers This proves the torus tail estimate used in (PT27) and shows explicitly why an absolute bound on the uncut infinite-volume derivative sum is unnecessary. The annular quadrature limit used above follows from a fixed-support dominated-convergence argument. Fix s > 0 and write cm = Nmhm, an integer for the reporting schedule. Physical Fourier sample points have spacing c−1 m . The normalized annular integrand is jointly continuous in its lattice parameter t = hms, its physical displacement, and its Fourier variable, including t = 0. It vanishes outside a fixed ball of radius 2s/(2π) and is uniformly bounded there when 0 ≤t ≤1. Choose an integer R containing that support. Eventually 2Rcm < Nm, because this inequality is 2Rhm < 1. Thus every integer Fourier address in the supporting box is its own centered torus representative. The annular torus sum is exactly the unrestricted grid quadrature over this box. Let rm(k) be the grid representative of k, so ∥rm(k)−k∥≤4/cm. If Fm is the annular integrand, define the step function Fm(rm(k)). Its integral is the quadrature, since each cell has volume c−4 m . The step functions converge pointwise to the continuum annular integrand by joint continuity, t →0, convergence of the physical displacement, and rm(k) →k. They are bounded by a common constant times the indicator of the ball of radius R + 5: a cell whose representative lies in the support is within distance four of it. Dominated convergence proves convergence of their integrals. The exact phase identity underlying this representation is e2πi⟨hm lift(xm),lift(p)/cm⟩= χp(xm). The normalized radial variable is also exactly the lattice radial variable, since cm = Nmhm. This proves the annular Riemann-sum statement with moving displacements. For clarity the continuum oscillatory bound can be obtained uniformly in the heat parameter. Let MD(q) be the normalized continuum annular symbol and HD,a(q) = e−a∥q∥2MD(q), a ≥0. Its support is contained in the compact annulus 1/2 ≤∥q∥≤3. For each fixed derivative order j, the derivatives through order j are bounded by a common constant times the indicator of this annulus, uniformly in a ≥0. To prove the heat-factor part, put y = √a q. Its i-th derivative is ai/2 times the i-th derivative of e−∥y∥2. When ∥y∥≤1, the lower bound ∥q∥≥1/2 implies √a ≤2, so compact derivative bounds suffice. When ∥y∥≥1, use √a ≤2∥y∥and the rapid decay of the Gaussian derivatives. Leibniz’s rule and bounded derivatives of MD give the stated common bound. Integrating by parts in the continuum Fourier integral now gives, for every integer j ≥0, ∥x∥j Z e2πik·xHD,a(k) dk ≤CD,j, a ≥0. For example select a coordinate with |xν| ≥∥x∥/2, integrate j times in that coordinate, and bound the resulting derivative in L1. There are no boundary terms because the integrand is smooth and compactly supported. Rescaling k = (s/(2π))q in the continuum annular integral contributes s|D|+2; taking j = |D| + 3 leaves a factor s−1. The rescaled heat parameter remains nonnegative, so the bound is uniform during heat removal. This proves the continuum annular bound and the summable dyadic tail asserted after (PT27). 16 The centered trace and its non-freeness test The trace is obtained from actual finite current and stress functions, not from an assumed anomaly formula. Localizing those functions introduces explicit finite residuals. The coefficient detector, 385 ## Source page 388 A Wilson Yang–Mills construction J. Rodgers common scaling panels and equation marker control those residuals, and a two-coordinate correction then yields an exactly conserved physical stress tensor. Its centered trace is the field whose positive spectral measure gives the final asymptotic and comparison exclusions. 16.1 The finite trace candidate, its equation identity and its compensation The local current and stress entering this chapter are finite configuration functions. For a base site o, let Cab(o, U) = curv(Ua(o)Ub(o + ea)Ua(o + eb)−1Ub(o)−1). The curvature map is the real color projection specified in the startup construction, so this expres- sion is defined on every group configuration. Gauge transformations act by the adjoint at o. Define a link-valued current and a scalar tensor by (Yb(o, U))(x,i) = 1{x=o}Cbi(o, U), Jab(o, U) = 1 4δab X i,j ⟨Cij, Cij⟩− X i ⟨Cai, Cbi⟩. (TF0a) Color coordinates are converted back to the actual link-coordinate space by the fixed color isometry. At the identity every C vanishes. Its derivative in a logarithmic link direction A is precisely cab(o, A) = Aa(o) + Ab(o + ea) −Aa(o + eb) −Ab(o). Thus Y (1) = 0; J(1) = 0 and DJ(1) = 0. Its Hessian is obtained by replacing a product ⟨C, C′⟩ by ⟨c(A), c′(B)⟩+ ⟨c(B), c′(A)⟩. These formulas supply the actual quadratic jets, rather than an unspecified continuum stress approximation. Initially J is symmetric and has identically zero trace. The divergence used for the finite Taylor calculation is the fixed high-order antisymmetric stencil Daf(o) = 4 X l=1 dl{f(o+lea)−f(o−lea)}, (d1, d2, d3, d4) = (4/5, −1/5, 4/105, −1/280). (TF0b) It has first moment one and its odd moments three, five and seven vanish. Substitution of the four rational coefficients proves these identities. Taylor expansion therefore reads the ordinary derivative exactly on polynomials of degree at most eight. Physical spacing factors are inserted only when the finite tensor is rescaled. During a stationary descent retain the full tuple (J, Y, E, Q, Ξ), with exact identities X a Jaa = X a ∇aQa, Jab −Jba = X i ∇iΞabi, X a DaJab −Eb ≡DYbP through Taylor order four. (TF0c) Here ∇is the unit forward difference, and the last equivalence includes orders zero through four at the identity. Initially Q = Ξ = 0, and E is defined as the difference on the last line, so that it is an equality. The Hessian of P is the retained covariant quadratic form; differentiating DY P, using Y (1) = DP(1) = 0, gives its two cross Hessian pairings. The local analytic change of variables preserves these four jets by the chain rule. The scalar rows J, E, Q, Ξ remain analytic, gauge invariant, and zero through order one. To see why the identities persist under blocking, denote the main averaging stencil by M, the directional flux stencil by Fa, and its correction by Ta. Their finite coefficient identities are Fa = M + ∇aTa, L5MDa = DaL4Fa. 386 ## Source page 389 A Wilson Yang–Mills construction J. Rodgers Pull each fine scalar function back along the stationary section. Transport Jab with L4Fa, Eb, Yb with L5M, and the difference potentials with their degree-three edge stencils. Add L4TaJaa to the transported Qa, and add 1{i=a}L4TaJab −1{i=b}L4TbJba to the transported Ξabi. Substitution proves all three identities in (TF0c); the two added terms cancel precisely the change from the main stencil to the flux stencils. The current is transported by the derivative of the stationary section, so the chain rule also preserves the last identity. No exact symmetry or zero trace of each blocked local representative is presumed. Localization is performed on the scalar and current functions themselves: gauge fix along the prescribed finite axial paths, discard coordinates beyond the chosen radius, and retain the Taylor polynomial through order four in those local coordinates. Denote the localized tensor by Jloc, and localize Q, Ξ by the same operation. Restore its exact trace and its six antisymmetric divergences by defining eJ00 = X i ∇iQloc i − X i̸=0 Jloc ii , eJab = Jloc ab (a < b), eJba = Jloc ab − X i ∇iΞloc abi (a < b). (TF0d) Other diagonal entries are unchanged. The six pairs are 01, 02, 03, 12, 13, 23; their identities follow individually by subtraction. Summing the four diagonal entries proves the trace identity. The same construction has bounded support, enlarged by one difference step. The localization error of a differentiated tensor is controlled at the actual stencil sites: (TF0b) uses only shifts of length at most four, so its error is bounded by P l|dl|(A+A) if the undifferentiated error is at most A. Summing the four directions gives four times this allowance. Support and anchor weights are enlarged by four at this step. The corresponding derivative and tail allowances are summed with the same absolute coefficients. Exponential path decay pays the discarded tails including their first moments; the stationary four-jet identity pays the local Taylor difference. These are the separate errors retained when the localized divergence is issued as a degree-five insertion. Finally average the scalar tensor and its divergence over signed permutations and conjugation, with weight 1/|Signed×{0, 1}|, multiplying each component by its tensor or vector sign. Reindexing this finite average proves reality and the required signed covariance. The vector average kills the identity-configuration constants: the axis reflection reverses the vector sign and pairs identical constants. It does not increase a common rooted bound, because the sum of the absolute averaging weights is one, every transformed chamber has the same rooted seminorm, and the rooted triangle inequality applies. Transport commutes with the same transformations. Thus the issued vector and tensor have the correct exact finite symmetries before any coefficient limit is taken. We first specify the finite objects whose limits enter the trace argument. This prevents a formal Taylor coefficient from being confused with a continuum insertion. At a fixed finite representative let Yb be its rooted infinitesimal link-current vector field, DY f the directional derivative of a configuration function along its left logarithmic link increments, and divν Y its divergence for product Haar measure. In exponential coordinates this divergence includes both the ordinary coordinate divergence and the derivative of the logarithm of the Haar density. Write PA for a finite partial sum of the classical action, LQ for a finite partial sum of the ordinary logarithmic corrections, and EA,Q = −βPA −LQ, Rb = β(Bb −DYbPA). (TF1) Here Bb is the rooted lattice stress expression, with the same mean and curvature rows as in the finite stress construction. The finite sums are taken from the actual source dictionaries, rather than from new representatives of their limiting functions. 387 ## Source page 390 A Wilson Yang–Mills construction J. Rodgers The protected configuration change along Yb, with step u > 0, gives a finite difference identity. If Cu,b denotes the resulting density-difference expression, its exact decomposition on the protected healthy region is Cu,b u = Zb −βBb + (divν Yb)(1), (TF2) Zb = Rb + {divν Yb −(divν Yb)(1)} −DYbLQ + Ku,b + X γ∈F Hu,b,γ. (TF3) The constant denoted by evaluation at 1 is evaluated at the identity configuration. The fourth term is the finite Taylor remainder of the changed action and Jacobian after subtracting their first directional terms, divided by u. The last sum is the finite correction for the chosen finite shelves of activity terms: each summand is the corresponding changed activity difference with the coefficient and the factor u−1 dictated by that difference. Thus (TF3) defines a literal configuration function. To prove (TF2), expand the change of density, distribute DY (−βPA−LQ), and use (TF1); the βDY PA terms cancel. Splitting the activity change into the finite retained shelves and their remainder gives exactly the last sum. This is an algebraic equality before any limit or expectation is taken. The correction shelves can be enlarged to approximate the full changed expression uniformly on configurations. The estimates for rooted activity sums above give nonnegative numbers gγ with each summand regular on its actual support Pγ, and X γ∈F mass(Pγ)gγ ≤εcorr. (TF4) The finite four-term part of (TF3) is regular on the common neighborhood and has derivative allowance 2csmall + 2cloop. The sum in (TF4) is included with its own supports, not paid as if every term had the largest common support. The conversion of a regular function f into a coefficient insertion is particularly explicit. Let Π≤5f be its Taylor polynomial along the local analytic section at the identity, projected to the degree-five frame, and weight its homogeneous components with the prescribed threshold powers. Call the resulting coefficient a(f). Choose the primary local section s(a(f)), and put the masked difference f −s(a(f)) into one regular activity slot, with every other slot zero. The resulting formatted insertion equals f on the protected region, has zero extra scalar term, and retains locality and symmetry. If rread is the norm of the finite Taylor reading map and rsect the norm of this section construction, a bound c for the original section gives the two respective bills mass(P)c(rread + rsect), mass(P)c((1 + csect)rread + rsect). (TF5) These are finite-dimensional operator bounds and the ordinary triangle inequality for f −s(a(f)). Add this construction for the four terms and the finite correction sum. Taylor reading is linear, so the actual candidate coefficient is the sum of the four weighted Taylor readings plus the weighted readings of Hu,b,γ. Equation (TF4) bounds the size of the latter by (rread +rsect)εcorr. This supplies the coefficient and its paid remainder together. The equation identity for this candidate is also literal. Differentiate the finite change of variables in the normalized Haar integral, or integrate its density difference before dividing by u. A specta- tor supported outside the changed links is unchanged. The resulting expectation is zero. Apply (TF2), subtract its identity-configuration constant, and then let the shelves increase and u follow its admitted small-step schedule. Their uniform allowances justify these passages by dominated convergence. The same identity with a spectator derivative gives the finite lattice Ward contact 388 ## Source page 391 A Wilson Yang–Mills construction J. Rodgers term when that spectator meets the support. In particular equation directions have zero separated readings, precisely the equation cancellation already used in constructing the physical quotient. There is a possible degree-five vector component in this procedure. The raw-to-free vector map has exactly the kernel RE proved in (FC9)–(FC10). Choose its real signed-equivariant section by the corrected polynomial lift and subtract the unique remaining multiple of E. This is finite- dimensional subtraction before the coefficient limit. It explains why the trace candidate can be compared in fixed vector coordinates despite its original equation component. The compensation by the quiet logarithmic part is exact at the coefficient level needed here. In logarithmic link coordinates the vector-current diagnostic is a sum, over d ̸= i, of the quadratic and cubic Taylor coefficients with two equal fixed-color variations in direction i and opposite lattice directions d, −d. Its cubic part is the contraction denoted crestq; denote the quadratic contraction by suited,i floftd,i,q. Define Hq(X) = crestq(X) −1 2 X d̸=i floftd,i,q(Xi). (TF6) The quiet logarithmic contribution is constructed with two moment cancellations; gauge invariance alone would not establish the required zero in (TF6). For each fixed ordered pair of plaquette types and color coordinates, let cγ be the finite two-slot path coefficient and zγ the displacement of the second slot from the first. Put C0 = X γ cγ, Ca = X γ cγ(zγ)a. (TF6a) Append a zero-displacement two-slot term with coefficient −C0+P a Ca, and four terms of displace- ments ea with coefficients −Ca. Repeat this for every plaquette and color pair. The augmented finite family then satisfies exactly X γ cγ = 0, X γ cγ(zγ)a = 0 (a = 0, 1, 2, 3). (TF6b) Its appended coefficients remain color invariant because that condition is a real linear subspace. The quiet logarithm is the signed and translated sum of this augmented family. The difference between it and the retained ordinary loop sum is precisely the finite appended part. Its coefficient sizes are bounded by the zero and first moments of the omitted path tail; exponential path decay bounds these moments, giving the small remainder already paid in (TF4). Here is why this construction cancels (TF6). Let T = Lquiet. Its Hessian at the identity vanishes when one direction has bounded support and the curl of the other is affine on the enlarged support neighborhood. In fact the Hessian of a two-slot term is the sum of the two products of its linearized curl slots. With the first direction fixed, the other slot evaluated at displacement zγ is A + F(zγ), with F additive. Summing over the family gives A P γ cγ + P a F(ea) P γ cγ(zγ)a = 0 by (TF6b). The same calculation applies to the reversed slot order and each signed translate. The nonwrapping and support hypotheses ensure that these displacements are the actual integer displacements; in the application 2(t + 2 + M) + 6 < n, where t bounds the current support and M the retained path length. For Fb = DYbT, differentiate twice at the identity. The current vanishes there, and the first derivative of a gauge-invariant scalar there is zero. The product rule consequently reduces the Hessian to D2Fb(0)[X, Z] = D2T(0)[DYb(0)X, Z] + D2T(0)[DYb(0)Z, X]. (TF6c) 389 ## Source page 392 A Wilson Yang–Mills construction J. Rodgers The directions DYb(0)X have the bounded support of the current. Choose X, Z to be the polynomial link directions of degrees at most two that occur in the coefficient test. Their curls are affine on that neighborhood. The preceding Hessian cancellation makes both terms in (TF6c) zero. Terms with a constant pure-gauge direction vanish by the differentiated gauge identity. Expanding the cubic coefficient in the two position variables of the test uses just these Hessian pairings, so its centered-link reading is zero. One must still convert the centered-link reading into the actual link reading. If the two position arguments are z0, z1, the change is multiplication by e i((z0)i+(z1)i)/2. The constant and first coefficients of the original reading vanish. The normalized third coefficient of this product is therefore its original third coefficient plus i((z0)i+(z1)i)/2 times its second coefficient. Apply the finite coefficient functional in the d, b variables. The latter term is zero unless i = b; when i = b, its value is −1 2 floftd,i,q. This follows by expanding the quadratic coefficient on (ed, −ed) and using i2 = −1. Summing over b and d ̸= i gives exactly (TF6). Its centered-link version has just been shown zero, so Hq(DY Lquiet) = 0. The factor 1/2 thus comes from this link-centering phase and the normalized Taylor coefficients. Subtracting the paid appended part gives the stated bound for the ordinary loop contribution. No quiet contribution is left undetermined. The needed vanishing of the quadratic part can be derived without setting a mass regulator equal to zero inside an integral. For 0 < μ ≤1, the local resolvent estimates below give, for the massive comparison expression Lμ, | cobd,i,q(Lμ)| ≤Cμ−1. (TF7) The proof differentiates the Fourier-resolvent expression twice; its pointwise integrand is bounded by C(|p| + μ)−5. Counting lattice shells and multiplying by the momentum cell volume gives Cμ−1 in four dimensions. This applies to both the physical and comparison resolvents, hence to their sum. At each fixed positive μ, the resolvent identity identifies the unregularized current coefficient with μ2 cobd,i,q(Lμ), up to an error tending to zero in the retained large-volume limit. Given ε > 0, choose μ = min(1, ε/[2(C + 1)]) first. The massive term has magnitude at most Cμ ≤ε/2; take the volume large enough to make the remaining error below ε/2. The unregularized quadratic coefficient therefore tends to zero. The Taylor remainder bounds give the same limit for its finite-difference reading. Summing the finitely many d, i shows that (TF6) has the same limit as its cubic part. That cubic limit is −12 sG, where sG is the one-loop slope constant in the coupling drift. Here is how the logarithmic slope fixes it. For the regulated determinant expression A(log μ), the deter- minant estimates give |A(x) + (−12sG)x| ≤C, x ≤0. (TF8) The regulated response is −A′(x), with x = log μ. The real mean value theorem on an interval [a, b] ⊂(−∞, 0] supplies an intermediate regulator at which this response differs from −12sG by at most 2C/(b −a). First choose b −a large, then use the local resolvent difference estimate to compare that response with the unregularized cubic current coefficient; this error tends to zero when the upper endpoint b is sufficiently negative and the retained volume is sufficiently large. The intermediate regulators stay in the fixed positive interval [ea, eb], so the large-volume estimates apply uniformly there. The triangle inequality proves the limit. Together with (TF7), it proves Hq →−12sG for every color q. These are finite rooted Haar-Jacobian coefficient tests, not an assumption about a mollified continuum field. Finally the passage through many adjacent coefficient transports retains a relative error. Put ωi = exp{−2 5 RLog( bβi)} −→0, k = k(i) > i. (TF9) 390 ## Source page 393 A Wilson Yang–Mills construction J. Rodgers The chosen calendar ensures every adjacent transport error between i and k is at most ωi/[2(k −i)] times the next coefficient size. If vj denotes that coefficient, backward induction proves ∥vj −vk∥≤ωi(k −j) k −i ∥vk∥≤ωi∥vk∥, i ≤j ≤k. (TF10) Indeed the induction hypothesis and ωi ≤1 imply ∥vj+1∥≤2∥vk∥; adding the next adjacent error gives the asserted bound at j. The calendar inequality follows from the positive reserve −2/5 + 1/2 = 1/10: logarithms and the number of intervening steps are eventually dominated by the exponential with exponent 1/10. Thus this accumulated relative error vanishes. The finite candidate, its equation identity, the quiet compensation and its scalar diagnostic consequently survive on the same retained field and enter the following trace comparison. 16.2 The interacting-to-Gaussian comparison used for the trace The radius selection is exact as a geometric statement: the requested radius s equals a purchased station radius ρk(b + u) for some 0 ≤u ≤1. This retains every comparison estimate bought on that interval. For the actual moments the conclusion remains approximate, for example |u01 − (m0m1 + z)| ≤(1 + |m0| + |m1|)c. It becomes exact in the specified limit as c →0. This distinction is needed when arbitrary sequences of radii are brought back to the same physical table and the same selected phase. This comparison concerns the interacting Wilson family and its selected continuum phase. It is distinct from the calculation of a Gaussian constant coefficient in the preceding chapter. We derive the passage between them. At a fixed retained comparison station take two physical lattice insertions A0, A1, supported in its strict chamber, with their coefficient representatives converging along the common comparison filter. Let m0, m1 be the corresponding one-point Gaussian readings and let z be the centered Gaussian two-point reading. The corrected singleton and pair polynomials are Dj = Aj −mj, D01 = (A0 −m0)(A1 −m1) −z. (IG1) We next derive the finite interacting bound for these functions. This also specifies which Gaussian quantities are subtracted. For n marked insertions let f be the cumulant polynomial of the Gaussian profile at the rounded positions. The corrected cylinder is the coefficient polynomial obtained by multiplying the insertion generating polynomial by exp(−f), in the finite algebra in which each marker has square zero. Its singleton coefficients are Aj −mj, and its pair coefficient is exactly D01 in (IG1). Thus the subtraction includes the Gaussian means as well as its connected pair coefficient. Let k be a comparison station, M ≥n its fixed maximum number of marks per chamber, and Hk its monitor. Define Ik = log Hk 16M , Rk = e2Ik, εk = e−Ik. (IG4) Here Hk →∞, so εk →0. The initial marked density is the Wilson density times the product of the reflected copies of the corrected cylinder over all chambers of the tiling. Push this density along the same exact renormalization course as its unmarked Wilson density. In particular its vacuum integral is the original Wilson partition function. The comparison uses this density identity, not a replacement of the interacting law by a Gaussian law. We describe the bounds on this push. At the middle stop its regular marked sources have values Wα(U), with specified fired marker sets and support cubes. They occur together with the forced 391 ## Source page 394 A Wilson Yang–Mills construction J. Rodgers sources and the unmarked background in the finite density expansion described in the marked- integration chapter. If Cα is the corresponding coefficient of the negative Gaussian cumulant, then on the healthy local chart |Wα(U) + Cα| ≤b−31/32(t), |Cα| ≤b1/4(t). (IG5) Here t is the station coupling, ba(t) = exp(a boil(t)), and boil(t) →∞. To obtain the first bound, expand the actual insertion product by its finite cumulant expansion. The local Gaussian com- parison of its calibrated chart coefficients gives an error at most b−3/2(t) for each coefficient; the finite exponential map from cumulants to moments is continuous on this fixed finite coefficient space. Its chosen neighborhood makes the cumulant-polynomial discrepancy plus that error at most b−31/32(t). The exact negative cumulant coefficient cancels the corresponding free coefficient. The second bound follows because the rounded Gaussian profile stays in a compact subset of the generic position domain, where its finitely many coefficients are bounded, while b1/4(t) →∞. Here is the polynomial estimate behind the local Gaussian comparison just used. Denote coef- ficient l1 size by coin, and let k0 bound the degree of each insertion polynomial. After the actual linear substitution into the common key and residual coordinates, write these polynomials as Pi; write Fi for their calibrated free polynomials, included in the same coordinates. The calibration is the exact identity that their weighted truncations through the assigned degree di agree. If the substitution has weighted size at most b, the original polynomial has size at most ai, and the weight radius is ρ ≥1, every remaining monomial has weight at least ρdi+1. Hence coin(Pi −Fi) ≤aibk0ρ−(di+1) ≤ξ. (IG8) This is the high-degree tail estimate; it uses the actual solved calibration maps. If coin(Fi) ≤B0, B0 ≥1, telescoping a product over any subset J of the n insertions gives coin Y i∈J Pi − Y i∈J Fi ! ≤|J|ξ(B0 + ξ)|J| ≤nξ(B0 + ξ)n. (IG9) Substitution of the Gaussian variables commutes exactly with their Wick functional: the substituted covariance is the pullback of the original covariance and its mean is the pullback of the original mean. These are equalities of finite Gaussian moments, including the residual coordinates, not limiting approximations. Let all means and covariances have absolute value at most B ≥1, and let E bound their discrepancies from the zero-mean continuum free data in the key coordinates. Put N = nk0 and CB,N = [2B(N + 1)]N. Wick’s recursion bounds the functional on a degree-N polynomial P by CB,N coin(P), and its change under these mean and covariance discrepancies by ECB,N coin(P). One proves both assertions by induction on the monomial degree: remove one variable, pair it with each remaining variable or replace it by its mean, and, for the difference, telescope the resulting products. The at most N + 1 choices at each step and the bound B fit the displayed constant. Combining this bound with (IG9), and then comparing the free-key polynomials under the two Gaussian data, gives total error at most CB,N nξ(B0 + ξ)n + Bn 0 E . (IG10) The common covariance estimates and small-mean estimates in the Gaussian comparison chapter give precisely these bounds on the pulled-back data, uniformly over the allowed small middle-stop means. The station choices make (IG10) at most b−3/2(t). Thus the estimate feeding (IG5) follows 392 ## Source page 395 A Wilson Yang–Mills construction J. Rodgers from the actual covariance comparison, exact calibration and a finite polynomial calculation. It is not an additional interacting-to-free hypothesis. These coefficient estimates are weighted in the density, as follows. A regular source in a support cube of radius QB has support charge at most exp(e(2Q + 1)4), since that cube occupies at most (2Q + 1)4 budget blocks. Its marker weight at radius Rk is at most RM k = b1/8(t), by the identity between the monitor and the station coupling. Consequently multiplication of (IG5) by the support charge and marker weight fits the prescribed regular-source bound. The same calculation with the additional unit of support charge fits the Gaussian compensation source bound. The forced sources already satisfy their full-configuration bounds. The one-step marked integration estimate therefore propagates this compensated density from the middle stop down to the width stop, preserving the unmarked background and bounding the total source weight per marker and the total forced weight per site by 1. This is finite induction over the stops: the initial insertion estimates give the terminal case, and at each earlier stop the chart transport estimate and its forced-source estimate give the next case. The delay conditions put every coupling in this finite induction in the same interval [t/2, t], so the chosen budgets apply throughout. For completeness, the final conversion of these density bounds to a rooted Wilson bound can be written explicitly. Let Z be the number of width-stop sites, which is also the number of reflected chambers. If q ̸= ∅is a fired subset of the n markers, the weighted coefficient estimate for the pushed density and the positive partition-function lower bound give ∥⟨ Y chambers copy Dq⟩Wilson∥RZ|q| k ≤exp{ZP −4 dim g log g} . (IG6) The parameter g > 0 here is the width-stop small-field parameter, with βg2 = 1; P is its per-site protection bound with source count at most 1 + n ≤1 + M. The partition estimate applies to the actual positive course density. Its comparison with the reference density uses the two-sided exponent error bounded by Zδ: one side bounds the numerator and the other side bounds the positive denominator, so their errors are both included in P. Taking the Z-th root in (IG6) is legitimate because Z > 0. The width-stop allocation was chosen so that P −4 dim g log g Z ≤Ik. The rooted chamber seminorm therefore obeys ∥Dq∥root ≤eIk R|q| k ≤eIk Rk = e−Ik = εk. (IG7) The middle inequality uses |q| ≥1 and Rk ≥1. The allocation is eventually available because the finite source-count ticket at the width stop is bounded by Ik; this is the delayed station choice, made simultaneously with the finitely many chart, insertion and transport inequalities. Applied to the two singleton modes and the pair mode, (IG7) is the asserted common bound c = εk for (IG1). These are bounds for the actual Wilson law, before any character is selected. Here is why this gives the same bound for the character reading of each corresponding continuum polynomial. Let D be any finite linear combination of distinct-point words with the rooted bound c, and let B be its compressed whole-word operator. For every integer n ≥1, place 2n copies of D, with alternating reflected conjugation, in chambers separated by a distance parameter t. At sufficiently large volume these are distinct chambers. The finite chessboard inequality bounds the absolute expectation of their product by c2n. At fixed n, t, the ordered lattice limit preserves this inequality; the comparison filter lies below that same ordered-limit filter and the coefficient 393 ## Source page 396 A Wilson Yang–Mills construction J. Rodgers representatives converge there. Then let t →∞. The long-word compression limit, expanding the finite linear combinations without commuting their operator factors, gives ⟨Ω0, (B∗B)nΩ0⟩≤c2n. The character decomposition therefore gives Z |χ(B)|2n dμ(χ) ≤c2n. (IG2) It follows that |χ(B)| ≤c for every character. Indeed, if |χ0(B)| > c, continuity provides an open neighborhood of χ0, a number a > c, and a positive measure h of that neighborhood, on which |χ(B)| ≥a. Full support gives h > 0. Then (IG2) implies ha2n ≤c2n for every n, an impossibility as n →∞. This includes c = 0. The two limits here have their stated order: first the lattice limit at fixed copy count and separation, then separation to infinity, then the integer n →∞. Let uj and u01 denote the continuum one- and two-point readings in any one character. Applying this argument to (IG1), and using the unit reading 1, gives |uj −mj| ≤c, |u01 −m0u1 −m1u0 + m0m1 −z| ≤c. The exact decomposition u01 −(m0m1 + z) = [u01 −m0u1 −m1u0 + m0m1 −z] + m0(u1 −m1) + m1(u0 −m0) proves |u01 −(m0m1 + z)| ≤[1 + |m0| + |m1|]c. (IG3) Thus the Gaussian comparison follows from three specified interacting rooted estimates, not from an assertion that the interacting measure converges to a Gaussian measure. At comparison index i, the station construction supplies these rooted estimates for every pair of basis labels through degree i, with c = εi →0, uniformly over its fixed position window. The Gaussian readings mj, z are the singleton and pair coefficients of the Gaussian moment profile. In particular m0m1 + z is its full two-point coefficient, by the defining cumulant identity. Write Qi(x) for the rounded position tuple and riQi(x) for the actual physical tuple. If x is generic and has a neighborhood in the comparison window, then Qi(x) →x, so it is eventually generic and stays in that window. Smoothness of the free hierarchy implies convergence, and hence boundedness, of its two singleton readings at Qi(x). Equation (IG3) yields convergence of the interacting two-point reading to the free two-point reading at x. The finite-cutoffrepresentatives of a free label converge, along the retained ultrafilter, to a linear lift Yd,i. Projecting that lift gives the continuum map Ud,i. The finite station bounds pass to these limits because the lattice moments with converging finite-dimensional coefficient labels have the calibrated ordered limits. The station construction also preserves the unit, conjugation and signed motions. The free detector proves eventual injectivity of Ud,i; the field-dimension bound gives surjectivity, so these maps are eventually isomorphisms at each fixed degree. Replacing finitely many exceptional initial maps by any fixed isomorphism leaves all limits unchanged. The conclusion also holds for convergent moving free labels vi,0, vi,1. To see this without assum- ing a uniform norm bound on all interacting bilinear forms, fix a basis ea of the degree-d free space and expand Bi(vi,0, vi,1) = X a,b v a i,0v b i,1Bi(ea, eb). 394 ## Source page 397 A Wilson Yang–Mills construction J. Rodgers Each basis-pair value converges by (IG3), and every displayed coefficient converges. There are finitely many summands. Their limits are therefore exactly the free bilinear form evaluated at the limiting labels. This is the moving-label comparison used for the normalized centered trace. We finally justify the universal radius selection used in (TLim1). Given an arbitrary positive sequence sn →0, a fixed finite degree horizon, any lower bound on the sequence index and station depth, and any positive radius ceiling, the station construction supplies a later sequence member below that ceiling whose value is exactly the station’s scale radius. Its purchased comparison bounds include the three modes in (IG1), the bounded coefficient lifts and the smeared comparison estimates through that horizon. All their error thresholds can be met together because only finitely many basis pairs and modes occur at this horizon. Recursively increase the horizon to i, require a larger sequence index and station depth, and set the next radius ceiling equal to half the radius just chosen. This produces a strictly increasing subsequence map, strictly decreasing chosen radii, and station depths tending to infinity. At each fixed degree all its comparisons hold eventually. The exact radius matching has an elementary interval argument. Write Mk ≥1 for the calen- dar’s magnification monitor and ρk(b) = akMb k for the station gap. Eventually Mk ≤Mk+1, and the purchased block bound gives Lk ≤Mk+1. Thus, for b ≥0, ρk(b) = ak+1LkMb k ≤ak+1Mb+1 k+1 = ρk+1(b + 1). Also ρk(b) →0 for each fixed b. Given a lower station index J, choose i ≥J beyond these eventual bounds. For 0 < s ≤ρi(b), take the first k > i with ρk(b) ≤s. Minimality gives s ≤ρk−1(b) ≤ρk(b + 1), also when k = i + 1. The continuous function u 7→ρk(b + u), 0 ≤u ≤1, therefore attains s. The station family was purchased for this entire interval of exponents; its other charges and interpolation parameters stay fixed over the interval. Consequently this adjustment of u preserves its estimates. Given a sequence sn →0, choose a later sequence index below the radius ceiling supplied by this argument, and repeat with a larger J. This proves the asserted simultaneous escape of sequence and station indices, with exact equality of radii. The bounded lifts are then extracted along the already retained ultrafilter and projected as above. Their finite-dimensional limits assemble into one scaling family with exactly those chosen radii. The trace and local-coordinate estimates apply to every such compatible scaling family; they retain the previously fixed physical table and phase. Hence this construction works for the arbitrary sequence supplied to it, including the violating sequence in the proof of the full trace limit. 16.3 One calibration matrix for the three panels The three comparison panels used in (WS1)–(WS17) differ in their point and smear geometry, but have the same finite Taylor order and the same calibration test system. We justify the common matrix, rather than choosing three unrelated solutions. At a fixed degree and sufficiently late stop, the detector matrix of that calibration system is invertible by (PC1)–(PC13). Select the solved chart at one anchor panel. Its defining equations prescribe its pairings with that test system. For any other panel, transport the same local representative and the same Taylor card to its allowed geometry. The support, threshold, and budget equalities make its equations identical to the anchor equations. Uniqueness of their solution then gives the same field matrix T at that panel. Signed covariance follows by transporting the solved equations under a signed permutation and again invoking uniqueness. This proves simultaneously the tested-chart and signed-intertwining properties needed in (WS1). More concretely, the inverse chart is T −1 after the field dilation and its engineering-weight factor. Applying the defining calibration equations to a basis proves that it solves those equations; the unit equation gives the unit, and conjugating or turning the equations gives conjugation and signed 395 ## Source page 398 A Wilson Yang–Mills construction J. Rodgers covariance. Finite dimensionality bounds the finitely many inverse coefficients after the detector lower bound has been paid. These are precisely the coefficients used in the rooted comparison estimate, including on the smear panel. There is no separate choice of an inverse for each spectator word. For a fixed finite horizon each panel has finitely many local demands, all eventually paid. Take their maximum together with the common detector and monitoring thresholds. At this stage the anchor solution transfers to all three panels at every finer cutoffand every admitted volume. Choose the stops by the cofinal advance already used in (WS16), satisfying these finite maxima and each earlier horizon. Passage to the common coefficient subsequence preserves the tested equations because their finite pairings converge. Therefore the point, displaced-point and smeared comparisons have the same limiting field identification and may be combined in the trace and Ward arguments. 16.4 Common scaling maps and convergence of every derivative The comparison of a physical field with a free field must use one map for all its readings. Here is the construction, including the compactness and interpolation steps. Fix a finite maximum number of slots, a finite field horizon and a bounded separated configuration window. During this construction these bounds may be enlarged to include detectors and derivatives. The physical table, its null quotient and its limiting correlations remain those already constructed. Choose radii rk ↓0 and lattice levels nk with δk = ank rk −→0. (WS1) For each free coefficient basis vector at each requested degree, the insertion construction gives a column of physical reference coefficients. Its bound is finite for each fixed k, d, uniformly in the late mesh and large period. These bounds are not asserted uniform in k. Extract the finitely many coordinates of each column on their compact balls, first in the mesh limit at a retained period and then in the period limit. Diagonalize over the countable pairs (k, d). The resulting maps Uk,d : F free d →Fd initially need not be invertible or commute exactly with the inclusions. Reality, the signed permutation action and the unit pass to the limit since they are finite linear equalities. The estimates that permit this extraction also supply numbers Ak ≥0, for every fixed word length n, with δkAn k −→0. (WS2) The center and comparison estimates are imposed at levels m ≤j satisfying am < rk/Q, 6naj < am, aj < rk/(4Q). The insertion supports are therefore separated before the comparison is taken. On the rescaled lattice, the aligned comparison tends to the free correlation. On a configuration window in which all the required coordinate projections have a fixed strict gap, the long finite-difference estimate gives |S(Ukv; rkx)| ≤An k, |∂xl,αS(Ukv; rkx)| ≤4QAn k. (WS3) Rounding to the rescaled lattice changes a coordinate by O(δk); the resulting error vanishes by (WS2). Thus the correlations converge to the free ones on these coordinate-generic windows. All bounds here are for a fixed finite word and fixed window. No uniform bound on all fields or all degrees is being taken. 396 ## Source page 399 A Wilson Yang–Mills construction J. Rodgers To fit the coefficient maps, use compactly supported smooth tests in the coordinate-generic part of the separated configuration space. Their readings separate the free quotient. Indeed, a field whose every such reading vanishes has zero correlations there by the fundamental lemma for test functions, and then on the entire separated space by continuity and density. Finite dimension now gives finitely many separating readings at each receiving degree. Spectators in these readings can have larger degree than the field being detected; enlarge the common horizon accordingly. If Ed denotes this finite injective reading map, the physical readings of Uk,d converge, in matrix norm, to Ed. Injectivity is open for maps on a finite-dimensional domain: choose a left inverse Ld of Ed; when ∥Ld(Ek,d −Ed)∥< 1, the matrix LdEk,d is invertible. Thus Uk,d is eventually injective. The previously constructed spanning family gives dim Fd ≤dim F free d , so it is eventually an isomorphism. We make explicit the finite-horizon estimates preceding these readings. A centered budget is the reflection-positive seminorm for every signed or conjugated copy of the one insertion in a reporting chamber. For a word of n insertions, take n chambers whose centers may be shifted within a finer common chamber. If distinct points have at least one coordinate difference larger than rk/Q, the inequalities preceding (WS3) permit these shifts so the inspected insertion cells are disjoint. Apply Cauchy–Schwarz in the positive reflection form successively in the separating coordinate directions. Each insertion contributes its centered seminorm at most Ak; no factor depends on the volume or the microscopic cutoff. Thus the finite Wilson word has magnitude at most An k. Take its ordered mesh and period limits at a coordinate-generic tuple. The fitted coefficients converge in their finite-dimensional balls, and multilinearity permits them to vary along the limit. The same bound passes to the limiting correlation. On the open set of tuples with some separating coordinate for each pair, the correlation is continuous and the generic tuples have full measure and are dense. Therefore the bound extends to every tuple in that open set. This proves the first part of (WS3) on the entire separated window, including coordinate ties. For a coordinate-generic tuple with all coordinate gaps larger than 1/Q, move one coordinate a positive rescaled distance t < (4Q)−1. On the microscopic lattice perform the integer displacement ⌊rkt/aN⌋in that coordinate. Its physical length tends to rkt. The long finite-difference estimate for the same centered budgets, with available gap rk/(4Q), bounds the difference of the two Wilson words by An k rkt rk/(4Q) = 4QAn kt. (WS8) Both endpoint readings converge with their original placements and coefficients. Pass through the ordered limits, divide by t, and let t ↓0 through endpoints that remain generic. Smoothness identifies the limit with the coordinate derivative and proves the second bound in (WS3). This estimate is uniform on a fixed strict coordinate-gap window. It is not used with Q tending to infinity near a projection tie. The small quantity in (WS2) has the following precise use. The finite suppliers provide it separately for every word in a fixed finite-degree basis and all its signed and conjugated variants. Taking a maximum over finitely many words preserves convergence to zero. Indeed, if ak,w ≥ 0 and δkan k,w →0 for each of finitely many w, then δk maxw an k,w = maxw(δkan k,w) →0. The polynomial prices of the solved columns are (PC12)–(PC13); the rooted insertion and aligned Gaussian estimates provide these centered budgets and the aligned word comparison on the same station. The finite estimates must supply these products at their actual station radius and grid; the grid cannot be altered independently after a comparison has been purchased. Thus varying labels in a bounded finite-dimensional set costs a fixed finite sum of these basis bounds, rather than requiring uniformity over an infinite set of labels. Here is the scale and price bookkeeping in that last statement. For each requested station k, 397 ## Source page 400 A Wilson Yang–Mills construction J. Rodgers all degrees through the horizon use one earlier index a(k) ≤k, with a(k) −→∞, sk −sa(k) ≤log bβk, aa(k) ak ≤bβk. (WS11) The last inequality follows by exponentiating the preceding one, since ai = e−si. At the comparison radius rk the actual rescaled mesh is aa(k)/rk = mag−1 a(k). The common finite matrices are taken to sublimits along the same comparison filter below the retained ordered filter. Each pair of degrees, each spectator and each placement therefore uses these same matrices and this same radius. The reverse coefficient transport from a level m to k ≥m has price quotak(v) ≤(am/ak) quotam(v). (WS12) Indeed one step costs the column allowance Ri+1 ≤Li; multiplying those inequalities gives Qk−1 i=m Li = am/ak. The solved readout at a(k), followed by its exact field-number and engineering dilations, consequently supplies a preimage y for every actual coefficient v, with ∥y∥≤2a−d k quotak(v). To check the factor two, its approximate matrix differs from the exact awake coefficient by at most half the awake coefficient size, as in (PC7); the triangle inequality bounds the approximate size by twice the awake size. The field shrink is at most one and the comparison magnification at least one, so undoing their positive dilations cannot increase the weighted coefficient size. Formula (WS12) then gives the displayed bound. This argument applies to the full coefficient space, including equation directions, before restricting to the physical quotient. The magnification and each fixed finite sum of its engineering and field-number powers have rate zero relative to log bβk. For example the label price is bounded by 2 X α (2 monitora(k))fαmag rα a(k). For every ε > 0, each fixed power is eventually at most bβε a(k) ≤bβε k; the finite sum has the same rate- zero property. These are the actual finite-horizon prices used with the rooted insertion estimates. The centered estimates assert jointly, for a word of length n, the existence of reporting levels m ≤j such that am < rk/Q, n(6aj) < am, aj < rk/(4Q), An k(aa(k)/rk) →0. (WS13) The first three conditions give the movable disjoint chambers used in (WS8); the last is exactly the price needed in (WS9). They are imposed together with the aligned moment comparison, not inferred merely from the two-field detector limit. In passing from a basis to bounded arbitrary labels, expand every slot in the fixed finite basis and sum the products of the coordinate norms. The finite sum preserves both the vanishing comparison error and the last limit in (WS13). The smeared station version says the same thing without committing to a grid interpolation: for every smooth test t in the stated separated window, with ∥t∥∞≤M, the ordered upper error is at most εM Q l∥vl∥. Its finite-cutofflift is bounded, real, signed-equivariant, and sends the unit to the actual reference unit. Its scale and normalization are fixed for all those tests. Enlarging the arity, window and degree and decreasing the error preserves the previous requests. The common-order selection first retains these finite data, then uses (WS4) and finite detecting matrices to obtain all derivative estimates. Thus neither the conditional formulation of a station estimate nor the existence of a two-point comparison is being used as a substitute for the joint estimates (WS13). 398 ## Source page 401 A Wilson Yang–Mills construction J. Rodgers The simultaneous centered supplier can be described explicitly; it is stronger than a two-point Gram comparison. Fix a degree cap D, an arity cap n ≥2, and a compact separated window. The actual construction provides three calibrated panels at exponents a, b, e, on one family and with the same coefficient matrices. The bounds used for a panel are its solved-column and marked insertion bounds (PC1)–(PC13), together with the rooted comparison of the entire requested finite word. Denote the common monitor by Mk, and write Hk(t) = Mt k. In the source this is bill(t, clim k); thus Hk(t)Hk(u) = Hk(t + u). A primary comparison radius is rk = akHk(a), its mesh spacing is δk = Hk(−a), and a small motion allowance is pk = Hk(c). Choose the exponents so that c < 0, b + 6 < a, e + 5 < a + c, (a −b)Dn + c < 0, (a −e)Dn −a −c < 0. (WS14) There is no obstruction to these finite numerical choices. Put x = 7Dn + 1, y = x + 6, a = 17+D+x+yDn+y, b = a−7, e = a−y, and c = −x. Then 7Dn−x = −1, e+5 = a−x−1 < a+c, and yDn −a + x = −17 −D −y < 0; also all three panel exponents exceed D + 6. These are the source’s finite choices. The calibrated panel supplier is chosen with these exponents before extraction. Here is how that supplier yields centered bounds, rather than merely ordinary moment bounds. For a single basis label, the rooted comparison applied to the singleton at the origin controls the observable minus its free scalar value by the word error εk →0. The centered root seminorm is subadditive, is homogeneous in scalar coefficients, and assigns norm one to the constant observable. Thus its full observable has root seminorm at most 1 + |rfree|, eventually. Interior support in the centered chamber holds at every translated origin by locality of the observable; translation of the chamber preserves this root seminorm. This gives a uniform centered bound at the panel’s own width, including its constant part. Changing the panel magnification from b to a multiplies a coefficient of engineering degree q ≤D by Hk(a −b)q. Expand in the fixed finite label basis. The triangle inequality therefore gives the centered price Ak = C Hk((a −b)D). (WS15) The same argument at panel e gives A′ k = C′Hk((a −e)D). To include every signed and conjugate variant, first apply these finite transformations to the free label, expand each in the same basis, and take the sum of their finitely many coordinate prices. The coefficient lift intertwines these transformations and sends the unit to the actual reference unit, so the resulting bound is for the required physical transformed token. All slots of the word are included before taking the maximum constant. Thus one reporting width works simultaneously for the word; it is not selected after seeing its offsets. The available plateau geometry gives, for jb the reporting width at panel b, n(6ajb) < ajb−1, ajb−1 ak ≤Hk(b + 6), aje ak ≤Hk(e + 5). (WS16) The first follows because one blocking factor tends to infinity. For the other two, the principal plateau length costs Hk(b) or Hk(e); the rounded length and two buffer lengths cost at most five further monitor powers, and passing to the preceding block costs one more. The block factor is bounded by the monitor and its tenth power dominates the fixed covering-size buffers. Multiplying these inequalities gives exactly the exponents b + 6 and e + 5, rather than changing the comparison radius. Equations (WS14)–(WS16) now give the simultaneous vanishing products needed in the moving- chamber proof: An kpk −→0, (A′ k)nδk/pk −→0, 4nδk ≤pk eventually. (WS17) 399 ## Source page 402 A Wilson Yang–Mills construction J. Rodgers Indeed their first two exponents are the last two negative numbers in (WS14); the third uses −a < c, which follows from the last inequality there. Also ajb−1 < rk/(2R) and aje < rkpk eventually, by the two strict panel gaps in (WS14). These two widths are the two centered budgets needed for the shift across a grid tie and for the comparison away from it. The rooted whole-word error is CRεk →0. The moving-chamber decomposition (WS8)–(WS9), with its strip of width pk, is therefore valid jointly in all slots. Integrating it against a test of supremum M gives the station error εM Q i ∥vi∥, including projection ties. This is the actual three-panel route to the joint station statement. The simpler conditions (WS13) record the part of that geometry used by its one-width formulation; no two-field estimate is being used to supply a whole-word centered estimate. For clarity about existence on the chosen trajectory, the dense calibrated supplies retain fixed three-panel exponents at each horizon, the foregoing inequalities, and the actual solved charts at a predecessor g(k) →∞satisfying (WS11). Their availability is paid by the same delayed local insertion and marked comparison schedule used for (PC1)–(PC13). The finite horizon does not depend on the requested extraction order. Passing their finite matrices to limits on a fixed common subfilter preserves those charts and gives the smeared comparisons for their continuum fields. At each horizon, the finite-dimensional derivative and inclusion adjustments in (WS4)–(WS7) act on these fields. Finally choose an increasing integer horizon whose every finite requirement has already become valid at index k: if KD is a threshold for horizon D, use the largest D ≤k with maxj≤D Kj ≤k. It tends to infinity. This is the common diagonal; it preserves the physical index k, the fixed family, and the inherited subfilter. It does not retune the grid for individual spectators. These same readings control the inclusions and derivatives without requiring a bound on the inverse in the physical coefficient norm. In a rescaled position variable, integration by parts is exactly Z t(x)S(rk∂αB, . . . ; rkx) dx = − Z (∂l,αt)(x)S(B, . . . ; rkx) dx. (WS4) The derivative of a compactly supported test has the same support constraint. Apply the converging injective reading matrix at the receiving degree to the two sides. A left inverse close to the fixed free reading matrix then proves U−1 k,d+1Iphys d,d+1Uk,d −→Ifree d,d+1, U−1 k,d+1rk∂phys α Uk,d −→∂free α . (WS5) Every inverse used in this deduction acts on a finite detector matrix with a fixed injective limit. It does not presume that the physical charts are uniformly conditioned. One can make the first line of (WS5) exact. Choose left inverses Pd of the free adjacent inclusions. Average them over the finite signed permutation group and then average with their complex conjugates. The averaged maps are still left inverses and now respect both symmetries. Define recursively Xk,0 = Uk,0, Xk,d+1 = Iphys d,d+1Xk,dPd + Uk,d+1(1 −Ifree d,d+1Pd). (WS6) Substitution of Ifree d,d+1v proves exact compatibility with the adjacent inclusion, hence with every inclusion by composition. Induction in d, using (WS5), gives U−1 k,dXk,d →1. Consequently these maps are eventually isomorphisms, preserve the unit and symmetries, and still satisfy the second line of (WS5). On the finitely many exceptional indices choose any isomorphisms; these choices do not affect a limit. Conjugation by matrices tending to the identity preserves all the finite-dimensional limits just obtained. 400 ## Source page 403 A Wilson Yang–Mills construction J. Rodgers It remains to remove coordinate-generic restrictions and to pass to all position derivatives. For a fixed word let Rk(x) = S(Xkv; rkx). ∂l,αRk(x) = S(Xkv1, . . . , XkDk,αvl, . . . ; rkx), Dk,α = X−1 k,d+1rk∂αXk,d. (WS7) The matrices Dk,α converge to the free derivative matrix. By multilinearity, readings with any converging finite list of coefficient vectors have the same limiting readings as their limits. Also (WS2) applies to the finitely many successor-degree basis words in (WS7). It bounds δk times every needed gradient by a quantity tending to zero on a slightly larger separated window. Interpolate from nearby sampled points to arbitrary separated points along short coordinate segments in that larger window. Their total length is O(δk). The fundamental theorem of calculus and (WS7) make the error tend to zero. The argument does not require a positive gap between distinct coordinate projections: the points themselves stay separated. Repeat (WS7) for each differentiation. For m derivatives use the m successor degrees for the differentiated labels and one additional degree for the interpolation gradient. Thus a horizon D proves the assertion when d + m < D. The finite sum over basis components proves convergence of the full derivative tensor in its norm. This convergence is uniform on every compact separated win- dow. Otherwise a sequence of points violating uniform convergence has a convergent subsequence in that compact set; the moving-point interpolation just proved gives a contradiction. Enlarging the initial finite request supplies every prescribed derivative order, word and compact window, on the same retained physical hierarchy. The interpolation near projection ties uses a different gradient bound from (WS8). On a slightly larger separated window each gradient in (WS7) is a finite sum of readings of successor-degree basis words, with coefficients from Dk,α. These coefficient matrices converge and hence are bounded. The first, all-separated part of (WS3) applied to those basis words and the finite-maximum argument above therefore give δk sup larger separated window ∥∇Rk∥−→0. (WS9) Round every coordinate to the rescaled mesh, then perturb the rounded coordinates by distinct bounded integer offsets to make every coordinate projection different. The number of forbidden equalities is finite for a fixed word, so one fixed finite list of offsets suffices. The displacement remains O(δk). For large k, all short joining coordinate segments stay in the larger separated window. The integral of the gradient along them is o(1) by (WS9). This proves interpolation also at coordinate ties without losing a power through a shrinking coordinate gap. There is one final sign choice when these maps are required to identify the normalized physical stress with the free stress. Each enlarged finite request allows one of the two signs of the physical conserved line. If one sign works for all requests, retain it. If a particular request excludes it, enlarge any subsequent request to include that one; its available map must use the opposite sign and restricts to a map for the subsequent request. Hence one fixed sign works for every finite request. Subsequent normalization fixes its amplitude as explained below. For completeness, a single sequence realizing all the finite requests is obtained as follows. At arity n, exhaust the separated space by Kn,j = {x : max l |xl| ≤j, min l̸=m |xl−xm| ≥j−1}, (WS10) with the minimum condition omitted for zero or one slot. Enumerate the countable requests consisting of an arity, a finite tuple of degree bounds, a derivative order and an integer j. At stage s, enlarge one finite request to contain the first s requests, their finitely many basis words, 401 ## Source page 404 A Wilson Yang–Mills construction J. Rodgers the detecting spectators, and one extra derivative degree. Use the finite construction on the same physical hierarchy and the same chosen stress sign. Along its available comparison sequence choose a station late enough that the radius is below 1/s, the rescaled mesh is below 1/s, all the first s derivative errors on their compact sets are below 1/s, and every required instance of (WS9) is below 1/s. There are finitely many conditions, each eventually true, so such a station exists. Pad maps above this stage’s horizon by arbitrary fixed isomorphisms; every fixed degree eventually lies below the horizon. The selected stage-s maps and radii form one common sequence. For a fixed enumerated request, all later stages include it and give errors at most 1/s; hence it has the claimed limit. Every compact separated set lies in some Kn,j, so this proves all the asserted compact limits. If ordered mesh-period coefficient extractions are refined during these constructions, retain their countable columns and inverse charts together by the ordered compact extraction already used in (WS1): at each period take mesh limits first, then refine the period sequence and diagonalize the countable coordinate list. This preserves the same table and stress. The station diagonal just described is taken after those ordered limits and does not identify them with an arbitrary joint cutoff-volume diagonal. At any one station only its finite horizon is asserted; every fixed request is eventually included. 16.5 Constructing the physical stress and its shell charges Centering removes the unit direction before stress rank is compared. Shell-flux representability is then proved for the conserved stress line, with the finite Ward residual displayed until its limiting estimate is consumed. Exact conservation of the eventual physical field does not imply vanishing of every finite localized divergence. We first detail the rank transfer on centered tensors. A scaling isomorphism need not take the free expectation functional to the physical one. Let c0(v) = v−⟨v⟩01 and c(v) = v−⟨v⟩1 be the two centering maps. Apply them componentwise to tensors. For a late scaling index, Uk,4 intertwines reality and signed symmetries and fixes the unit. It therefore induces the map Rk : K0 −→K, RkB = c(Uk,4B), (SR1) between the real centered symmetric signed tensor spaces. Its inverse is A 7→c0(U−1 k,4A). Indeed the first centering subtracts only componentwise unit multiples. The inverse isomorphism carries those units to units, which the second centering kills; the remaining term is the original centered tensor. Thus these are exact inverse maps even though the two expectations differ. The corresponding degree-five isomorphism Qk on real signed polar vectors needs no centering correction. Write D0, D for the free and physical divergence maps on these centered tensor spaces. On the fixed free spaces consider Zk = Q−1 k rkDRk. (SR2) Derivatives of the unit are zero, so the centering correction in (SR1) contributes nothing to (SR2). The derivative comparison (WS7) gives ZkB →D0B for every B. The free calculation makes D0 onto and its kernel one dimensional. Choose a right inverse G of D0. Then ZkG →I in matrix norm, by convergence on a finite basis. Eventually ∥ZkG−I∥< 1; the geometric series makes ZkG invertible, hence Zk onto. Since rk > 0 and both surrounding maps in (SR2) are isomorphisms, D is onto. The exact dimension equalities from (SR1) and Qk, followed by rank–nullity, now give dimR ker D = dimR ker D0 = 1. (SR3) This establishes the conserved centered line without identifying the expectation functionals through Uk. 402 ## Source page 405 A Wilson Yang–Mills construction J. Rodgers The physical stress is selected on the common field quotient, before its final amplitude is fixed. At weight four the real signed symmetric tensor space includes the vacuum tensor as well as (FC3), so it has dimension four. The rank comparisons transfer this dimension from the free table to the physical table and identify it with the image of the four real reference coefficients Jab(w) = ( w0 1 + w1 P i Qai,ai + w2 P i,k Qik,ik, a = b, w3 P i Qai,bi, a ̸= b. (SC1) In (SC1) take the actual physical classes of these reference labels, not their free polynomial values. The physical weight-five polar vector space has dimension at most two, by the same rank comparison. Differentiation on the quotient therefore gives a linear divergence map from a four-dimensional real space into a space of dimension at most two: (∇J)b = X a ∂aJab. (SC2) Its range is polar because the signed derivative transformation has one sign for each derivative index; summing over that index cancels its squared sign. Adjoin the real one-point reading of J00 to (SC2). The resulting map has target dimension at most three and thus has a nonzero kernel. Choose w in that kernel and normalize its Euclidean coefficient norm to one. The tensor map (SC1) is injective by the four-dimensional rank comparison, so J(w) ̸= 0. It is conserved and its 00 mean vanishes. Reality makes every diagonal mean real; translation makes each mean constant in position; unsigned permutations make all diagonal means equal; reflection in one of two different indices negates an off-diagonal mean. Thus every mean vanishes. This constructs a centered nonzero real symmetric conserved tensor. It does not yet select its sign or claim its Ward charge is the unit translation generator. Here is the normalization of this line along the comparison curve. For any nonzero tensor J in it and the fixed nonzero free tensor T 0, put Bk = U−1 k,4(r4 kJ), ξk = ∥Bk∥ ∥T 0∥> 0, eBk = ξ−1 k Bk. (SR4) The inverse and norm here are on the full tensor coordinate space, including its scalar directions. The norm of eBk is exactly ∥T 0∥, so finite-dimensional compactness gives a convergent subsequence with limit B of that norm. No positive lower bound on ξk is needed. Closedness of the real signed symmetric space puts B in that space. Its free mean is zero: the one-point comparison for bounded converging labels applies to eBk, while its physical image is the positive scalar multiple ξ−1 k r4 kJ, whose expectation is exactly zero. Applying the converging derivative matrices (WS7) to eBk similarly passes the exact physical divergence zero to the free divergence zero. Thus B ∈ker D0. By its one-dimensionality, B = cT 0 for real c; equality of norms gives |c| = 1. Choose the single sign of J that makes c = 1. Restricting the comparison parameter to this subsequence preserves all earlier limits and exact compatibility laws. This proves the normalized stress comparison used above; its eventual Ward amplitude is fixed separately below. The finite lattice Ward identity associated with this construction retains its residual explicitly. At spacing aN, write T N,p ab (x) for the calibrated cylinder (SC1). Define its symmetric divergence, a clover-based directed Wilson variation, and its residual by diva Tb(x) = (2aN)−1 X a [Tab(x + ea) −Tab(x −ea)], ∆bF(x) = (2a5 N)−1 X i [contactx,+i(Xbi, F) −contactx,−i(Xbi, F)], Ab(x) = diva Tb(x) −ZδBb(x). (SC3) 403 ## Source page 406 A Wilson Yang–Mills construction J. Rodgers Here Xbi is the signed clover: one quarter of the four oriented plaquette curvature readings, with signs +, −, −, + for (+b, +i), (+b, −i), (−b, +i), (−b, −i). All are transported to the common base. Its adjoint covariance and antisymmetry follow by reversing the corresponding plaquette words. In a color orthonormal basis ec, the contact is P c XcDx,±i,ecF. The corresponding equation insertion is β⟨X, forcex,±i⟩−P c Dx,±i,ecXc; Bb is its same signed sum divided by 2a5 N. Right or left Haar invariance in that one directed link gives * β⟨X, force⟩− X c DcXc ! F + = *X c XcDcF + . (SC4) Differentiate the change of variables in the finite Haar integral; the derivative of the Wilson density gives the force, and the product rule gives the derivative of X and of F. Compactness and continuity of the finite word derivatives justify moving that derivative through the integral. Linearity over the color basis proves (SC4) for the group-valued clover vector field. For a lattice cutoffχ, a vector field V , an enclosed insertion F and an exterior word E, let Wab(x) = (χ(x + ea) −χ(x))Vb(x + ea) + (χ(x) −χ(x −ea))Vb(x −ea) 2aN . (SC5) The finite flux is −a4 N P x,a,b Wab(x)⟨Tab(x)FE⟩. Its exact decomposition is Flux = a4 N X x,b χ(x)Vb(x)Zδ⟨∆bF(x)E⟩+ a4 N X x,b χ(x)Vb(x)⟨Ab(x)FE⟩. (SC6) Use (SC4), the product rule, and finite symmetric summation by parts. The term with the discrete derivative of V vanishes for the constant and skew-linear Killing fields in question, by symmetry of T. The shifted V ’s in (SC5) are essential for this algebra. The derivative of E vanishes when each active Ward center avoids its full local insertion footprint. This is a support condition on the whole cylinder, stronger than just excluding its marked point. Taking absolute values in the second sum gives its exact error bound. No smallness of A, including at the enclosed insertion, follows from this identity alone. The continuum charges are instead supplied by the paid short-distance product estimates. Their input is the actual isolated product r hIJab(⌈y/aN⌉)IB(0) −Iu(0), D + h = d + 4, (SC7) with B of weight d and u in the one fixed horizon D. At a fixed shell its coefficient bound may depend on that shell. It is common to every exterior test at that shell. The estimate is a rooted cylinder estimate on the reporting cell and on all its specified aligned seams and offsets, not only a two-point expectation. The paid product construction and weighted mesh interpolation give, for each finite family of exterior tests annihilating every D-field, sup y∈{rm/4<|y|<2rm} X ν cνrh m⟨Jab(y)B(0) Oν⟩ −→0. (SC8) The schedule uses rm > 0 tending to zero and a finer mesh level j(m) with sj(m)/sm →t > 1. The product price chooses t = 1 + 1/[64(d + 12)], so (d + 12)(1 −t) = −1/64 > −1/32; its exponential gain dominates the growth of the two insertion columns. The rooted remainder after the exterior stencil estimate is at most 2 stakes(k)/ pitts(m), where pitts(m) →∞. The common all-order mesh bounds transfer this from the fine reporting grid to the whole annulus, with the finer mesh 404 ## Source page 407 A Wilson Yang–Mills construction J. Rodgers small compared with rm/128. These are the annular product estimates used here; the finite Ward residual in (SC6) is not substituted for them. Here are the rate choices that supply (SC7), rather than leaving its annular approximation as an input. Write qi = log(a−1 i ). The calendar has qi+1/qi →1. For t > 1, let j(i) be the first index with qj(i) ≥tqi. Minimality gives qj(i)−1 < tqi, so squeezing between these inequalities and qj(i)/qj(i)−1 →1 gives qj(i)/qi →t; also j(i) > i + 1 eventually. A positive sequence of logarithmic rate b here means log fi/qi →b. Products add rates, quotients subtract them, and composition with j multiplies them by t. The estimates remain valid with arbitrarily small rate losses. For D + h = d + 4, the two-field product and shell rescaling bounds give the available local approximation radius a logarithmic rate −D + M, M = (d + 12)(1 −t). (SC8a) The d and 4 are the field weights, the h is the shell rescaling, and the additional eight is the pair of four-dimensional placement bounds. The adjacent insertion remainder starting at level k obeys the explicit induction bound mk,i ≤b  ai ak D+1 fi fk , (SC8b) where fi > 0 is its retained error allowance. The induction step is exactly the allowance inequality ηiLD+1 i fi ≤fi+1, combined with ai = Liai+1. Choose t = 1 + [64(d + 12)]−1. Then M = −1/64 > −1/32. The test normalization ci = eqi/32 has rate 1/32, while the accumulated coefficient- normalization gain from level k to i is e(qi−qk)/8, of rate 1/8. Substitution of (SC8a)–(SC8b) into the finite product approximation leaves the strict inequalities 1 32 −(D + 1) < −D + M, 1 32 −M −1 8 < 0. (SC8c) All column counts, fixed-frame norms and logarithmic factors have rate zero. The two strict margins in (SC8c) therefore pay them eventually, give a positive admissible approximation radius, and make its rooted error at most 2 stakesp,k /ci, the bound in (SC7). The support requirements hold as well: the shell radius tends to zero, and the finest mesh times every finite stencil count tends to zero relative to the fixed physical chamber size. Thus the finite two-field approximation applies at every point in the required annulus, with bounded coefficient representatives. This is the asserted source of (SC7). Why do the approximants in (SC7) disappear from (SC8)? For a finite exterior family, let LN : ED →C be its combined finite-cutoffreading. Its limiting reading is zero by hypothesis, and convergence on a basis of ED gives ∥LN∥→0. At each fixed shell ∥uN∥≤A, so |LN(uN)| ≤ A∥LN∥→0, although the approximant itself need not converge. The rooted estimate bounds the remaining product discrepancy against each exterior word by reflection positivity and its spectator cell bounds. The mesh interpolation then gives (SC8). This argument integrates annihilators, rather than selecting a new field at every integration point. Let χ be smooth, compactly supported, one near the enclosed point and zero near the other insertions. For a Killing field V , define QV (B; χ) = − X a,b Z (∂aχ)(y)Vb(y −x0)⟨Jab(y)B(x0) O⟩d4y. (SC9) Its integrand is supported on a compact shell disjoint from all insertions, so it is an ordinary integrable smooth function. Conservation and symmetry imply independence of the cutoff: for their difference integrate the divergence of (χ1 −χ2)Vb⟨JabBO⟩. The divergence of J is zero, and 405 ## Source page 408 A Wilson Yang–Mills construction J. Rodgers the symmetric tensor contracts to zero with the skew derivative of V . The difference vanishes near every insertion and has compact support, so this integration by parts has no contact or infinity boundary term. Take V homogeneous of degree i = 0 or 1 and shrink χ to radius rm. Changing variables y = x0+rmz in (SC9) gives the factor r3+i m : four powers from volume, minus one from differentiating the cutoff, and i from V . Choose h = 3 + i in (SC8). Its bound times the finite integral of |∂aχ(z)Vb(z)| tends to zero. Therefore every finite linear relation among the ordinary D-field exterior readings is satisfied by the flux readings, with D = d + 1 for translations and D = d for rotations. Finite-dimensional linear algebra now represents all these flux readings by one field. Indeed, if functionals Aν ∈E∗ D and values bν satisfy every finite relation P cνAν = 0 ⇒P cνbν = 0, define a linear functional on their span by Aν 7→bν, extend it to E∗ D, and use E∗∗ D = ED. Faithfulness of the quotient makes the representing class unique. Initially use generic grid configurations; translation and continuity of the shell integral and of the kernels extend the same identity to every separated configuration, including coordinate ties. Linearity of (SC9) in B and faithfulness prove that the representing field depends complex linearly on B. Its rotation charge depends real linearly on the skew matrix, by the same argument. These are the actual unnormalized translation and angular charge maps on the physical quotient. Their normalization and full rotation laws are established by the subsequent angular-flux and comparison arguments; centering and the finite-dimensional existence argument alone do not fix them. The shell functional also has an ordinary face-flux interpretation. Let a closed coordinate box contain x0 strictly in its interior and every exterior insertion strictly outside the closed box. Integrate each a-component of the current on its upper face minus its lower face, with three- dimensional Lebesgue measure in the remaining coordinates. This sum equals (SC9). To prove it choose a smooth window supported strictly inside the box and equal to one near x0. Multiply the current by 1−χ. It is zero near the only enclosed insertion and smooth at all points of the box. Its divergence equals −P a ∂aχ Fa, by conservation and the Killing identity. The ordinary coordinate divergence theorem applies to this smooth truncated current. On every face χ = 0, so its face flux is the original one. The volume integral is (SC9), and cutoffindependence gives the equality for any allowed window. This argument treats faces with their outward orientations and does not integrate with four-dimensional ambient measure on the surface. The bounded coefficient in (SC7) is required at each fixed shell before its lattice limit. It need not be uniform as the shell shrinks. The order is: fix the finite exterior annihilator and a reporting level; buy a sufficiently fine seam level; fix a shell index; then take the ordered mesh and period limits. At that shell the finite-dimensional annihilator norm tends to zero and the approximant coefficient is bounded. Only after this limit is taken does the shell radius go to zero. The rooted error tends to zero by (SC8a)–(SC8c). The same aligned-cell estimate applies on every seam and offset, so it permits moving a shell node freely in the annulus. This is the joint two-field approximation required for the shell construction. Equation terms may be restored before this estimate is used. For one chamber their difference has zero expectation against every function supported outside that chamber, by the link Haar identity and avoidance of all the active links. Replace one chamber factor at a time in the full replicated array. All other factors are supported in complementary chambers, so each replacement leaves the integral unchanged. Finite induction replaces every chamber, including the conjugated reflected copies. Taking the reciprocal-cardinality power then gives equality of the two rooted seminorms. An equality only for a single unreplicated expectation would not suffice. No contact error at overlapping stencils is suppressed in this argument: overlap is excluded by the full footprint 406 ## Source page 409 A Wilson Yang–Mills construction J. Rodgers condition, and actual contacts are kept in (SC4)–(SC6). 16.6 The centered stress trace The positive two-point representation gives monotonicity in positive Euclidean time before the short-distance limit is known. The comparison can be selected inside any proposed sequence of radii on the same table and phase. Monotonicity sandwiches nearby generic distances around each selected radius; applying this to a hypothetical violating sequence proves the full right-hand limit. A single favorable scaling subsequence would not suffice for this step. Use the selected character vacuum of the same hierarchy and its conserved physical stress tensor. Let T = P3 μ=0 Tμμ be its Euclidean trace, a degree-four physical field, and let Θ = T −⟨T⟩1. Its pinned reconstructed field is Hermitian and Lorentz scalar, and ⟨Ω, Θ(f)Ω⟩= 0, Θ(f) = T(f) −⟨T⟩ Z f I (TR1) on the common field domain. To check the identity operator in this formula, first insert a unit label in any Euclidean correlation and use the hierarchy’s unit law to remove it. The two corresponding strict complex-time functions agree on imaginary times and hence everywhere by holomorphic uniqueness. Smearing in that slot gives R f times the remaining function, by Fubini, and passing to the tempered boundary gives the same Wightman identity. Pair the resulting word vectors with arbitrary word vectors. Their dense span implies that unit insertion is ( R f)I on the common domain. Linearity then proves (TR1). Similarly the Euclidean one-point function of Θ is zero; holomorphic uniqueness and the boundary limit prove its vanishing vacuum matrix element. Reality and scalar covariance follow from the stress tensor and the real invariant unit. Let E(s) = ⟨Θ(−s 2e0)Θ( s 2e0)⟩E, s > 0. The normalization used in the source gives lim s↓0 s8 log2(1/s) E(s) = AG > 0, AG = 12d2(G)2 b0(G)2 dim g π4 , d2(G) = −b0(G) 2 . (TR2) Thus AG = 3 dim g/π4 in these conventions. Positivity follows from b0(G) > 0, dim g > 0 for the simple nonabelian Lie algebra, and the nonzero coefficient d2. These constants refer to the physical stress normalization and the Wilson-calibrated running charge. We record the mechanism establishing the full positive-distance limit from the scaling con- struction. Orthogonal invariance of Θ and translation invariance identify its separated two-point function at any two points with E of their Euclidean distance. An orthogonal reflection takes their difference to a time-axis vector of the same norm; a common translation then puts them at the opposed points above. The degree-four scaling comparison identifies the rescaled trace with the scalar free-field label multiplied by 4d2, after the logarithmic running normalization. The free scalar two-point contraction is 3 dim g π4 |x0 −x1|8 . The running normalization contributes b−2 0 /4, while the trace coefficient contributes (4d2)2, giving exactly AG|x0 −x1|−8. This proves the normalized limit along the extracted scaling radii and their nearby generic configurations. 407 ## Source page 410 A Wilson Yang–Mills construction J. Rodgers To remove the nearby-distance error, use the positive spectral representation E(s) = Z V + e−sp0 dμΘ(p). (TR3) It is real and nonnegative. Since p0 ≥0 almost everywhere for this measure, 0 < s ≤t implies e−tp0 ≤e−sp0, and integration proves E(t) ≤E(s). Finiteness at every positive s is part of the positive-time spectral representation. Thus this monotonicity is available before proving the short- distance limit; it is not inferred from that limit. We give the precise selection statement needed here. Fix the physical table, its phase decompo- sition and its stress tensor. The trace construction selects a comparison ultrafilter below the table’s mesh-then-volume filter, its retained local data, and one free hierarchy. For every sequence sn > 0 tending to zero, it then provides a strictly increasing map j : N →N, a scaling family on this same physical table, and its comparison data, whose radii satisfy the exact identity ri = sj(i). (TLim1) The scaling comparison applies to these radii. In particular the statement is not merely that one favorable sequence ri →0 exists. Nor does it select a different physical hierarchy or phase when the sequence sn changes. The construction and this universal sequence property were established in the preceding interacting-to-Gaussian comparison. First prove convergence along the radii of any such retained comparison. For 1/2 < u < 2, put x(u) 0 = 0, x(u) 1 = (u/2, u/2, u/2, u/2). The four coordinates of the two points are distinct, so this is a generic two-point configuration, and its Euclidean separation is u. Its maximum-coordinate separation is u/2 > 1/4, while its configuration norm is less than 2. Consequently it has a neighborhood in the fixed comparison window with norm bound 100 and maximum-coordinate separation at least 1/100. Let bx(u) i be the rounded configurations in that comparison, and set di(u) = |bx(u) i,0 −bx(u) i,1 |. Round- ing convergence gives di(u) →u. The actual physical configuration is exactly ribx(u) i , not just an asymptotic replacement of it. The interacting comparison, followed by the running-clock identity and the free scalar Wick contraction, gives c(ri)E(ridi(u)) −→AGu−8, c(r) = r8 log2(1/r). (TLim2) The centering and scalar homogeneity used in this identity are retained in the trace-label calculation below. The same normalization c(ri), independent of u, is used on both sides of the coming inequalities. Fix a < 1 < b in (1/2, 2). Eventually 0 < di(a) < 1 < di(b). Monotonicity and c(ri) ≥0 give c(ri)E(ridi(b)) ≤c(ri)E(ri) ≤c(ri)E(ridi(a)). By (TLim2), their outer limits are AGb−8 and AGa−8. Given ε > 0, first choose a, b close enough to 1 that both outer limits differ from AG by less than ε/2. Then choose i large enough for the two convergences and the bracketing inequalities. This proves c(ri)E(ri) →AG. The complex pairing has zero imaginary part by (TR3), so this also proves complex convergence. Finally, suppose the full right-hand limit at zero failed. There would be ε > 0 and, for every n, a radius 0 < sn < 2−n such that |c(sn)E(sn) −AG| ≥ε. 408 ## Source page 411 A Wilson Yang–Mills construction J. Rodgers Apply the universal selection statement to this particular sequence. Equation (TLim1) retains this inequality at every ri, whereas the preceding sandwich proves that its left side tends to zero. This contradiction proves (TR2) for the full small-scale limit. No density or ratio condition on a single favorable sequence of radii is being assumed. The positive measure in (TR3) is unique and is the trace spectral measure. With the Fourier sign and normalization used in the reconstruction, it satisfies ∥Θ(f)Ω∥2 = Z | bf+(p)|2 dμΘ(p), ⟨Θ(f)Ω, Θ(g)Ω⟩= Z bf+(p)bg+(p) dμΘ(p). (TR4) Its support lies in the forward cone, it is Lorentz invariant and of temperate growth, and every positive time damping has finite total mass. The full two-point Wightman distribution is its relative Fourier transform, also on joint Schwartz tests. Existence follows from the reconstructed joint momentum measure on the dense family of smeared trace vectors; these quadratic forms patch to a single positive measure. Uniqueness follows by testing all | bf+|2, polarizing, and using compactly supported test functions to identify the measure. The regularized point-vector construction (TR5) and the measure argument (TR10)–(TR17) below give the patching and ultraviolet limit explicitly. The spectral construction can be made directly from the regularized point vectors. Let vs, s > 0, be the single trace shot with positive imaginary-time height s, centered at the origin. The shot identities already proved give vs+t = e−tHvs, W2 at heights (−s, s) = ⟨vs, U(x1 −x0)vs⟩. Let μs be the diagonal joint-momentum measure of vs. Functional calculus gives μs+t = e−2tp0μs. Define dμΘ(p) = e2p0 dμ1(p). (TR5) The semigroup identity shows equivalently dμΘ = e2sp0dμs for every s > 0: compare both heights through their sum. Hence e−2sp0μΘ = μs is finite, and its characteristic function is the regular- ized two-point function. This proves (TR3), with s/2 in the shot identity. Smearing the same identity and taking the previously established boundary limit yields (TR4) and the full joint-test Fourier representation. The momentum cone support is inherited from the shot measures, and scalar Lorentz covariance identifies each Lorentz pushforward with the same unique measure. The polynomial growth estimate for these measures follows from the common polynomial shot-height bound; this estimate will be detailed below. The free contraction constant in (TR2) has a direct finite calculation. For x ̸= 0, put Rai(x) = δai −2xaxi/|x|2. This is an orthogonal reflection, since multiplying its two rows and summing the common index gives δab. The free curvature covariance has the form Curlabij(x) = D(x) RaiRbj −RajRbi , D(x) = 4 normal |x|−4. Orthogonality yields X a,b,i,j Curlabij(x)2 = 24D(x)2. Indeed summing first over j, then i, gives D2(2 −2δ2 ab); summing the a, b indices gives 24D2. The scalar curvature-square field has coefficient 1/4 in each sum, two Wick contractions between its two quadratic factors, and dim g color choices. Thus its two-point function is (1/16) 2 dim g 24D(x)2 = 409 ## Source page 412 A Wilson Yang–Mills construction J. Rodgers 3 dim g/(π4|x|8) with the source’s Green-function normalization. This supplies the free constant used above. We next derive the ultraviolet measure limit from (TR2). Write Z(ρ) = ρ−8 log2 ρ, νρ = Z(ρ) (p 7→p/ρ)∗μΘ, ρ > 1. These are positive Lorentz-invariant cone-supported measures. For s > 0, Z e−sp0 dνρ = Z(ρ)E(s/ρ) −→AGs−8, (TR6) because Z(ρ)/Z(ρ/s) →s−8. Lorentz invariance extends (TR6) to every strictly future timelike damping vector. The comparison measure is dν∞(p) = AG 1536π (p2 0 −|p|2)21{p0≥|p|} d4p. (TR7) Its time-axis Laplace integral is AGs−8, as can also be checked directly: integrate spatial radius r ∈[0, p0], obtaining 4π Z p0 0 (p2 0 −r2)2r2 dr = 32π 105 p7 0, and then R ∞ 0 e−sp0p7 0 dp0 = 7!s−8. The product is 1536πs−8. Positivity, cone support and finite timelike Laplace transforms allow the Laplace convergence theorem to identify the compact-test limit with (TR7); its proof is unfolded next. There is a uniform polynomial bound strong enough for Schwartz tests. Since e−p0/R ≥e−1 on p0 ≤R, νρ{p0 ≤R} ≤e Z e−p0/R dνρ ≤CR8, R ≥1, (TR8) uniformly for sufficiently large ρ. For the last inequality use (TR2) at (ρR)−1, and Z(ρ) ≤R8Z(ρR). The same bound holds for ν∞. On the forward cone, Euclidean norm is comparable to p0; dyadic an- nuli and Schwartz decay therefore make the tails of every Schwartz test uniformly small. Combining this with compact-test convergence gives Z(ρ) Z φ(p/ρ) dμΘ(p) −→ Z φ dν∞ (φ ∈S(R4)). (TR9) For joint two-point tests use their relative wave transform. The normalized position dilation fρ(x) = ρdf(ρx) satisfies bfρ,+(p) = bf+(p/ρ) by change of variables. The same identity for the joint wave transform carries (TR9) to the source’s relative and joint Wightman distribution limits. We now prove the polynomial growth and Laplace continuity assertions used in (TR6)–(TR9), then the finite mass-shell obstruction. The subsequent stress correction and Ward calculation derive the trace coefficient and its Wilson normalization. 16.6.1 Polynomial bounds and the Laplace continuity argument Which topology the spectral passage controls The spectral reader is first defined at positive time, where exponential damping makes its integral finite. Passing from these Laplace readings to a tempered distribution requires uniform polynomial growth bounds on the rescaled positive measures and a uniqueness argument on the allowed tests. The proof below supplies those bounds before taking the Schwartz limit. Convergence at a few positive-time values would not by itself justify arbitrary Schwartz smearing or the later spectral comparison tests. 410 ## Source page 413 A Wilson Yang–Mills construction J. Rodgers Here are the measure arguments used above. They apply to positive measures; no assertion about convergence of arbitrary signed measures is needed. Suppose μ is supported in the closed future cone and μ{p : p0 ≤R} ≤ARk (R ≥1). (TR10) The damped point-vector estimate supplies this inequality directly. Indeed, on the energy slab the weight e−2p0/R is at least e−2. If the squared norm of the vector at height R−1 is at most A0Rk, its spectral representation gives (TR10) with A = e2A0. A Euclidean ball of radius R is contained in that energy slab. On the dyadic sets 2j ≤1+∥p∥≤2j+1, the mass is at most A2k(j+1). Consequently Z (1 + ∥p∥)−k−1 dμ(p) ≤2k+1A. (TR11) The sets may overlap on their boundaries, which only enlarges this upper bound. This proves the polynomial temperateness needed for Schwartz testing. For completeness, let μi, ν be supported in the future cone C, with finite Laplace transforms at every u in its open dual cone C◦, and suppose Z e−u·p dμi(p) −→ Z e−u·p dν(p) (u ∈C◦). (TR12) The pairing here is the real Euclidean pairing used for the damping parameter. On the one-point compactification C ∪{∞}, the functions su(p) = e−u·p, extended by zero at infinity, are continuous: for p ∈C, u · p ≥1 2(u0 −∥u∥)∥p∥. Their products satisfy susv = su+v. Together with the constant function they separate points. They distinguish infinity from finite points by positivity; if two finite points have equal pairings with every u ∈C◦, perturbing the time unit vector by a sufficiently small multiple of any real vector shows that the two points coincide. The real Stone–Weierstrass theorem therefore makes their linear span uniformly dense in the continuous functions on this compactification. Fix u ∈C◦and damp all the measures by e−u·p. Their total masses converge by (TR12). Integrals of the constant and of every sv converge, since the latter are precisely the Laplace trans- forms at u + v. Uniform approximation and the bounded total masses give convergence against every continuous function on the compactification. If f is continuous and compactly supported in the ambient space, eu·pf(p), restricted to C and extended by zero at infinity, is such a function. Its damped integral equals R f dμi. This proves convergence against real compactly supported continuous tests, and then against complex tests by taking real and imaginary parts. If (TR10) holds with common A, k for all these measures and the limit, this convergence extends to every Schwartz function. To see this explicitly, put q = 2k+1A and choose B with |f(p)| ≤ B(1 + ∥p∥)−k−2. Let χR be a smooth cutoffequal to one on the radius-R ball, supported in the radius-R + 1 ball, with 0 ≤χR ≤1. For R ≥1, Z |f −χRf| dμi ≤Bq/R, Z |f −χRf| dν ≤Bq/R. Choose R to make both tails smaller than a prescribed third of the error, then apply compact-test convergence to χRf. This proves (TR9) from (TR6)–(TR8), with all tail estimates uniform. The same density argument also proves uniqueness of the positive cone-supported measure from its all-future Laplace transform. 411 ## Source page 414 A Wilson Yang–Mills construction J. Rodgers 16.6.2 The finite polynomial comparison class and its obstruction The mass-shell exclusion is a finite comparison statement. The allowed functions are the finite complex linear combinations of finite products of the specified mass-shell integrals, with the stated integer and homogeneity restrictions. Their radial expansions have the prescribed power and loga- rithmic form, incompatible with the inverse square logarithm of the trace. The separate Maxwell argument retains its own locality, sector and comparison hypotheses. The polynomial-field exclusion has no bound on the number of colors or on the degree of the two chosen polynomials. Each elementary contraction, however, has a finite sum of fixed mass-shell kernels with finite polynomial momentum numerators. More generally, the proof permits a real continuous homogeneous spatial numerator P of nonnegative integer degree j, an energy power k ≥0, and a mass m ≥0. Write ωm(p) = q m2 + |p|2, a(p) = ωm(p)kP(p) 2ωm(p) , p+ = (ωm(p), p). (TR13) Values at the isolated massless zero momentum do not affect the integrals. A contraction track with n elementary factors acts on a total-momentum test by L(φ) = Z (R3)n n Y α=1 aα(pα) φ n X α=1 (pα)+ ! n Y α=1 dpα. (TR14) The empty track is evaluation at zero. Allowed comparison kernels are finite complex linear combi- nations of these tracks. This includes units, arbitrary constant contraction coefficients and statistics signs; positivity of the comparison kernel is unnecessary. Indeed, expand each polynomial into finitely many monomials. A cross-vertex pairing is a bijection between the occurrences at the two vertices. For each arc, expand its finite sum of elementary shell kernels, and then expand each polynomial numerator into monomials in energy and spatial coordinates. Each resulting term is (TR14); all arc coefficients, monomial amplitudes and pairing coefficients multiply into its constant coefficient. Summing the finite choices proves the asserted inclusion for any two finite polynomial fields. An infinite set of available generator labels causes no difficulty, since only finitely many occur in the two fields being compared. At positive imaginary time s, a track is absolutely damped and its value factors: L(e−sp0) = n Y α=1 Z R3 aα(p)e−sωmα(p) dp. (TR15) Polar coordinates and homogeneity reduce an elementary factor to 1 2 Z S2 P(θ) dθ  Z ∞ 0 rj+2(m2 + r2)(k−1)/2e−s √ m2+r2 dr. (TR16) These formulas also give absolute integrability for compact total-momentum tests: such a test is bounded by a constant times e−sp0 on its support. The small-s radial expansion used here has the following exact meaning. For every integer N, the radial factor differs by O(sN) from a finite sum of terms sa(log s)b, where a ∈Z and b ∈N. Finite sums and products preserve this property. For the product statement, first bound each factor by some integer power of s, then take the other factor’s expansion far enough to pay that bound. Thus the real part of every finite combination of (TR15) admits, in particular, such a finite expansion with bounded remainder. The radial expansion itself will be derived below from the massive integral; it is not an assumption on the comparison fields. 412 ## Source page 415 A Wilson Yang–Mills construction J. Rodgers No real function with such a bounded-remainder expansion can have a nonzero finite limit after multiplication by s8 log2(1/s). To prove this, the bounded remainder contributes zero. Combine equal terms in the finite expansion and set s = 2−n. A nonzero term becomes a constant times  2−(8+a)n nb+2. (TR17) Order the finitely many terms lexicographically by their exponential base and then their polynomial degree. If the largest base is below one, every term tends to zero. If it is at least one, its term diverges in magnitude and every lower term divided by it tends to zero. Dividing a supposed finite limit by this largest term would force its nonzero coefficient to vanish. These two cases prove the assertion. It remains to justify that equality only on the product tests stated in the theorem would imply equality of these imaginary-time values. Every smooth compactly supported momentum test v can be represented as the relative wave transform of a tensor product of two Schwartz position tests. Choose the first Fourier test equal to one on a ball containing supp v; choose the second by Fourier surjectivity so that the relative product is v. Equivalently one may multiply v by the first Fourier test and its conjugate, whose squared modulus is one on this support. Hence equality on the indicated tensor tests gives equality of R v dμ and the finite track combination for every such v. Choose smooth compact cutoffs χn tending pointwise to one, with 0 ≤χn ≤1, and apply this equality to vn(p) = χn(p)e−sp0. On the trace side, finite damped mass permits dominated convergence. On each track side, the absolute damped integrability in (TR15) permits the same passage. Therefore equality of the tested two-point functions would imply E(s) = M X l=1 clLl(e−sp0) (s > 0). Taking real parts and applying (TR17) contradicts the positive limit (TR2). This proves the stated exclusion for every finite polynomial comparison of this class. The promised radial expansion follows from a convergent-integral argument, so it does not require summing a formal asymptotic series. Put Mj,k(s) = Z ∞ 0 rj+2 √ m2 + r2 (m2 + r2)k/2e−s √ m2+r2 dr. For s > 0 these integrals are finite. The cases k = 0, 1 are dominated respectively by rj+1e−sr and rj+2e−sr; higher cases follow from the exact recurrence Mj,k+2 = m2Mj,k + Mj+2,k. (TR18) The gamma integral gives Mj,0(s) ≤Γ(j + 2)s−j−2. Differentiation under the integral is justified locally at s > 0 by the integrable bound with damping s/2, and gives M′ j,k = −Mj,k+1. Integration of the derivative of rj+3e−s √ m2+r2 gives sMj+2,0 = (j + 3)Mj,1, d ds sj+3Mj,1  = −m2sj+3Mj,0. (TR19) The radial boundary terms vanish at zero and infinity. The derivative is integrable by the preceding bounds. Here is the finite-order integration rule used with (TR19). Say that f has expansion to order a ∈Z if f is a finite linear combination of integer powers of s times nonnegative powers of log s, 413 ## Source page 416 A Wilson Yang–Mills construction J. Rodgers up to O(sa). If f′ = g, g has expansion to order a, and a ̸= −1, then f has expansion to order a + 1. Every finite expansion has an antiderivative of the same kind: integrate s−1(log s)b directly, and use integration by parts for all other powers. After subtracting this antiderivative it remains to treat |f′| ≤Csa. If a < −1, integrate from s to a fixed positive point to obtain f = O(sa+1). If a > −1, the derivative is integrable at zero, so f has a finite limit and differs from that constant by O(sa+1). The arbitrary integration constant is itself an allowed expansion term. This proves the rule. Start with the bound at order b = −j −2 for Mj,0. If it has expansion to some integer order a ≥b, multiply by sj+3, apply the integration rule to the second identity in (TR19), and divide back by that power. Since j+3+a ≥1, the exceptional integration exponent is avoided. This yields an expansion of Mj,1 to order a + 1. Its negative is the derivative of Mj,0; integrating improves the latter to at least order a + 1. If a + 1 = −1, first weaken the derivative remainder to order −2 and integrate that remainder, obtaining the required order −1. Otherwise the integration rule gives order a + 2, which can be weakened to a + 1. Induction supplies expansions of Mj,0 to every integer order. The preceding deduction then does the same for Mj,1, and (TR18) proves it for every k. This proves the expansion asserted after (TR16), and closes the polynomial-field exclusion. 16.6.3 The Maxwell comparison and the reduction to local sector estimates The Maxwell conclusion concerns a specified comparison of representations. For m copies use the photon measure dλ(k) = d3k/(2|k|), with its irrelevant zero-momentum value assigned arbitrarily, and the Hilbert direct sum Famb = M n≥0 M s:{1,...,n}→{1,...,m}×{1,2,3} L2((R3)n, λ⊗n). (TR20) The physical subspace consists of waves symmetric under simultaneous permutation of momenta and labels and transverse in each slot. Transversality means invariance under applying Πab(k) = δab −kakb/|k|2 to that slot’s spatial component. Translation is multiplication by the plane wave at total momentum P(k) = P i(|ki|, ki). The vacuum is the unit wave in sector zero. The comparison hypothesis is a complex linear isometry U : H →Famb whose image is physical, which takes the selected vacuum to this vacuum and intertwines translations. It requires neither surjectivity nor an identification of all fields. For a list h of real Maxwell electric test functions, its one-particle wave is zh(c, a; k) = X b Πab(k)|k| bhc,b,+(|k|, k). Its coherent wave has sector values E(h)n,s(k) = e−∥zh∥2/2 in √ n! n Y α=1 zh(sα; kα). (TR21) Schwartz decay and λ({|k| ≤R}) = πR2 make zh square integrable. The sector norm sum is the exponential series, so (TR21) defines a physical Hilbert vector. Put σ(f) = UΘ(f)Ω. The excluded condition is the vacuum matrix identity ⟨E(−h), σ(f)⟩= ⟨σ( ̄f), E(h)⟩ (TR22) for every compact smooth f and compact smooth real list h whose supports are spacelike separated from that of f. The proof actually needs this only for f supported in one fixed ball of any positive radius. Thus it yields the precise small-region variant as well. 414 ## Source page 417 A Wilson Yang–Mills construction J. Rodgers Here is the initial spectral reduction, which uses only the comparison hypothesis. Let |Uv|2 dΛ denote the squared-wave measure on the direct sum (TR20). Its pushforward under P is the translation spectral measure of v: their characteristic functions are both ⟨v, T(b)v⟩. For v = Θ(f)Ω, this measure is | bf+|2 dμ. Polarization, also after bounded momentum multipliers, therefore gives bg+(P) σ(f) = bf+(P) σ(g) almost everywhere on the wave space. (TR23) For clarity, the squared norm of the difference equals the corresponding norm in L2(μ), where the two products are identical. This proves (TR23) without pointwise evaluation of a Hilbert vector at exceptional momenta. The remaining sector argument constructs common amplitudes qn,s from (TR23), proves from (TR22) that they satisfy the local position identities, and uses their finite dilation spectrum. Its numerical end is particularly useful to state now. Write Rj = 2j, Bn,j = {k : |ki| ≤Rj ∀i}, and let νq = |q|2λ⊗n. Positivity gives νqn,s(Bn,j) ≤μ{p : p0 ≤(n + 1)Rj}. (TR24) The trace asymptotic implies that the right side, multiplied by R−8 j , tends to zero. The local sector estimate proves that every sector with n ≥4 vanishes and that each of the remaining finitely many labeled sectors has νqn,s(Bn,j) ≤Cn,sR6 j. (TR25) Summing gives μ{p : p0 ≤Rj} ≤CR6 j, since an energy slab is contained in each sector’s correspond- ing barrel. This contradicts the R8/ log2 R high-energy law already proved. The common-amplitude construction and the local sector estimate are unfolded next. Choose a countable sequence of Schwartz functions vj whose positive Fourier transforms have modulus one on the radius-j ball. At each wave-space point choose the least j with bvj,+(P) ̸= 0, and define q = σ(vj)/bvj,+(P). (TR26) This is measurable, since the selection is made from a countable measurable partition. For any fixed f, intersect the countably many full-measure sets in (TR23) for g = vj. On that intersection, σ(f) = bf+(P)q. Thus the construction gives one amplitude for all f, with the usual separate almost-everywhere interpretation for each test. Weighted spectral equality then implies μ = P∗(|q|2dΛ), μ(B) = X n,s Z {P(k)∈B} |qn,s(k)|2 dλ⊗n(k). (TR27) To remove the Fourier weights, use the same vj: their squared weights equal one on an exhausting sequence of momentum balls. Equality on each ball and countable exhaustion prove equality of the measures. This proves (TR24), since P 0 = P i |ki| ≤nRj on Bn,j. The harmless n + 1 in that bound also covers the vacuum sector. We record explicitly the algebraic and measure-theoretic part of the local sector estimate. Treat amplitudes as measurable functions modulo almost-everywhere equality. For a slot i, let (Diq)(k1, . . . , kn) = q(k1, . . . , 2ki, . . . , kn). The operators Di commute. The photon measure scales with degree two: the pushforward under k 7→rk is r−2λ. For a parcel B(d) = {|ki| ≤di} this gives νDiq(B(d)) = 1 4νq B(d1, . . . , 2di, . . . , dn) . (TR28) 415 ## Source page 418 A Wilson Yang–Mills construction J. Rodgers The space of amplitudes with finite mass on every dyadic barrel and R−8 j νq(Bn,j) →0 is a real vector space, stable under every Di. Addition uses |q + r|2 ≤2(|q|2 + |r|2); stability follows from (TR28) and containment in the next barrel, with the fixed loss 26. Thus all finite polynomials in the Di preserve this growth restriction. The local distribution argument gives an integer K, common to the finitely many slots, such that K−1 Y a=0 (Di −2a−1)q = 0 (1 ≤i ≤n). (TR29) Here the exponent a−1 includes the electric-test factor |ki|. To split these identities while preserving the growth restriction, work entirely in the preceding invariant real vector space. A linear operator annihilated on a vector by a polynomial with distinct real roots decomposes that vector into its finitely many eigencomponents. One direct induction takes v = Q a 0, and each coefficient is zero. One can justify coefficient uniqueness without an analytic-series theorem: after the first N coefficients vanish, divide by tN; the tail is dominated for 0 < t ≤1 by the absolutely summable series at t = 1, so t ↓0 recovers the next coefficient. 418 ## Source page 421 A Wilson Yang–Mills construction J. Rodgers Each Lx is multilinear and symmetric. Symmetry follows by permuting the momentum variables and the labels together, using physical symmetry of x and invariance of the product photon measure. The diagonal identity for real lists therefore polarizes to arbitrary real lists. For example, expand the diagonal polynomial at h+tu; its derivative replaces one slot by u, and symmetry identifies the sum with n times that term. Induction on n recovers all mixed terms. Compact tests supported spacelike to the trace support form a complex vector space, and every such list is a complex linear combination of real lists by its real and imaginary parts, with the same support restrictions. Complex multilinearity then gives the polarized identity for all these lists, with complex conjugation of the lists on the conjugated side. Transversality removes the polarization matrices in these integrals. For a fixed slot expand zg = Π zraw g ; move the real symmetric matrix Π onto the wave component in the finite sum over spatial labels. That component is unchanged because x is transverse. Repeat in every slot. More explicitly, exchanging the selected component label and the summed matrix index is a bijection of the finite label-index pairs; all other slot factors remain fixed. No regularity of wave representatives is used, since there are only finitely many almost-everywhere transverse identities at each particle number. Finally take in slot i a list having just the component si nonzero, with scalar test gi. After the projectors are removed, all label sectors except s vanish in the sum. The surviving polarized identity is Z xn,s(k) Y i |ki|bgi,−((ki)+) dλn = Z yn,s(k) Y i |ki|bgi,+((ki)+) dλn, where x = σ(f) and y = σ( ̄f). The Fourier conjugation identity b ̄g+(p) = bg−(p) fixes the signs. Taking the real shrinking trace bump as described above yields (TR32). Together with (TR26)– (TR36) and the high-energy contradiction, this completes all the Maxwell exclusions asserted here. 16.6.4 Passing a running trace label through the scaling limit There are two separate ingredients in the trace limit: convergence of the running label in its finite-dimensional field space, and convergence of the moments evaluated on that label. The latter passage does not require a uniform convergence assertion on an infinite-dimensional space. Let Ed be the fixed free field space of degree d, choose a basis ea, and let Ui : Ed →E(i) d be the scaling identifications. The source proves, for every pair of basis vectors, ⟨Uiea(xi)Uieb(yi)⟩χ −→⟨ea(x)eb(y)⟩free (TR38) along the same chosen refinement and in every selected phase under consideration. The rounded scaled configuration tends to the target configuration; its rounding error is at most 4n/mi, where the mesh magnification mi →∞. Generic separated configurations stay in their open admissible region. The comparison error is bounded by oi(1 + M(xi, yi)), where oi →0 and the finite profile norm M is continuous at the separated target. This proves (TR38). If vi, wi ∈Ed tend to v, w, expand them in the chosen finite basis. Every coordinate converges, and bilinearity expresses the moment as the finite sum X a,b (vi)a(wi)b⟨Uiea(xi)Uieb(yi)⟩χ. Taking the limit term by term gives the free moment of v, w. This is the passage used for the running centered trace, rather than an interchange of an uncontrolled varying field with a distributional limit. 419 ## Source page 422 A Wilson Yang–Mills construction J. Rodgers The centering convention is also harmless at this step for a specific reason. If Tμν is the constructed stress tensor, replace it by Tμν −⟨Tμν⟩1. This preserves symmetry, reality, conservation and its charges: derivatives of the unit vanish, and a constant symmetric unit tensor has zero charge contribution in the stated Ward construction. Its trace differs from the phase-centered trace by a constant multiple of the unit. Under the inverse scaling map the latter contributes Bir4 i c 1, which tends to zero by the running-band estimate. The scaling maps identify the unit exactly. Thus the tensor normalization can first be proved with the convenient centered tensor and then transferred to the exact selected-vacuum trace. In the source’s coupling convention b0 = 11 Cadj 48π2 , βslope = 2b0, d2 = −b0 2 . (TR39) The last identity follows from the stress coefficient d2 = forceR after substituting the displayed slope; its derivation from the stress comparison is treated with that comparison below. Once the running centered trace converges to 4d2 Oscale, the preceding finite-basis argument gives its two- point limit. The free Wick calculation yields 3 dim g/(π4|x −y|8). The running charge contributes (2b0)−2 after multiplication by log2(1/ri). Consequently the constant is (2b0)−2(4d2)2 3 dim g π4 = 12d2 2 b2 0 dim g π4 = 3 dim g π4 , as claimed in (TR2). The powers and logarithms cancel before taking the limit: the two running labels each carry Bir4 i , and the prefactor is (log(1/ri)/Bi)2. The running-clock cancellation used here follows directly from the calibrated clock, including the extra magnification in the scaling sequence. Write bβi = β∗+ g ti, ri = e−tiai, g = 2b0 > 0. The chosen magnifications satisfy ai ≥1 eventually and 0 ≤log ai ≤Gate( bβi), where Gate(x)/x → 0. The calibrated coupling sequence and the band charge satisfy βi/ bβi →1 and βi/Bi →1. Therefore log(1/ri) Bi = ti −log ai bβi ! bβi βi βi Bi −→1 g. (TR40) Here ti/ bβi = (1−β∗/ bβi)/g, and the magnification term tends to zero by its bound. Also log(1/ri) → ∞, because ri ↓0 along the scaling sequence. The elementary exponential limit le−4l→0, with l= log(1/ri), gives Bir4 i = log(1/ri)r4 i log(1/ri)/Bi −→0. This proves both the logarithmic factor and the disappearing unit contribution used in the trace argument. 16.6.5 The three finite candidates and their common extraction All three finite tensor candidates, their individual stencil derivatives and their divergences are retained in one extraction. Thus a divergence is the sum of derivatives of those same finite functions. 420 ## Source page 423 A Wilson Yang–Mills construction J. Rodgers The correction does not combine unrelated limiting candidates whose coefficient coordinates happen to agree. Here are the actual finite functions behind the three columns used below. Write Cab(x) = curv(Hol□(a,b)(x)), with the curvature coordinate and normalization of (SQ1)–(SQ12), and put Pija(x) = ⟨Cia(x), Cja(x)⟩. At a fixed issued ticket, the three tensors are J∅ ij(x) = Meld(J)ij(x), Jsh ij (x) = ( 0 i = j, P a Pija(x) i ̸= j, Jsc ij (x) = δij 4 X a,b Paab(x). (TF11) The first is exactly the restored, localized principal tensor (TF0d), including its six antisymmetric differences and its scalar forward difference. The other two use elementary plaquette holonomies; there is no new stationary current in their definition. Their free coefficient vectors in the three- coordinate tensor basis are respectively the principal vector, (0, 0, 1), and (0, 1/4, 0). In the source’s three-coordinate convention the principal free vector is −weyl. Its physical sign is fixed together with the Ward charge in (TR41)–(TR44); it cannot be changed independently of that normalization. All sixteen entries of each tensor, all sixteen individual stencil derivatives, and all four diver- gences are extracted on the same filter. Explicitly these are the origin functions Jb ij(0), elmi Jb ij(0), and P i elmi Jb ij(0), for b = ∅, sh, sc. The tensor entries have degree-four coefficient records and the derivatives have degree-five records. These records include both the finite-cutofflabel and its retained limit; extraction of the joint correlation book gives the corresponding fields Li, Ωi, ∆i. Thus extraction of a divergence is extraction of the sum of its actual finite derivatives, not an independently selected vector limit. The local bounds needed here have a direct source. A plaquette has four links. If each matrix link has derivatives through order k bounded by the chart price d, its holon- omy has price travelCost(G, k, 4, d). Applying the curvature map gives price z(d) = curvatureCost(G) travelCost(G, k, 4, d). The bilinear product has price 2k∥⟨·, ·⟩∥z(d)2. Sum the finitely many axes and multiply the scalar branch by 1/4. This gives a bound independent of the torus and the ticket, supported in radius four. One stencil derivative enlarges the radius to eight. The scheduled small-field extension makes these bounds into the coefficient and correlation bounds used in extraction. The issued principal tensor has these same bounds by (TF0d) and its localization estimates. The condition 20(high(x) + rim(x) + 10) < n makes all integer offsets used in the coefficient tests nonwrapping. Taylor projection commutes with the stencil divergence on these records: expand the finite translates and use the stencil moments from (TF0b). The degree-four stress coordinate therefore maps to its degree-five divergence coordinate. For the principal branch the antisymmetric differ- ence corrections have zero relevant tensor coordinate; the scalar difference has zero trace coordinate. Consequently its free divergence scalar coordinate vanishes. This is the specific low-degree state- ment used later; it does not say that its entire finite divergence vanishes. The finite action, localized current, surgery polynomial and coefficient equations are issued by one ticket, so the six terms of (LCE6) concern this same first branch. The two elementary branches supply the correction columns in (TR41). With YiT = r4 i U−1 i T and Pi = BiYi, their normalized limits and correction are precisely (TR41)–(TR44): solve the two divergence coordinates for ci,0, ci,1, then set eTi = BiLi+ci,0Ωi+ci,1∆i. This specifies the finite origin, normalization, and limiting meaning of all three candidates and both 421 ## Source page 424 A Wilson Yang–Mills construction J. Rodgers correction objects. 16.6.6 Correcting the stress tensor and fixing its amplitude The two correction coordinates are solved by an eventually invertible finite-dimensional detector matrix. Its kernel agrees with the true divergence kernel on the stated stress space. This converts small detector errors into an exactly conserved tensor, while the weighted scalar sensor fixes the trace amplitude. The local estimates following the inverse argument supply the quantitative bounds needed for this stronger conclusion. The coefficient in (TR39) is earned by an exact finite-dimensional correction, not by inserting an assumed trace anomaly. The three lifted stress candidates are denoted Li, Ωi, ∆i. Their free limits are respectively the Maxwell stress tensor, a traceless correction, and a scalar tensor with trace 4Oscale. All lie in the centered symmetric, real, rotation-covariant stress space. Let Ki be the two-component detector map for their divergences, and define Si(c0, c1) = c0Ωi + c1∆i, Xi = BiKiLi. The maps KiSi converge to an invertible free map. In finite dimensions this gives eventual invert- ibility and a uniform inverse bound: identify matrices in fixed bases, write the nearby matrix as A(I + A−1Ei), and use the convergent geometric inverse when ∥A−1Ei∥< 1/2. Solve (KiSi)ci = −Xi. (TR41) The available detector estimate bounds ∥Xi∥≤CRi, hence ∥ci∥≤C′Ri, with Ri/Bi →0. The detector map has exactly the same kernel as the true divergence on this stress space. Consequently eTi = BiLi + Sici is exactly conserved for every sufficiently large i. The distinguished scalar detector li satisfies, for a nonnegative control ti with Ri ≤ti, li(Xi) →−g, tili(KiΩi) →0, ti li(Ki∆i) + 4  →0. (TR42) The coefficient bound therefore kills the two error products. Applying li to (TR41) gives 0 = li(Xi) + ci,0li(KiΩi) + ci,1li(Ki∆i). Equivalently, 4ci,1 = li(Xi) + ci,0li(KiΩi) + ci,1 li(Ki∆i) + 4 . Thus ci,1 →−g/4 = forceR. The factor four is the exact scalar detector value, and the sign follows from the displayed equation. The primitive comparisons supplying (TR42) are unfolded with the detector construction. Write YiT = r4 i (U−1 i T), component by component, and Pi = BiYi. The normalized lifted candidates satisfy PiLi →TMaxwell, PiΩi →TΩ, and Pi∆i →T∆. Therefore Yi eTi = PiLi + ci,0 Bi PiΩi + ci,1 Bi Pi∆i −→TMaxwell. (TR43) For the scaled trace, the stronger traceless estimates give Bi tr(PiLi) →0 and ci,0 tr(PiΩi) →0. The remaining trace tends to Bi tr(Yi eTi) −→4 forceR Oscale. (TR44) Only the first correction coefficient requires a growth bound here; the scalar coefficient has the actual finite limit just proved. 422 ## Source page 425 A Wilson Yang–Mills construction J. Rodgers The conserved centered stress space is one-dimensional. To see why this rank statement follows from the same construction, KiSi being onto makes Ki onto. Its kernel equals the divergence kernel. The real dimension of the centered stress space agrees with the corresponding free dimension by the scaling and unit identifications. The detector target has the free divergence-target dimension. Rank–nullity and the free calculation, whose divergence kernel has dimension one, therefore show that the physical divergence kernel has dimension one as well. Let R be the actual centered stress tensor with its translation Ward charges. It is nonzero. Indeed the free derivative map is nonzero: its divergence target contains a nonzero coordinate and is attained; if every observer vanished on every first derivative, the separating observers would make that target zero, a contradiction. Choose an observer with a nonzero derivative value y. Its free translation-flux Ward identity has value y. The corresponding physical shell-flux test of R, with positions, cutoffand fields scaled together and with the single factor ri prescribed by the derivative, tends to y. If R = 0, every such test would be zero, contradicting y ̸= 0. Since the kernel is one-dimensional, eTi = viR for real vi eventually. The shell-flux test of R tends to y, while its test on eTi also tends to y, by (TR43) and the tested derivative comparison. Linearity makes the latter vi times the former. Divide by the nonzero limiting value to obtain vi →1. Then vi ̸= 0 eventually, and multiplying (TR43) and (TR44) by v−1 i transfers both limits to R. Centering back to the selected-vacuum trace uses the vanishing unit term proved in (TR40). This proves the running trace-label limit required for (TR2). There is also an exact scalar-coordinate formulation. The invariant scalar degree-four space has the real basis 1, 4Oscale, with injective coordinate map (e, d) 7→e 1 + 4d Oscale. A linear left inverse exists and is continuous in this finite-dimensional space. Applying it to the running trace limit gives coefficients ei, di with U−1 i (r4 i Θ) = ei 1 + 4di Oscale, Bidi →d2, Biei →0. The exact scalar membership, followed by this left inverse, justifies these simultaneous coefficients without selecting unrelated subsequences. The two correction columns have a concrete free-coordinate description. The free divergence detector is a surjective map V : R3 →D with ker V = R(1, −1 4, 1). The two columns are Ω= (0, 0, 1) and ∆= (0, 1 4, 0). Their images are independent: if xΩ+ y∆= c(1, −1 4, 1), coordinate zero gives c = 0, and the other coordinates give x = y = 0. They span the quotient: for w ∈R3, (w2 −w0)Ω+ (4w1 + w0)∆= w −w0(1, −1 4, 1). (TR45) Thus their detector images form a basis of D. This is the free invertibility used in (TR41). Here are the precise error transfers behind (TR42). Let si be the moving map from the finite bare degree-five coordinate space to the free divergence target, with si →s∞in operator norm. For a lifted candidate Ai, its detector value pi = KiAi and its bare divergence coordinates ui satisfy HN i (pi −siui) →0 for every fixed N ≥0. (TR46) The increasing clock Hi dominates the positive control ti. For the two correction candidates, ui →u∗, with s∞u∗= V Ωor V ∆, respectively. In particular (TR46) at N = 0 gives the two column limits. 423 ## Source page 426 A Wilson Yang–Mills construction J. Rodgers The scalar functionals li are uniformly bounded and approximate a fixed bare functional L with the stronger rate sup ∥u∥≤1 |ti(li(siu) −L(u))| −→0. (TR47) For each correction candidate the bare comparison also gives ti(L(ui) −L(u∗)) →0. Decompose the full scalar error into the three terms ti(li(pi) −L(u∗)) = tili(pi −siui) + ti(li(siui) −L(ui)) + ti(L(ui) −L(u∗)). The first tends to zero by boundedness of li, ti ≤Hi, and (TR46) at N = 1. The second tends to zero by (TR47) and boundedness of ui. The third is the stated bare comparison. The exact free scalar readings are L(uΩ) = 0 and L(u∆) = −4, giving the last two limits in (TR42). For the principal candidate, a renormalized bare divergence vi satisfies Xi −sivi →0, ∥vi∥≤CRi, Ri ≤ti, L(vi) →−g. (TR48) Although vi need not stay bounded, (TR47) still pays its growth: |li(sivi) −L(vi)| ≤εi ti ∥vi∥≤Cεi →0. Uniform boundedness of li handles the first difference in (TR48), proving li(Xi) →−g. Operator- norm boundedness of si, the bound on vi, and Ri ≥1 eventually give ∥Xi∥≤C′Ri. Thus the same estimates supply both the scalar limit and the norm bound needed to solve the correction equation. The elementary absorption principle used throughout this step is: if gi ≥0, |ai| ≤Cgi, and gixi →0 in a normed space, then aixi →0. It follows immediately by taking norms. In (TR48), the first difference is obtained by applying this principle to the two preexisting comparisons: the physical detector differs from the candidate’s bare coordinate reading by an error smaller than every inverse power of Hi, and the candidate’s renormalized divergence differs from vi by an error of the same strength. Since Bi ≤2Hi, both errors remain negligible after multiplication by Bi. This explains why growing running factors can be inserted in (TR42)–(TR44). The scalar membership used after (TR44) follows from the same free degree-four classification. A real scalar u, fixed by all signed orthogonal symmetries, gives a diagonal tensor Jab = δabu. It belongs to the classified free symmetric stress space. Writing it in that space’s real coordinates w, its trace has the form tr J = 4w0 1 + 4(w1 + 4w2)Oscale = 4u. Thus u = w0 1 + (w1 + 4w2)Oscale, as required. The scalar-coordinate map is injective: expectation kills Oscale and sends the unit to one, forcing the unit coefficient to vanish; the scalar field is nonzero by its positive free two-point norm, forcing the remaining coefficient to vanish. The running inverse scaling preserves conjugation and signed symmetries, so it preserves this exact scalar membership. The physical map Ki has the exact formula (KiT)a = Bir5 i U−1 i,5 (div T)a . (TR49) The real symmetry restrictions make this a map between the stated real tensor spaces. Its prefactor is positive and Ui,5 is invertible, so KiT = 0 if and only if div T = 0. This proves the kernel identification used above. The bare-to-physical degree-five map si is likewise the componentwise 424 ## Source page 427 A Wilson Yang–Mills construction J. Rodgers renormalized lift followed by U−1 i,5 . Equivariance under conjugation and signed symmetries makes its image lie in the free divergence target. Applying the componentwise divergence comparison to these exact formulas yields (TR46). We give the finite-dimensional repair which supplies (TR47). This is needed because the limiting degree-five map has a one-dimensional kernel. Let E, H be finite-dimensional real spaces, f : E → H a surjection with ker f = Rx, x ̸= 0, and choose a right inverse U. Suppose si : E →H have exact right inverses ui : H →E eventually, with ui →U. Suppose also that wi →x and siwi →0. There is a linear functional A : E →R such that y = Ufy + A(y)x. Indeed y −Ufy lies in the one-dimensional kernel and its scalar coordinate is linear. Set Vi = uif + A ⊗wi. Then Vi →IE, hence V −1 i exists eventually and tends to IE. Also siViy = fy + A(y)siwi −→fy. Multiplication by V −1 i gives si →f in operator norm. Pointwise convergence is sufficient in this argument because all operator spaces are finite-dimensional: evaluate on a fixed finite basis and reconstruct the operator continuously. Let L : E →R be the bare scalar functional and ei the required precision scale. Assume the stronger two estimates eisiwi →0, eiL(wi) →0. (TR50) Define li = Lui. These functionals converge to LU, so they are uniformly bounded. Put Qi = ei(lisi −L). On the vector Viy = uify + A(y)wi, the right-inverse identity cancels the first part exactly and gives QiViy = A(y) [Lui(eisiwi) −eiL(wi)] −→0. Since V −1 i →IE, it follows that Qi →0 in operator norm. This proves (TR47) with ei = ti. It also proves the convergence of si needed for (TR46)–(TR48); no prior uniform convergence on its near-kernel direction was assumed. In the present application x is the nonzero free equation-of-motion vector, the kernel of the free degree-five map. The exact right inverses ui are obtained by lifting free target vectors at the same retained scale and undoing the renormalized coordinate map. The projection-lift identity makes siui = I exactly; the common scaling comparison gives ui →U. The near-kernel vectors wi are the issued equation-of-motion coordinates. Their two paid estimates are precisely (TR50). The bare scalar functional annihilates the limiting equation-of-motion vector. Thus this argument constructs the detector used in (TR42), with the precision scale ti = Age1/5 i . For the correction candidates the sharper scalar estimate in (TR47) is compatible with their qualitative coordinate limits for an additional reason. Before taking the continuum limit their issued coordinates differ from the ideal orbit coordinates by at most lossi = Age−1/4 i . The ideal scalar reading equals the target scalar reading exactly. Bounded linearity therefore bounds the scalar discrepancy by a constant times lossi. Multiplication by ti = Age1/5 i gives the decaying power Age−1/20 i . This proves the third error limit used after (TR47). The qualitative joint degree- four and degree-five coordinate limits follow by taking the same finite-level orbit comparison, whose coefficients approach the prescribed vector, and using the vanishing unweighted loss before passage along the original extraction filter. 425 ## Source page 428 A Wilson Yang–Mills construction J. Rodgers For the principal divergence coordinates, the explicit prelimit estimate which supplies the final limit in (TR48) is ∥vi∥≤2∥O∥Age1/6 i , |L(vi) + g| ≤∥M∥Age−1/4 i + 4∥M∥Age−1 i + 1 12H−30 i + 1 12H−100 i . (TR51) Here O is the fixed orbit map on degree-five coordinates and M = LO. To pass this bound to the continuum coordinates, first compare the issued representative with its projection to the retained coordinate core, with error Age−1/4 i . The retained representative has norm at most Age1/6 i . Apply the bounded orbit map for the first bound and the bounded functional for its error contribution. The finite-level scalar calculation gives the remaining three terms and the value −g. Continuous linear maps and closed norm inequalities preserve these bounds along the extraction limit. Every term on the right of the second inequality tends to zero. The scalar finite-level calculation is developed below, and (TS1)–(TS5) supply the weighted near-kernel estimates required in (TR50). The scalar functional has an explicit invariant-coordinate formula. For the bare divergence vector parametrized by w ∈R3, ρ(bareVector w) = −4(w0 + 4w1). (TR52) The coordinate parametrization is an isomorphism for simple g. The equation-of-motion vector is the image of (1, −1/4, 1), which (TR52) annihilates. More generally, if the vector belongs to the equation ideal, its coordinates satisfy 2(w0 −w2) = 0 and 2w1 + 1 2w2 = 0; these force ρ = 0. The field-space functional L is ρ after the degree-five coordinate isomorphism. This proves both its equation-of-motion annihilation and the readings 0, −4 on the two columns of (TR45). At finite level the same reading can be computed before quotienting. For a real regular local vector V = (Va), the coefficient functional M on its degree-five cards satisfies M  θ−2(altoD Va)a  = 1 12 Re Holt(V ). (TR53) Here θ is the actual cutoffthreshold. The local-card identity gives altoD V = θ2 DCard V ; the factors cancel exactly. The card reading identifies the coefficient contraction with the local polyno- mial contraction Holt. Reality and nonwrapping of the support justify that identification without periodic alias terms. Thus (TR53) is a literal coefficient identity with the stated cutoff, rather than a remainder estimate. Here are the local estimates underlying the coefficient calculation. Write x for the large coupling parameter and put Ωa(x) = exp{a(log log x)2}, θ = x−1/2Ω1(x), cx = θ2Ω−2(x), tx = θ2Ω1/8(x), σx = θ2x−1/100. (LCE1) All assertions in this paragraph concern sufficiently large x > 1. Products of the Ωa add their subscripts; for every fixed a and ε > 0, Ωa(x) ≤xε eventually. This follows by dividing a(log log x)2 by log x, whose ratio tends to zero. The regular local fields entering the exact split have the following derivative envelopes on the common healthy chart: the nonlinear principal discrepancy and the finite Taylor surgery error have envelope cx; the principal comparison field and its loop field have envelope tx; the finite-loop tail has envelope cx. These are envelopes through the prescribed chart order, on the stated local support, including measurability, gauge covariance and chart regularity. The large-x threshold is common to the stated local estimates. At each fixed admitted x, exhaust the finite loop and tail 426 ## Source page 429 A Wilson Yang–Mills construction J. Rodgers shelves after specifying the required positive error allowance. Absolute convergence makes their errors arbitrarily small. These shelves may depend on x; a single finite truncation uniform for all large couplings is not needed. For clarity, the nonlinear estimate has a quantitative Taylor proof. Let s = θ(1 + high(x))P , where P is the fixed stage power. In exponential link coordinates the discrepancy K between the nonlinear curvature expression and its linear comparison has derivatives through order four at zero bounded by x−30. Its derivatives through the required higher order on ∥y∥≤Ds are bounded by 2Ω1/200(x); D ≥1 is a fixed chart radius. The change from color coordinates to link directions has norm at most one, so these low-jet bounds persist under that change. Taylor’s formula with integral remainder, after rescaling y = sp, gives, uniformly for ∥p∥≤D, ∥minors(βK)∥Ck ≤4x 2Ω1/200(x) C5,Ds5 ≤σx, k ≥5, |β| ≤4x. (LCE2) The low-order terms fit this same estimate because x−30 ≤2Ω1/200(x)s5−j for 0 ≤j ≤4. The final inequality follows from s5 = x−5/2 times a fixed subpower factor, whereas σx = x−1−1/100Ω2(x). Thus the fifth-order Taylor remainder has a strictly decaying power to spare. The finite-difference surgery uses h = θx−1/24. Since h/s ≤x−1/24, choosing its order m ≥48 makes every fixed subpower multiple of s(h/s)m+1 at most σxh eventually. This controls the surgery error after division by its difference step. The support margins and nonwrapping conditions used here are the finite chart conditions already imposed when choosing the station; none is a new continuum hypothesis. We spell out how these local bounds survive projection and summation. Fix the degree-five frame and its local section, and denote their fixed reading prices by r ≥1 and v ≥1. If the section has growth exponent p ≥0, put K0 = (1+p)r +v. A field with derivative envelope cx has weighted coefficient size at most rcx. This is the coefficient projection estimate: each coordinate is a bounded derivative reading, its fixed reading weight is included in r, and summing the coordinate absolute values bounds the coefficient norm. For the secondary surgery terms indexed by a finite set I, let Qi be their supports and gi their derivative envelopes before multiplying by the difference coefficient. The actual summable tail bound is X i∈I es0 cover(Qi) teasi h gi ≤σx. (LCE3) Here s0 is the fixed support weight of the local budget, distinct from the chart radius s. Multiplying each summand by its projection price at most r+v and using the triangle inequality gives coefficient size at most (r + v)σx for their sum. The same summation constructs its local representative and bounds its approximation remainder; this is why a bound for each individual term is not used in place of the required bound for their sum. The principal local support has bounded weighted cover, so its support weight is at most a fixed A = exp(P0 TileNo). Its derivative envelope is 2cx + 2tx ≤4tx. Combining it with (LCE3) bounds both the total projected coefficient and its local representative error by K0(4Atx + σx) ≤(4A + 1)K0θ2Ω1/8(x) ≤θ2Ω1/6(x). (LCE4) The last inequality follows from Ω1/8/Ω1/6 →0. Similarly, K0σx ≤cx, rcx ≤θ2Ω−1(x). (LCE5) Indeed their respective ratios are K0x−1/100Ω2(x) and rΩ−1(x), both tending to zero. These in- equalities prove the total envelope and all four small coefficient envelopes used next. They also show explicitly why the local errors still vanish after division by θ2. 427 ## Source page 430 A Wilson Yang–Mills construction J. Rodgers We identify those six contributions in terms of the finite functions, to avoid ambiguity between this split and the six antisymmetric tensor differences in (TF0d). At the current scale let Yb be the localized current, eJ the restored tensor, PA the finite principal action, and LQ the finite ordinary loop sum. Write QQ for its augmented quiet sum, with the zero and first path moments removed as in (TF6a)–(TF6c). Define DY f by differentiating f along the left logarithmic link current, and write divν Y for its Haar divergence. The six objects are Wb = altoD{β(P a Da eJab −DYbPA)}, Bb = divν Yb −(divν Yb)(1), Kb = DYbQQ, Db = DYb(QQ −LQ), Eb = altoD{h−1Tm,h(Yb, −βPA −LQ)}, Fb = X γ∈F weightedPγ Hγ. (LCE6) In this display W, E, F already denote coefficient vectors; B, K, D are scalar configuration functions until altoD is applied. The sign identity is DY LQ = K −D, so the loop contribution is −K + D. The operator altoD means the degree-five Taylor reader with its actual threshold weights, on the common local support; the secondary readers retain their own supports Pγ. The current here is the localized version of (TF0a), transported through the stationary descent, not an independently chosen continuum vector. The term Tm,h is a concrete finite-difference remainder. If ti > 0 are the distinct nodes and li their Lagrange polynomials on m nodes, use wi = li(0)/ti, Dm,hf = X i 0 is the issued path width; K, P depend on G and the fixed derivative order four, and the geometric and path prices are fixed by the stage. Cofinal finite path cuts give the tail derivative bound T. The weighted product rule for contracting a current with an action derivative bounds (TF13) by V (x) = vocationG(v, p, U) + vocationG(v, q, T). (TF15) The prices in (TF15) have the following explicit meanings. Set cG(a) = # Loc(G, 1) e2a6! 24 a6 , gG(a, k) = (1 + cG(a))k, vocationG(a, C, D) = 16 cG(a/5)C gG(a/5, 3)D. (TF20) Here Loc(G, 1) is the fixed finite set of first-order link direction types (axis and color); its cardi- nality has no volume dependence. The lattice shell count bounds the sum of e−a distance over these directions by cG(a). Expanding a multilinear derivative in its coordinate directions and summing these decays gives gG(a, k). For a current-action contraction, write DY F = P lYl∂lF. The lo- calized current has its first three derivatives bounded by Ce−(a/5) distance(l,0)gG(a/5, 3); the action derivative has four derivatives bounded by D and vanishes at the identity. In its fourth product derivative the term requiring a fourth derivative of the current is multiplied by that zero. The remaining binomial coefficients sum to 24 = 16. Summing in lgives exactly (TF20). The vanishing action derivative follows because its one-color linear coefficient is invariant, and simplicity excludes an invariant color vector; it holds for both the full action and its invariant finite tail. This proves the contraction bound in (TF15) using only three current derivatives and four action derivatives. The scheduled radii and cuts give T = O(x−98), V = O(x−65), and x−100gG(σ/5, 4) + x−70gG(v/5, 4) + V (x) = O(x−64) ≤x−30 (TF16) eventually. These bounds are uniform in the admitted finite torus and hold for every sufficiently large finite cut at that fixed stage. Equations (TF12)–(TF16) supply the second, third, and fourth derivatives as well as the zeroth and first ones. They are the quantitative hypothesis used in the rescaled Taylor remainder (LCE2). In that application the higher derivative bound and x−30 ≤ 2 thinM(x)s5−j, 0 ≤j ≤4, give the order-five rescaled remainder, with s = θ(1 + high)stage power. Summing the Taylor terms gives the stated θ2x−1/100 envelope. The quiet derivative K has zero scalar contraction by the two path-moment cancellations (TF6a)–(TF6c), whose full-family justification is given immediately below. The derivative of the appended quiet correction is D. The term B keeps the Haar Jacobian, and its centered constant is explicitly removed. The last term is obtained by expanding each finite surgery exponential into support factors, with coefficient wi/h; its entire weighted sum is bounded in (LCE3). Thus the four small terms are exactly W, altoD D, E, F, and the two non-small terms have their actual scalar calculations. For completeness, the retained quiet moments are small because the full issued loop family has zero moments. For each ordered pair a of plaquette types and pair b of color indices, define ca,b γ = 1{types of γ=a} coeffγ(b), zγ = offsetγ,1 −offsetγ,0 . (TF17) The path bounds include absolute summability of these coefficients and of each ca,b γ (zγ)l, together with their normalization X γ ca,b γ = 0, X γ ca,b γ (zγ)l = 0 (l = 1, . . . , 4). (TF18) 429 ## Source page 432 A Wilson Yang–Mills construction J. Rodgers These are curvature loop value and first-jet normalizations, distinct from the potential jet normal- ization. They are carried in the issued path data as the summable loop terms and their zero sums; joining path families preserves them by absolute summation. The path-data construction retains these identities along with the quadratic metric normalization, so (TF18) applies to the family actually used in (LCE6). Let L be the retained finite set. Subtracting its sum from (TF18) gives X γ∈L ca,b γ = − X γ /∈L ca,b γ , X γ∈L ca,b γ (zγ)l = − X γ /∈L ca,b γ (zγ)l. (TF19) Since |zγ,l| ≤length γ and the coefficient is bounded by its atom budget, an omitted bound P γ /∈L budgeta(γ) (1 + length γ) ≤A bounds every retained moment in (TF19) by A. Thus the coefficients appended at displacement zero and the four unit displacements in (TF6a) have total coefficient magnitude at most 5 #{color pairs}#{type pairs}(A + 4A). Indeed there are five dis- placement choices for each type pair, and each coefficient has magnitude at most A + 4A; the unit-displacement coefficients individually have the sharper bound A. Their paths have length at most ten, so a weighted atom budget at width a costs at most an additional e10a. Fixed-order local function and derivative bounds follow by the finite plaquette product estimate used for (TF11). The exponential path estimate gives A = q + e−σr/4 momentPrice1(σ/2)B after a sufficiently large cut and radius r, for width σ > 0 and full loop price B. This proves the small appended-tail bound used for D. The augmented finite family, meanwhile, has exactly zero constant and first moments by its explicit coefficients. Its Hessian against a supported direction and a direction with affine curl therefore vanishes by (TF6b), proving the scalar cancellation for K. Smallness of the correction and exact cancellation of the augmented family are separate consequences of (TF18) and (TF6a), respectively. The issuance is constructive: for each of these regular scalar functions subtract the local section of its weighted Taylor card and put the masked difference in one regular activity slot, as in (TF5). Add the four principal formatted insertions and the finitely many secondary ones. Their coefficient is exactly the sum of their Taylor cards by linearity, and their support-weighted prices add. Equations (LCE3)–(LCE5) then prove both the bound on the issued coefficient and the paid remainder of its representative. This proves existence of the candidate with its coefficient identity and its budgets simultaneously. The principal issued divergence decomposes into six vector contributions: V = W + altoD B −altoD K + altoD D + E + F. (TR54) The decomposition is exact because the local split satisfies y + D = K, and the regular-card map is additive on this regular split. The central term K has zero Holt contraction by the finite-level cancellation identity, so its scalar reading is zero by (TR53). The term B gives Re Holt(B)/12. Its difference from the finite-level scalar payment midPay /12 is at most calorie/12. Each of the four remaining terms W, altoD D, E, F, after multiplication by θ−2, has coefficient norm at most Wave(−1). Applying the operator norm of M and summing gives M(θ−2V ) −midPay 12 ≤4∥M∥Wave(−1) + calorie 12 . (TR55) The total vector’s coefficient-size envelope is θ2Wave(1/6), so ∥θ−2V ∥≤Wave(1/6). These esti- mates use the coefficient-size bound, which dominates its norm, and the actual positive threshold to cancel θ2. The scalar payment comparison | midPay −landPay | ≤shock, with landPay /12 = −g, gives the last two terms and limiting value in (TR51). This reduces that limit to the local scalar 430 ## Source page 433 A Wilson Yang–Mills construction J. Rodgers cancellation, the four explicit local envelopes and the payment comparison; it introduces no inde- pendent continuum assumption. The finite scalar payment itself is obtained by differentiating a massive comparison, with the mass parameter kept strictly positive until after the large-volume estimates. Let Ri(μ) be its scalar resolvent primitive and Si(μ) its differentiated shell quantity. The source proves d dx Re Ri(ex) = −Re Si(ex), |Re Ri(ex) + Lx| ≤C (x ≤0), (TR56) eventually for every fixed x, where L = −12g and the same C works for every such x. The derivative is a derivative of a convergent finite-volume resolvent expression; strict positivity of its massive matrix gives its inverse throughout this calculation. For any fixed a < b ≤0, the real mean value theorem gives a point ci ∈(a, b), hence μi = eci ∈ [ea, eb], with | Re Si(μi) −L| ≤ 2C b −a. (TR57) Only the two endpoint estimates in (TR56) are used, so their separate eventual assertions can be intersected. No uniform convergence in the mass parameter is needed. The mean-value choice may depend on the finite-volume index. Let Xi be the local payment before its final second-derivative correction. The comparison supplies Xi −Si(μi) −μ2 i Ki(μi) →0, |μ2 i Ki(μi)| ≤C1μi, (TR58) for every moving mass in any fixed interval [δ, 1], δ > 0. To prove Re Xi →L, first choose b ≤0 so negative that C1eb < ε/4. Then choose b −a large enough that the right side of (TR57) is at most ε/4. The interval lower endpoint ea is now fixed and positive, so (TR58) applies to the mean-value masses. Its vanishing error and the two preceding bounds prove the desired convergence. The choice order is essential: the volume limit is taken with this fixed positive lower mass, after choosing the error tolerance. We derive the local resolvent bound in (TR58). Use a periodic lattice of side n, angular mo- mentum spacing h = 2π/n, and normalized momentum weight n−4. Write kx = h lift(x) for its centered momenta. If R ≥h, any subset of momenta with |kx| ≤R has at most (3R/h)4 (LR1) members. Indeed their centered integer coordinates have absolute value at most ⌊R/h⌋. The lift is injective, and the integer coordinate box has at most (2⌊R/h⌋+ 1)4 ≤(3R/h)4 points. This upper bound remains true when the coordinate box wraps on the torus. For h ≤μ ≤1, group momenta according to 2jμ ≤|kx| + μ ≤2j+1μ, j ≥0. Assign boundary points to one of their adjacent groups. A group has at most (6 · 2jμ/h)4 points by (LR1); its summand (|kx| + μ)−5 is at most (2jμ)−5. Therefore n−4 X x (|kx| + μ)−5 ≤ 64 (2π)4μ X j≥0 2−j ≤2 · 64 (2π)4μ. (LR2) Only finitely many groups are nonempty at a given n; extending their sum by zeros justifies the infinite geometric majorant. 431 ## Source page 434 A Wilson Yang–Mills construction J. Rodgers Here is the integrand estimate to which (LR2) applies. At z = |k| + μ > 0, scale an external momentum direction p by z. The zero-field matrix, one-field vertex and two-field vertex have sizes z2, z, 1 with bounded scaled derivatives through the finite orders in use. These follow from their vanishing origin jets and the analytic symbol bounds proved in the massive-symbol section. Ellipticity gives the inverse size z−2. Differentiating A−1A = I gives D(A−1) = −A−1(DA)A−1, D2(A−1)[u, v] = A−1(DA[v])A−1(DA[u])A−1 + A−1(DA[u])A−1(DA[v])A−1 −A−1(D2A[u, v])A−1. These formulas, and one further differentiation when needed, retain the same inverse size under scaled derivatives. Hence the contracted resolvent-insertion integrands have scaled analytic-jet bounds of size z−2, with constants depending on the fixed directions and finite matrix dimensions but not on k, μ, n. For such an analytic integrand V , let [V ]j(p) = DjV (0)[p, . . . , p]/j! be its homogeneous Taylor coefficient. The scaled bound gives z3|[V ]3(p)| ≤Cz−2/3!, hence |[V ]3(p)| ≤C′z−5. For the second coefficient it gives |[V ]2(p)| ≤C′z−4. The angular cube and μ ≤1 give a fixed upper bound z ≤R∗, so this too is bounded by C′R∗z−5. Summing the finite matrix entries and applying (LR2) bounds both contracted coefficients by a constant times μ−1. The local readings are explicit finite coefficient selectors. For a scalar function on integer arguments set Pan f = 1 2(f(1) −2f(0) + f(−1)), Flop f = 1 12[8(f(1) −f(−1)) −(f(2) −f(−2))]. On powers of degree at most two, Pan selects the quadratic coefficient; on powers of degree at most three, Flop selects the linear coefficient. The mixed selector is i Pana Flopb, evaluated on the specified two-direction momentum contour, with the corresponding powers of h−1. Its fixed finite coefficients bound it by a finite sum of the directional third coefficients just estimated. The local second selector is Pan on the opposed-momentum contour, with h−2. Analytic Taylor remainder bounds show that each selector differs from its indicated derivative coefficient by a term tending to zero as h →0, with mass restricted to a fixed [δ, 1]. For δ > 0, the massive matrices have a common positive lower bound, so their analytic inverse neighborhoods and the higher derivatives needed for these remainders are bounded throughout this interval. This also allows a mass μn depending on n. Absorb a final selector error at most 1 into μ−1 n , since μn ≤1. The physical local lens is the prescribed half-weighted physical resolvent contraction minus its gauge comparison contraction. Both have the preceding bounds; their matrix tickets identify the actual finite matrices with the Fourier inverse matrices. Linearity and the triangle inequality therefore prove, with constants fixed before the positive lower mass cutoffis requested, ∃Cd,a,b ≥0 ∀δ > 0 ∀(μn)n, δ ≤μn ≤1 ∀n  =⇒|Kd,a,b,n(μn)| ≤Cd,a,bμ−1 n eventually. (LR3) The eventual threshold may depend on δ and the moving mass family. Once δ is fixed, h ≤δ ≤μn eventually, as required in (LR2). Sum (LR3) over the finitely many tensor indices. Multiplication by μ2 n yields C1μn, the remainder bound in (TR58). For the local second-derivative correction, its exact massive comparison has error tending to zero at each fixed 0 < μ ≤1, and its massive term is at most Cμ2μ−1 = Cμ by the second-selector version of the same argument. Given ε > 0, choose 0 < μ ≤1 with Cμ < ε/2, and only then take 432 ## Source page 435 A Wilson Yang–Mills construction J. Rodgers the volume large enough that the comparison and selector errors total less than ε/2. This proves vanishing of that correction, including the actual second Taylor coefficient rather than merely its finite difference. Consequently midPayi = Re  cresti −1 2 X d,a floftd,a,i  −→−12g. (TR59) The crest comparison identifies its limit with that of Xi; each displayed correction tends to zero. This proves the scalar payment limit used to choose the finite-level accuracy in (TR55). The exact quiet-loop cancellation in (TR54) is local. For a finite roster of gauge-invariant loop functions pα, their periodized sum is P(U) = creset X α X seats s pα(s · U). (TR60) Each summand is analytic on the permitted group charts, so the finite sum is analytic and gauge invariant. If Ya is an analytic gauge-covariant vector field supported in a nonwrapping finite neigh- borhood, its directional derivative ra = YaP is again analytic, local and gauge invariant. At the identity it vanishes, since the vector field does. Gauge covariance follows by differentiating the identity for P after conjugating the infinitesimal direction; locality follows by differentiating only links in the common support. The quiet roster is chosen so that the Hessian of (TR60) pairs to zero between a direction supported in that neighborhood and the prescribed distant polynomial directions of degree at most two. The loop lengths are bounded by M; the separation condition 2(t + 1 + 1 + M) + 6 < n ensures this statement uses actual separated links rather than wrapped copies. The derivative of Ya at the identity maps every direction into the same supported neighborhood. Differentiating ra = YaP twice therefore expresses its Hessian pairing as two pairings of the Hessian of P, each with one supported direction. Both vanish. In the third local difference quotient entering Holt, the remaining terms with constant-direction legs vanish by gauge invariance and the local Ward identity; the other terms are exactly these zero Hessian pairings. The finite difference quotient is consequently zero. The local coefficient identity expressing Holt as the finite sum of these third differences proves Holt((YaP)a) = 0. To obtain the quiet roster from the original loop sum, the construction adds its finite boundary compensation roster. If S denotes the original truncated loop sum and U this compensation, the quiet sum is exactly P = S + U. Differentiation gives YaS = YaP −YaU, the split y = K −D in (TR54). The compensation has the small boundary envelope calorie; the original truncated derivative has its larger local envelope. The compensated roster retains gauge invariance and the common loop-length bound, so the preceding cancellation applies. This accounts for the exact zero term and for its separate small boundary correction. The derivative in (TR56) can also be read directly from its determinant definition. Its local primitive is a finite contraction of 1 2 log det(A(U) + μ2I) −log det(B(U) + μ2I). (TR61) The first matrix contains the physical quadratic term and gauge-fixing term; the second is its gauge comparison matrix. For a fixed positive μ, both are invertible near the identity. The real matrix derivative formula gives d dμ log det(F + μ2I) = 2μ tr(F + μ2I)−1. (TR62) 433 ## Source page 436 A Wilson Yang–Mills construction J. Rodgers Joint analyticity in the mass increment and the link chart permits the finite local coefficient deriva- tives to commute with this mass derivative. Apply (TR62) to (TR61), then the normalized local contraction yardd,a = |sites|−1 cobd,a, and sum over the ordered off-diagonal axis pairs d ̸= a. The resulting derivative is −μ times the corresponding resolvent contraction. Substitution μ = ex supplies the second factor of μ, proving the derivative identity in (TR56). For its bound, the period-limit coefficient comparison identifies the normalized local contraction with the massive momentum integral up to an eventual error at most one. The latter obeys |α(μ) + g log(1/μ)| ≤C0 (0 < μ ≤1), where C0 is independent of μ and of the allowed regulator segment. The ordered-axis contraction is twelve times this coefficient. Hence | Re Ri(μ) + 12g log(1/μ)| ≤12(C0 + 1) eventually for each fixed μ, which is (TR56). The momentum-integral coefficient and its uniform bound follow from the logarithmic momentum integral and weighted massive-symbol calculation below. The roster condition behind that Hessian zero has a simple exact form. For every pair of plaquette axes and color labels, its charges cα and relative offsets δα satisfy X α cα = 0, X α cα(δα)a = 0 for each axis a. (TR63) These are zeroth- and first-moment cancellations. For any additive real function F on the offset lattice, P α cαF(δα) = 0, by expressing F in its four coordinate values. The Hessian of each two-leg loop term is the sum of its two cross-leg curvature products. Shift the site summation so its first leg is at the same site for every α. Where the supported direction has nonzero discrete curvature, that site lies in its one-link enlargement. The other direction’s curvature, on this neighborhood enlarged by the loop length, is affine: curl T(x + δα) = C + F(x) + F(δα). Consequently summing the charge-weighted products gives a fixed first-leg factor times (C + F(x)) P α cα + P α cαF(δα) = 0. Both leg orders vanish in this way. The same argument ap- plies to each signed rotation; the supported neighborhood grows by at most one link. Summing over seats and rotations proves the Hessian cancellation used above. Thus the zero in (TR55) ulti- mately rests on the explicit finite moment identities (TR63), affine curvature of the test directions and the stated nonwrapping margins. 16.6.7 The logarithmic momentum integral The massive resolvent estimates use a positive mass lower endpoint chosen after the error tolerance and before the volume limit. Shell counting separates the near-zero lattice momenta from the remaining uniformly controlled modes. The later moving-mass statement is obtained by combining those estimates with its scalar payment and selector bounds. It does not interchange the massless and volume limits without control. We now derive the uniform bound used after (TR62). Use real angular coordinates k ∈ (−π, π]4 = B, Euclidean radius r = |k|, and normalized momentum weight c4 = (2π)−4. The 434 ## Source page 437 A Wilson Yang–Mills construction J. Rodgers actual massive two-point symbol gives a continuous scalar integrand Fμ(k), its half second deriva- tive in the chosen external momentum direction. The momentum coefficient has the exact integral formula α(μ) = c4 Z B Re Fμ(k) d4k. (TR64) To obtain this formula from the torus symbol, parametrize the unit-period torus by its half-open fundamental cube and change variables k = 2πx; the Jacobian is (2π)4. The symbol is real analytic in external momentum and internal angular coordinates for fixed μ > 0. Its first and second external derivatives are bounded on the product of a compact external interval and the closed internal cube. Dominated differentiation therefore moves both derivatives under the integral. The half in the symbol’s second coefficient and the factor two in the definition of α give exactly (TR64). The massive symbol estimates have the following two forms, with constants A, D ≥0 indepen- dent of 0 < μ ≤1: |Fμ(k)| ≤ A (r + μ)4 , |Fμ(k) −PG(k)| ≤D r3 + Dμ2 r6 (μ ≤r), (TR65) where PG(k) = 4Cadj k2 1 r6 −4k2 1k2 0 r8 −1 r4 ! . (TR66) The distinct axes 0, 1 specify the chosen external direction and polarization. The comparison construction establishes the first estimate at scale r + μ, and the second on the subfamily μ ≤r at scale r; packaging all admissible symbols and masses into these two families gives common constants. The estimates apply to the same mass μ, without shifting it to a preferred radius. Let Aμ = {μ < r ≤1}. In four dimensions the unit ball has volume π2/2, so polar integration gives Z Aμ r−4 d4k = 2π2 log(1/μ) =: Jμ, Z Aμ r−3 d4k = 2π2(1 −μ), Z Aμ r−6 d4k = π2(μ−2 −1). (TR67) Rotational invariance determines the required angular moments without a sphere normalization convention. Equal diagonal moments and P a k2 a = r2 give R Aμ k2 a/r6 = Jμ/4. For a ̸= b, rotate a unit vector to (ea + eb)/ √ 2 and to (ea −eb)/ √ 2. Add their fourth-power identities; the odd terms cancel and give R k4 a/r8 = 3 R k2 ak2 b/r8. Summing the mixed moments over b then gives Z Aμ k2 ak2 b r8 d4k = Jμ/24 (a ̸= b). Substitution into (TR66) yields c4 Z Aμ PG(k) d4k = c4 4Cadj  1 4 −4 24 −1  2π2 log(1/μ) = −11Cadj 24π2 log(1/μ) = −g log(1/μ). (TR68) Split the entire cube into r ≤μ, Aμ, and B\{r ≤1}. The unit ball is contained in the cube. On the first region (TR65) bounds the integrand by Aμ−4, whose integral is at most Aπ2/2. On the 435 ## Source page 438 A Wilson Yang–Mills construction J. Rodgers last region it bounds the integrand by A, giving at most A vol(B). On the annulus the difference from the pole integrates, by (TR67), to at most 2Dπ2(1 −μ) + Dπ2(1 −μ2) ≤3Dπ2. All inequalities are absolute before summing the regions. Equations (TR64) and (TR68) prove |α(μ) + g log(1/μ)| ≤c4 h Aπ2/2 + A vol(B) + 3Dπ2i . (TR69) This supplies the mass-independent constant in (TR56). It also displays the origin of the factor eleven and the normalization g = 2b0. The pole (TR66) comes from explicit vector and ghost traces. For a normalized adjoint-color generator Tb, tr(T 2 b ) = −Cadj. This is the adjoint Casimir identity in an orthonormal color basis. For the external polarization in axis 1 and momentum te0, the continuum vector vertex is p(k)I4 + Q(t)  ⊗Tb, p(k) = −2ik1, Q(t)ab = 2it(δb0δa1 −δa0δb1). The ghost vertex is p(k)Tb. The free propagators are the scalar inverses r−2I, and the shifted denominator is D(t) = r2 + 2tk0 + t2. Spatial and color traces factor. Direct multiplication gives tr[(pI + Q(−t))(pI + Q(t))] = 4p2 −8t2. Thus the two bubble traces, near t = 0, are Hgh(t) = p(k)2(−Cadj) r2D(t) , Hvec(t) = (4p(k)2 −8t2)(−Cadj) r2D(t) . Their denominators are nonzero for sufficiently small t when r > 0. Differentiating twice, using D′(0) = 2k0, D′′(0) = 2, and p(k)2 = −4k2 1, gives H′′ gh(0) −1 2H′′ vec(0) = 2PG(k). (TR70) This proves the pole and its sign, including the half weight of the vector determinant. For the actual massive symbol the same logarithmic determinant differentiation separates each second coefficient into an upper term and a lower bubble term. Denote their second derivatives by Uv, Lv for the vector matrix and Ug, Lg for the ghost matrix. The exact scalar combination is Fμ(k) = 1 2 h 1 2(Uv −Lv) −(Ug −Lg) i . (TR71) This equality is before any bound. The upper terms and bubble terms each have common bounds of order T −4 in the admissible scaled family. Since multiplication by 1/2 decreases the absolute value, (TR71) yields the first bound in (TR65) upon taking T = r + μ. On the family μ ≤r, take T = r. The more precise estimates are |Uv| ≤u/T 2 + vμ2/T 6, |Ug| ≤x/T 2 + yμ2/T 6, |Lv −H′′ vec(0)| ≤A(T + μ2/T 2)/T 4, |Lg −H′′ gh(0)| ≤B(T + μ2/T 2)/T 4. (TR72) Combining (TR70)–(TR72) bounds |Fμ −PG| by uT + xT + A + B T 3 + (A + B + v + y)μ2 T 6 . The angular cube has T ≤R∗= 1 + 2π, so D = max(uR∗+ xR∗+ A + B, A + B + v + y) is a valid common constant in the second bound of (TR65). All constants are chosen on the family of all admissible regulator symbols, masses and internal momenta; restricting afterward to μ ≤r preserves those choices. This proves the scalar assembly of the two bounds. Their upper-term and bubble estimates are the matrix estimates (TR72), whose propagator and derivative bounds are developed next. 436 ## Source page 439 A Wilson Yang–Mills construction J. Rodgers 16.6.8 Weighted matrix derivatives for the massive symbols Weighted matrix differentiation retains the order of factors in the finite determinant and symbol calculations. The vector and ghost contributions are combined only with their prescribed multiplic- ities and Fourier measures; the angular factor eleven belongs to this normalization. The subsequent position Ward tests determine the measured coupling rather than assuming that a bare coupling has already been measured. Uniformity for moving positive masses The regulator mass can vary along the retained sequence, but it remains strictly positive when the finite inverse and determinant are differentiated. The weighted symbol estimates state how their constants depend on that mass and the annular momentum scale. Only after the corresponding uniform bounds and volume errors have been proved is the mass sent toward zero. This order is what permits assembling the vector and ghost contributions without exchanging a singular limit with an uncontrolled inverse. We describe the matrix calculus behind (TR72). All matrices here have fixed finite index sets. Use a submultiplicative matrix norm ∥· ∥∗for which | tr M| ≤n∥M∥∗. An analytic one-variable matrix family A(t), at scale T > 0 and size L ≥0, has the required weighted bound if T j∥A(j)(0)∥∗≤CL (j = 0, 1, 2), (TR73) with one common C for the whole indexed family. Product differentiation gives sizes L1L2, with constant at most 4C1C2: the second derivative has the four terms A′′B, A′B′, A′B′, AB′′. Sums add constants, and scalar multiplication multiplies the size by a bound on the scalar. If A(0) is invertible, A has size L > 0, and ∥A(0)−1∥∗≤B/L, then the inverse has size L−1 at the same scale. Indeed, (A−1)′ = −A−1A′A−1, (A−1)′′ = 2A−1A′A−1A′A−1 −A−1A′′A−1. The three weighted bounds are at most B/L, CB2/L, and (2C2B3 +CB2)/L. Their sum is a valid common constant. Analyticity of the inverse near zero follows either from its determinant formula or from a local geometric inverse. For two invertible families the exact identity A−1 −B−1 = −A−1(A −B)B−1 (TR74) transfers a relative error size EL into inverse error size E/L. Products of several differing factors are compared by replacing one factor at a time; (TR73) controls every term. These statements retain the raw derivatives, including their binomial coefficients. The underlying symbol has a zero-field matrix A(k), one-field vertices V (q, k) and their incoming counterparts V (−q, k + q), and the two-field vertex W(−q, q, k). Along q = te, ∥e∥≤1, the symbol estimates give sizes T 2, T, 1, respectively, in (TR73), for both shifts k ± te. The shift-independent zero-field matrix has the same T 2 bound. Products of symbol families preserve this collection of bounds: a one-field vertex of a product has its two Leibniz terms, and its two-field vertex has the four terms corresponding to both placements and both cross placements. Inversion preserves it with overall size inverted, by (TR74) and the differentiated inverse formulas. This accounts for all placements of the external legs. These sizes follow from precise Taylor cancellations at the origin. The zero-field matrix differs from |k|2I by a function whose jets of orders below three vanish. The one-field vertex differs from its continuum linear vertex by a function whose jets of orders below two vanish. Their derivatives through orders three and two, respectively, are uniformly bounded on the fixed angular ball. Taylor’s estimate therefore makes their remainders of orders T 3 and T 2, including their 437 ## Source page 440 A Wilson Yang–Mills construction J. Rodgers weighted derivatives through order two along the admissible rays. The continuum one-field vertex itself is linear, so has size T. The two-field vertex has uniform derivatives through order two at the unscaled size one. The symbol’s fixed analytic-strip bounds give all these derivative constants by Cauchy estimates, with chain-map norms included. For instance, bound the highest required derivative on the ball, use the vanishing lower jets, and integrate successively along the segment from zero to k to obtain ∥Djf(k)∥≤C|k|u−j for j ≤u. This proves the stated scaled Taylor bounds. The admissible scale obeys |k| ≤T ≤|k| + μ, μ ≤T, 0 < μ ≤1. On the angular cube the lattice momentum magnitude s(k) satisfies |k|/4 ≤s(k). Hence s(k)2 + μ2 ≥T 2/64. (TR75) The ghost massive matrix is this scalar times the identity. The physical vector quadratic form, including its gauge-fixing term, is bounded below by min(ε, 1)(s(k)2 + μ2)I, where ε > 0 is the common physical ellipticity constant. The elementary coercive inverse estimate gives both inverses norm at most a fixed constant times T −2. Thus the inverse calculus above applies uniformly, even as both momentum and mass tend to zero. Let G(k) denote the massive inverse and V−, V+ the incoming and outgoing one-field vertices. The lower bubble matrix is G(k)V−(−te, k + te)G(k + te)V+(te, k). Its size is T −2T T −2T = T −2. Taking its second derivative and trace by (TR73) gives CT −4, as used in (TR71). The upper matrix is G(k)W(−te, te, k). Split W into its massless vertex and mass-regulator vertex. In the massless part the first inverse is independent of t; its norm is O(T −2), while the raw second derivative of the massless two-field vertex is bounded without a scale loss. This part therefore contributes CT −2. The regulator’s two-field vertex has size μ2T −2 at derivative scale T, and multiplication by G(k) gives size μ2T −4. Its second derivative costs another T −2, yielding Cμ2T −6. This proves the upper estimates in (TR72), and the rough T −4 estimate follows from the full sizes and the bounded range of T. For the bubble comparison take T = |k|, μ ≤T, and compare with the continuum matrices. The zero-field matrix discrepancy has size T 3 + μ2. Write E = T + μ2/T 2, T 3 + μ2 = ET 2. Equation (TR74) gives inverse discrepancy size ET −2. The vertex discrepancy has size T 2+μ2/T ≤ ET; the continuum vertex has size T. Telescoping the four bubble factors therefore bounds their matrix difference at size ET −2. Its second derivative and trace are at most CET −4, exactly the last two estimates in (TR72). This completes the matrix power counting underlying (TR65); the origin-jet identities and common strip bounds invoked here are the corresponding symbol inputs. The origin cancellations used in this calculus can be stated independently of the family bounds. For a real analytic function, the statement f(tx) = O(tu) on every real ray implies that all its jets of degrees below u vanish. Indeed its first nonzero homogeneous Taylor polynomial would be nonzero on some real diagonal, by polarization, contradicting the ray estimate. Conversely, vanishing of these jets and a bounded order-u derivative give the uniform Taylor estimate used in (TR73). Thus ray constants may depend on the ray; only the final derivative bounds must be common. 438 ## Source page 441 A Wilson Yang–Mills construction J. Rodgers For the lattice difference symbol qa(k) = eika −1, qa(k) −ika vanishes to order two and qa(k) vanishes to order one. Therefore X a qa(−k)qa(k) −|k|2 vanishes to order three: write each product discrepancy as the sum of one second-order remainder times a first-order factor and one first-order linear factor times a second-order remainder. The ghost zero-field matrix is this lattice scalar times the identity, proving its required third-order cancellation. For the vector zero-field matrix, its exact gauge-fixed form is Avec(k) = d1(k)∗M(k)d1(k) + d0(k)d0(k)∗. The metric coefficient satisfies M(0) = I. Subtract the same expression with M = I. Its discrepancy is the product of two first-order exterior differences and the first-order metric discrepancy, hence vanishes to order three. The free exterior identity says that the expression with M = I is the lattice scalar times the identity. Combining with the preceding ghost calculation proves the vector third- order cancellation. This uses the actual quadratic symbol normalization, including its longitudinal gauge-fixing term. The one-field ghost vertex has the exact component expression h −e−i(ka+pa)qa(k) + qa(−k −p)eikai (Tb)xy. Rescale p, k by a real t, divide by t, and let t →0. The exponentials tend to one and qa(tk)/t →ika, giving −i(2ka+pa)(Tb)xy. Analyticity then makes the discrepancy from this linear expression vanish to order two. The gauge-fixing vector contribution has corresponding linear expression [−ikjδia −i(ki + pi)δja](Tb)xy. Adding the Wilson curvature vertex gives δij[−i(2ka + pa)] + 2i(pjδia −piδja), times Tb, exactly the continuum vertex used in (TR70). The Wilson curvature contribution is obtained by differentiating its group commutator to second order and then applying the third- order chain rule; the Hessian of a commutator at identity is the sum of the two products of first derivatives with their reversed products subtracted. Terms from a group factor with zero first jet vanish. This fixes the curvature-vertex sign. The permitted perturbing symbol has its one-field part vanishing to order two by its prescribed soft order three. Adding it therefore preserves the same continuum linear vertex. The common strip estimates on the base symbol and the perturbation give the derivative bounds required after these jet identities. This explains how the actual, possibly perturbed, symbol supplies the cancellations in (TR73), while retaining one continuum pole for the entire permitted family. 16.6.9 The determining position-space Ward tests The coefficient can be read in the position-space frame by a finite determining system. This step distinguishes the formal first even coefficient from the later real remainder estimate. Let O = Oscale, let T 0 be the normalized free Maxwell stress tensor, and write δv for the tensor δabv. For a real invertible degree-four frame Ps preserving the stated conjugation and symmetries, the physical tensor decomposition is P −1 s (s4Tab) = κsT 0 ab + dsδabO + esδab1. (TR76) 439 ## Source page 442 A Wilson Yang–Mills construction J. Rodgers Here all three coefficients are real and chosen simultaneously from the exact tensor identity. The physical coefficient result above gives ds/u2 s →d2 for any coupling us with the same leading normalization, in particular for the Wilson charge. Since us →0, it also gives ds →0. Use the separated two-point configuration x = (e0, 0), with one exterior field O. At the scaled configuration sx, pull both labels through Ps. Define Es as the first-point divergence of the moment with the stress label PsT 0; define Bs as the same divergence with Ps(δO). Thus Bs is just the time derivative in the first position of the two-scalar moment. These derivatives are with respect to the unscaled configuration x, so the chain rule supplies a factor s multiplying the physical differentiated field. For a smooth shell cutoffχ around the first point, define Rs as the corresponding translation shell-flux test with stress PsT 0, inserted field PsO and exterior field PsO; define Qs with stress Ps(δO) in its place. Conservation and the translation Ward identity give exactly κsEs + dsBs = 0, κsRs + dsQs = Bs. (TR77) To check the scale factors, the shell-flux integral changes by s3: its volume contributes s4 and its cutoffderivative contributes s−1. Multiplying (TR76) by this flux functional and using the chain- rule factor s on the translated field yields the same factor s4 on both sides, which cancels. The unit term has zero divergence and zero translation shell flux; these are exact unit identities. No coincident insertion or contact identity is used in (TR77). The position-frame jet comparison on compact separated configurations gives Bs →B0, Rs →B0, Qs →Q0(χ). (TR78) The second limit uses the free translation Ward identity. The first is explicit. Put cG = 3 dim g/π4. The free scalar moment on tx is cGt−8, so B0 = −8cG ̸= 0. (TR79) Consequently the second equation in (TR77), together with ds →0, implies κsRs →B0. Dividing by Rs →B0 ̸= 0 gives κs →1. Divide the first equation by u2 s and by κs; then Es u2s = −ds/u2 s κs Bs −→−d2B0. (TR80) Thus the tested conservation column is known through its first possible even coefficient; the other three columns are needed only at order zero. For clarity, the determining formal system uses an unknown order-zero stress amplitude κ and a two-entry invariant jet (d0, d1), the latter corresponding to powers 1, u2. Its verified columns are B = B0, E0 = 0, E1 = −d2B0, R = B0, Q = Q0(χ). The equations are κE0 + d0B = 0, κR + d0Q = B, κE1 + κ′E0 + d1B + d0B′ = 0 for every κ′, B′. (TR81) Since B0 ̸= 0, the first forces d0 = 0, the second forces κ = 1, and the third forces d1 = d2. Conversely those values satisfy every equation. In the signed parity convention this gives the trace jet (0, 0, −2b0, 0), because the trace multiplies the scalar coefficient by four and 4d2 = −2b0. This formal parity statement alone asserts no real order-four remainder; that stronger estimate requires the normalized frame construction below. 440 ## Source page 443 A Wilson Yang–Mills construction J. Rodgers 16.6.10 A normalized position coupling and its real remainder A finite formal parity jet is distinguished from a real remainder estimate. Here the exact normalized coefficient identities give a bounded quotient at order four, and hence an actual O(g4 X) error. The weaker frame class retains only the stated leading coefficient after its positive scalar branch and nonzero two-point normalization are imposed; the stronger remainder is not transferred to arbitrary frames. The degree-four frame can be adjusted by two independent positive real factors while fixing the unit. Let C be the centering projection and Π the rotation-invariant projection onto the scalar direction in the centered field space. They satisfy C2 = C, CΠC = ΠC, and ΠCΠC = ΠC. The free scalar obeys ΠCO = O, while the free Maxwell stress components obey CT 0 ab = T 0 ab, ΠCT 0 ab = 0. Define Ax,y = xΠC + y(C −ΠC) + (I −C), x, y > 0. (TR82) The three summands are complementary projections. Consequently A−1 x,y = A1/x,1/y. It fixes the unit, preserves expectation, commutes with rotations and conjugation, multiplies O by x and multiplies T 0 by y. These facts follow by applying the three projections successively; in particular none of the centered components feeds the unit component. Put P ′ s = PsAxs,ys. Multilinearity of the moments and their derivatives gives the exact trans- formations E′ s = ysxsEs, B′ s = x2 sBs, R′ s = ysx2 sRs, Q′ s = x3 sQs. (TR83) For example the shell test R contains one stress label and two scalar labels, accounting for its three factors. Let b = B0, which is real and nonzero by (TR79). Define, for sufficiently small positive s, xs = s b Re Bs , ys = b x2s Re Rs . (TR84) The quotients tend to one by (TR78), so xs, ys > 0 eventually and both tend to one. Outside that final interval take any total choice of frame; only the eventual identities are used. Equations (TR83)–(TR84) impose Re B′ s = b, Re R′ s = b (TR85) exactly. This establishes existence of the normalized frame. Conversely any positive adjustment of the form (TR82) satisfying (TR85) tends to the identity. Indeed x2 s Re Bs = b implies x2 s →1; positivity chooses xs →1. Then ysx2 s Re Rs = b gives ys →1. Thus all such choices preserve the limiting frame and its reality, symmetry, unit and centering properties. They also preserve every compact separated jet comparison. For any finite tuple of labels, expand P −1 s P ′ svj = Axs,ysvj in a fixed finite basis. Its coefficients converge to those of vj. Expand the moment multilinearly in that basis, commute its finite sum with the fixed position derivatives, and bound each difference by the sum of its coefficient error times the bounded free derivative and its bounded coefficient times the original comparison error. Compactness bounds the free derivatives uniformly on the separated compact set. Finitely many terms then give any prescribed error. This proves all the comparisons for P ′, without a further extraction. Set c = (−d2)b ̸= 0, λX(s) = Re E′ s c , gX(s) = q λX(s). (TR86) This is a measured position-space charge, defined from a separated derivative test rather than from the physical stress coefficient. First apply (TR80) with the independently calibrated Wilson charge 441 ## Source page 444 A Wilson Yang–Mills construction J. Rodgers gW . Equations (TR83) and xs, ys →1 show Re E′ s/gW (s)2 →c. Hence λX/g2 W →1, so λX > 0 eventually, gX/gW →1, log(1/s) gX(s)2 →(2b0)−1. (TR87) The positive square root fixes the branch. In particular this coupling is calibrated to the Wilson charge before it is used in the Ward equations. Transfer (TR76) to P ′, denoting its coefficients again by κ, d, e. The coefficient comparison survives this asymptotically identity change, so d/g2 X →d2. The exact Ward equations (TR77) still hold. Take their real parts, use (TR85), and write qs = Re Q′ s →q0 = Re Q0. They give κs −1 = −dsqs b , ds = d2gX(s)2κs. (TR88) The second equality uses the definition (TR86), not a truncated expansion. Therefore κs −1 gX(s)2 →−d2q0 b , ds −d2gX(s)2 gX(s)4 →−d2 2q0 b . (TR89) For the first limit use d/g2 X →d2 in the first identity; it also implies κs →1. The second follows by substituting the second identity into its numerator. Since a convergent real quotient is bounded, (TR89) proves ds = d2gX(s)2 + O(gX(s)4), 4ds = −2b0gX(s)2 + O(gX(s)4). (TR90) It also implies the signed cubic remainder is little o, by dividing the order-four bound by g3 X and using gX →0. No full perturbative series is required for these assertions. Finally take the trace and expectation of (TR76). Since the aligned scalar is centered, T 0 is traceless and the unit has expectation one, one obtains es = s4⟨tr T⟩/4. Subtracting the selected- vacuum expectation instead gives the exact centered-field comparison s4Θ = 4ds centerχ(P ′ sO). This identifies the scalar in the normalized frame with the actual physical trace used in the spectral and exclusion arguments. The projection in (TR82) is constructed by compact-group averaging. Average the real orthog- onal action on degree-four fields against normalized Haar measure, on the left and on the right as in the aligned frame construction, and compose the two averages. Haar invariance shows that its image is fixed by every orthogonal transformation and that it is unchanged when its argument is rotated. It fixes every invariant vector; hence it is idempotent. Since the action commutes with conjugation, the average does also. Invariance of expectation shows that averaging preserves ex- pectation, so it commutes with centering in the required identities. The scalar O is fixed. Each off-diagonal component of T 0 changes sign under reflection of just one of its indices; its invariant projection is therefore its own negative and is zero. The projected diagonal components are equal by rotation covariance. Their sum is zero because T 0 is traceless; all diagonal projections are consequently zero. This proves every projection property used in (TR82). The leading scalar coefficient also transfers to the weaker position normalizations asserted by the theorem. Such a frame need only be real, rotation-aligned, unit-preserving, have its centered scalar in the prescribed positive branch, and have the centered scalar two-point norm at the fixed separated test tend to the free nonzero value. The branch classification places its centered scalar on the same positive ray as that of a reference plotted frame: center(LsO) = asPsO, as > 0. 442 ## Source page 445 A Wilson Yang–Mills construction J. Rodgers The ratio of the two centered two-point norms equals a2 s. Both norms tend to the same nonzero free value, so a2 s →1, and positivity gives as →1. Taking the trace and centering in the tensor decomposition yields s4 center(tr T) = 4d(L) s center(LsO), and the analogous identity for P. The centered scalar is nonzero eventually by its nonzero two-point norm. Cancelling it gives d(L) s as = d(P) s , hence d(L) s /g2 s →d2 for every equally normalized coupling. Unit mixing in LsO disappears under centering. This proves precisely the leading transfer, without transferring the stronger order- four bound to arbitrary frames. All spectral pairings used in this discussion are ordinary integrals at each fixed positive scale. A dilated compact continuous test remains compact continuous and is integrable against a cone- supported measure with finite future Laplace damping, by the compact-test argument following (TR12). A dilated Schwartz test remains Schwartz and is integrable by (TR10)–(TR11). On the limiting cone measure the same bounds apply to its polynomial density (p2 0 −|p|2)2. Thus the compact and Schwartz versions of the spectral limit, and their relative and joint position versions, carry their integrability assertions as well as their limiting identities. 16.6.11 The same scalar diagnostic in the selected vacuum The scalar diagnostic is first normalized in the unselected hierarchy and then interpreted in the selected vacuum through an exact centered label identity and positivity. Convexity and secants control the selected scalar function. This argument keeps the phase fixed; it does not obtain a desired result by replacing the selected vacuum as the radius changes. The frame was positioned using the unselected hierarchy. Its interpretation in the selected vacuum follows from the exact centered label identity and positivity. Let vs = PsO, let ms = ⟨vs⟩χ, and write ws = vs −ms1. The coefficient comparison gives s4Θ = 4dsws, log(1/s) 4ds −→−1. (TR91) For any fixed separated two-point shape x, rotation and translation reduce the trace moment to E(s|x1 −x2|). The short-distance limit (TR2), with the fixed dilation of the logarithm accounted for, gives s8 log2(1/s)⟨Θ(sx1)Θ(sx2)⟩χ −→cG|x1 −x2|−8. Substitute (TR91) and divide by the square of its nonzero limiting coefficient. It follows that ⟨ws(sx1)ws(sx2)⟩χ tends to the free scalar two-point function. The exact unit remainder in the uncentered label comparison gives ms = O(s4/|ds|), hence ms →0, since s4 log(1/s) →0 and the second limit in (TR91) is nonzero. The unit insertion rule gives ⟨vs(sx1)vs(sx2)⟩χ = ⟨ws(sx1)ws(sx2)⟩χ + m2 s. (TR92) Thus the uncentered selected scalar moment has the same pointwise free limit at every separated shape. The derivative diagnostic requires a further argument; differentiating this pointwise limit would be unjustified. For fixed s > 0, the real function Us(t) = Re⟨vs(ste0)vs(0)⟩χ, t > 0, is convex. In fact (TR91) and (TR92) express it as Us(t) = s8 (4ds)2 E(st) + Re(m2 s), (TR93) 443 ## Source page 446 A Wilson Yang–Mills construction J. Rodgers for all sufficiently small s. The first prefactor is nonnegative, and E(st) = R e−stp0 dμ(p) is convex in t, since each real exponential is convex and its integrals are finite. The additive constant does not affect convexity. We use the following elementary consequence. If differentiable convex functions Us on (0, ∞) converge pointwise to V , and V is differentiable at 1, then U′ s(1) →V ′(1). For fixed a < 1 < b, convexity bounds U′ s(1) between the secant slopes from a to 1 and from 1 to b. Pass to the pointwise limit in these two fixed slopes, then let a ↑1, b ↓1. This proves the assertion with the correct order of limits. Here (TR92) gives V (t) = cGt−8, so U′ s(1) −→−8cG = b. (TR94) Smoothness of the selected separated moments identifies this derivative with the real first-position scalar derivative Bsel s . Apply physical conservation to (TR76), now pairing the resulting derivative identity with the selected scalar field. Derivatives of the unit vanish, exactly as in (TR77). Therefore κsEsel s + dsBsel s = 0. Using κs →1, ds/g2 W →d2, and (TR94) yields Re Esel s gW (s)2 −→(−d2)b = c. (TR95) This proves the selected-vacuum diagnostic claimed for the same measured fields. The argument needs only pointwise scalar asymptotics, convexity and the exact conservation identity; it claims no higher-rate comparison between selected and unselected diagnostics. 16.7 The quantitative near-kernel vector for the trace sensor The near-equation vector is an explicit linear combination of transported degree-five insertions. Its weighted sensor estimate is stronger than bare convergence to the equation subspace. The proof therefore constructs a funded local marker with a genuine finite density variation and replicated root bound. A formal integration by parts symbol, without its local representative and expense estimates, would not supply this weighted conclusion. The near-kernel vector required in (TR50) is a specific combination of two transported degree- five insertions. Write bβk for the prescribed clock and βk for its extracted coupling; the common calibration gives bβk/2 ≤βk. Let Ak(c) = AgeK(c, k). For every fixed c, Ak(c)/βk −→0. (TS1) Indeed the clock estimate gives Ak(c) ≤bβ1/2 k eventually, while β−1 k ≤2 bβ−1 k . Their product is at most 2 bβ−1/2 k →0. Let Pk be the real orbit projection of the principal degree-five logarithmic insertion, and Ek the corresponding orbit projection of the degree-five divergence insertion associated with the un- corrected stress. They are expressed in the same transported coordinate chart and use the same extracted coupling. Define vk = β−1 k Pk −Ek. (TS2) The principal bounds proved in the finite scalar calculation give ∥Pk∥≤CAk(1/6) and Lease(Pk) → −g. The divergence insertion satisfies Ek −→−Stem5, Ak(1/5) Lease(Ek) −→0. 444 ## Source page 447 A Wilson Yang–Mills construction J. Rodgers Here Lease(Stem5) = 0, because its bare vector is an equation-of-motion vector and the scalar sensor annihilates that equation subspace. The limit of Ek is obtained by projecting the same finite three-coordinate stress card; its coordinates tend to the negative equation vector. The scalar equality on these cards is exact, so its transported error has the displayed weighted bound. Equations (TS1)–(TS2) consequently give vk −→Stem5, Ak(1/5) Lease(vk) −→0. (TS3) For the second assertion expand the linear functional in (TS2). The principal contribution is (Ak(1/5)/βk) Lease(Pk) →0 by (TS1) and its finite limit; the divergence contribution is the pre- ceding weighted error. The other required estimate concerns the measured vector si(vki), where ki is the common retained scaling sequence. Before transport, (TS2) is β−1 k times the orbit of the coefficient difference Xk −βkZk. The exact finite local divergence relation makes this difference a served equation insertion. Its coefficient charts and actual lattice functions have common limits by the already established loga- rithmic insertion comparison. The degree-five core projection commutes with the orbit operation, and the coordinate change is linear, so (TS2) is exactly its transformed, β−1 k -normalized limit; this is an identity, not an asymptotic replacement. The retained equation estimate on these coefficients gives, for every integer p ≥0, bβ p ki si(vki) −→0. (TS4) Only the first component needs to be estimated before the finite real orbit projection; the remaining components are its prescribed symmetry transforms. In particular take p = 1. The clock estimate Aki(1/5) ≤bβki eventually and positivity give Aki(1/5)si(vki) −→0 by comparison of norms. Together with (TS3), this supplies both weighted leaks required by the finite-dimensional repair argument (TR50). That argument produces uniformly bounded function- als μi, convergence si →Vbase, and Aki(1/5)∥μi ◦si −Lease ∥op −→0. Thus the quantitative scalar sensor is obtained from an actual near-equation insertion, with its growing weight paid before the stress correction is solved. The passage from the retained equation insertion to (TS4) is detected by degree-four spectators. At each retained horizon choose the finite local coefficient family for Xk −βkZk on the same actual region. Its locality window contains both the equation surgery and the divergence film: the first is contained in the inspected radius-canvasR tower, enlarged by eight; the second in the radius-townD tower. Use Sliver = max(canvasR +8, townD +1). This window is subpower in the clock. Its site count and the total uniform norm allowance are eventually at most the fourth power of the clock. Thus both local representations fit in the common chamber before applying the retained insertion comparison. On the healthy region, the exact finite equation identity expresses the difference of the two films as the surgery marker plus its centered normalization. The surgery approximation error is 445 ## Source page 448 A Wilson Yang–Mills construction J. Rodgers bounded by its small normalization times e−prism; after division by that normalization it is at most e−prism. Outside the healthy region use the sum of the uniform film bounds, covered by the same fourth-power allowance. These are the local and global bounds used to establish the equation probe estimate; no continuity of a hard healthy-set indicator is asserted. For each fixed degree-four free spectator w, the resulting two-point estimate after taking the coefficient limit is S  xi; a5 i project(Xki −βkiZki), U(4) i w  ≤Tau(ji, −100). (TS5) The left-hand side is an actual fixed-placement hierarchy limit of its lattice two-point moment. The two lattice positions are the prescribed near point and the origin. They are distinct and have a common nonwrapping placement on the eventual volume tail. The spectator is its retained corrected lift, so the coefficient and correlation limits use the same refinement. The coordinate identity in (TS2) converts the first insertion of bβ p kisi(vki) under the scaling frame into bβ p kib5 i a5 i project(Xki −βkiZki), where the physical scale is aibi. It follows from linearity of the transported chart, the exact cancel- lation of β−1 ki βki, and the definition of the degree-five scaling factor. Consequently (TS5) bounds each tested response by bβ p ki tiltCap(ji)5 Tau(ji, −100) −→0. The clock and tilt factors have rate zero in the horizon variable, whereas the last factor has strictly negative rate. This proves the limit for every fixed p. Finally the real invariant-vector detector, obtained from these degree-four tests and the same frame comparison, converts their vanishing responses into convergence of the measured invariant vector to zero. This is (TS4); the estimate is proved on the exact degree-five near-kernel combination rather than inferred from its unweighted limit. The marker estimate in this comparison is a finite local sum. At the summit scale k = k(i), put x = bβk and let ρ(x) > 0 be the surgery parameter. For fixed surgery order m, the marker is Mm(U) = X α∈Am,x wm,α ρ(x) Lα(U). (TS6) Here Am,x consists of the surgery addresses in the prescribed finite box and their finite orbit indices; Lα is the local lens obtained from that surgery. The orbit census gives X α∈Am,x |wm,α| ≤cardB(x) wheat(m). The box census has subpower growth in x, and the surgery parameter bound consequently gives cardB(x) wheat(m)/ρ(x) ≤x2 eventually. On the chosen scale calendar this is at most T100000 i . Each lens has support in the same window of radius 2 coast(x), that window fits the comparison chamber, and its pullback to the finest lattice has rooted test norm at most T−1900000 i . The local lens estimate applies at every address in the sum because its beacon is in the common enlarged box and within the allowed distance from the origin. The rooted test norm is a seminorm on these chamber-supported functions. Pullback commutes with this finite sum and its scalar coefficients. Thus (TS6) has rooted test norm at most T100000 i T−1900000 i = T−1800000 i . 446 ## Source page 449 A Wilson Yang–Mills construction J. Rodgers The surgery’s pointwise bound also gives |Mm(U)| ≤x2 for every configuration. All these assertions hold eventually in i, uniformly in the finer cutoffand volume; m is fixed before that eventual threshold. We spell out the label transfer as well. Let xe, ye be the two local films used in the surgery, and Xe, Ye their billed degree-five labels. With fine comparison spacing ai, summit spacing ak, and actual coupling β, set q = (ai/ak)5, Ve = Xe −βYe, Je = q(xe −βye). The billing approximation, physical normalization, and transport give ∥back(Je) −I(a5 i Ve)∥root ≤T−200 i . (TS7) Indeed the two individual errors are at most T−430 i ; the triangle inequality multiplies their sum by at most 1 + β. Since 0 < β ≤2 bβk, the calendar bound 1 + β ≤T230 i pays this loss. Let Dd denote the signed finite orbit sum defining the relevant divergence component. It commutes with the physical insertion map, with coefficient transport, and with pullback on the common window. Applying its rooted seminorm estimate to (TS7) gives the same stated error for back(DdJ) −I(a5 i DdV ). The real orbit projection places a5 i DdV in the required invariant label subspace. To check its coefficient budget, each billed input has transported coefficient size at most 2, hence Ve has size at most 2 + 2β. If c ≥1 is the fixed degree-five orbit cost, its divergence has size at most c(2 + 2β). Eventually 6c ≤bβk = t, with t ≥1, so c(2 + 2β) ≤6ct ≤t2 ≤t4. Multiplication by a5 i gives exactly the physical coefficient budget. The previously obtained crown bound for DdJ, its chamber support, and (TS7) now give the required physical probe estimate for the corrected divergence label. In particular this transfer retains the coupling multiplying Y , its degree-five spacing factor, and the actual common readout. Here is how the local lens estimate entering (TS6) is obtained. A single lens is the Haar average of a local surgery difference. Copy it into every chamber of a tiling based at a coarser reporting level J, applying the chamber’s signed motion and conjugation. If the original surgery is based at z, each copied base lies at distance dist(0, z) from its chamber center. The embedded centers are separated by at least the block ratio between levels J and k. Choose that ratio larger than both 2 pierSize +2 + 2 dist(0, z), 64B b + 2B + 2 dist(0, z). The inspected surgery supports are then disjoint, the changed bonds are distinct, and the stronger separation needed for the marked expense bound holds. All rotations and optional conjugations preserve the local boundedness and gauge conditions. Integration in one component of a product Haar variable is its ordinary one-component Haar integral, so the marked coefficient at each chamber is exactly that chamber’s copy of the original lens. Introduce two marker slots per chamber and give each the same positive weight w(k). The surgery expense inequalities have the following explicit form. Put s for the support exponent, l for the support-cover bound, F for the bound on the inserted local function, and m for the surgery truncation. With the ordinary and tail tariffs denoted by Twing and Tlease, choose c ≥1 and r ≥0 such that 4ce(s−3)l ≤eb, Fe(s−3)l+2 w(k)2 c ≤r, 4e−mTlease ≤1, Twing ≤1. 447 ## Source page 450 A Wilson Yang–Mills construction J. Rodgers The remaining expense is t = r + 8ebe−mTlease + 2ebbforced, (2B + 1)4t ≤1, et −1 ≤uallowed. The local surgery expansion and its support census then give a served marked insertion with support exponent s −4 and budget 1. The factor w(k)2/c is included in its coefficient bound. The tariff summation is performed separately at each surgery base; disjointness of the inspected supports permits their combination with no additional factor equal to the number of chambers. Transport this insertion from k down to an intermediate level m0, retaining the two markers per chamber and weights satisfying w(j) ≤δjw(j + 1), m0 ≤j < k. At each scale the embedded chamber anchors are distinct. Hence the occupancy of the doubled marker set is at most two, which is within the available multiplicity rj ≥2. This proves the admissibility of the marker transport at every step. The retained insertion estimate and positive scalar normalization bound its final, fully marked coefficient. If v is the number of reporting chambers and Q the physical Wilson moment of the product of the copied lenses, the result is |Q| w(m0)2v ≤exp[| Sitem0 | treasury(m0) + 4v] . This coefficient identity follows also directly by integrating the restored marked insertion: the partition identity transports the intermediate integral back to the finest Wilson integral, and the same positive Wilson partition factor occurs on both sides and cancels. The chamber-count identity is | Sitem0 |/v = (aJ/am0)4. Taking the v-th root therefore proves the local rooted estimate |Q|1/v ≤exp (aJ/am0)4 treasury(m0) + 4  w(m0)2 . (TS8) Thus the bound is independent of the number of copied chambers. For the summit calendar take J = i and m0 its prescribed basin. The surgery window radius and the necessary separation radius are subpower functions of bβk; the calendar’s block ratio eventually exceeds them uniformly in the volume. Its treasury and weight choices make the right side of (TS8) at most T−1900000 i . The same window fits the reporting chamber. This establishes the local estimate used in (TS6), with the finite surgery expense conditions displayed above. The expense conditions in this argument are consequences of the same large-coupling choice; they are not additional assumptions on the final theory. To see the numerical allocation, fix a stage and the integer l. Set B = ⌈(log x)2⌉, P = B40 + 1100, M = B60 + 1100, ex = e−P/8. These are respectively the prism and surgery truncation used above, and M ≥4P eventually. For every fixed constant C and integer j, Cxj ≤eP/32 eventually. (TS9) Indeed log x ≤B eventually, while the exponent P has degree forty in B; taking logarithms proves the claim. The lease tariffhas an upper bound x3, uniformly over the admissible representations at 448 ## Source page 451 A Wilson Yang–Mills construction J. Rodgers this fixed stage. Its defining expression is a fixed finite sum of products of the coupling, threshold powers, path costs, signed-motion count, and the local site census. The coupling is at most 4x; the other growing factors have subpower growth. The expression therefore has a rate bound of exponent 2, and is eventually at most x3. In the expense inequalities take the truncation m = M, the separation count b = P, and c′ = eP/2, r = ex, 1 ≤w ≤eP/32. The fixed factor c0 = e(s−3)l satisfies 4c0 ≤eP/32 eventually. Thus 4c′c0 ≤eP . Likewise (TS9) gives xc0e2 ≤eP/32, and hence xc0e2 w2/c′ ≤e3P/32−P/2 ≤ex. Writing u = e−MTlease, we have 8eP u ≤8x3eP−M ≤ex. The forced allowance is the specified rapidly decreasing function smoke(x) = exp[−Wave(1/5, x)]. Its decay dominates eP , giving 2eP smoke(x) ≤ex eventually. Consequently the total expense t is at most 3ex. The site factor (2B +1)4 has subpower growth, so (2B + 1)4t ≤1 eventually. Finally et −1 ≤1, which is the required output allowance here. Also 4u ≤1 follows from 8eP u ≤ex ≤1. The wing tariffis the already imposed local surgery tariff. This proves all the displayed expense conditions simultaneously for every weight in the stated interval. For each fixed stage, the preceding bounds impose only one eventual lower bound on x. The stagewise threshold selection enlarges the original coupling-entry delay to exceed these bounds at every permitted stage transition. Taking the maximum with the other construction thresholds preserves them all; later delay increases preserve this choice. Thus the physical calendar used in (TS6)–(TS8) satisfies the expense bounds before the finer cutoffor volume is chosen. 16.7.1 Finite density surgery and the equation insertion The finite lens changes one bond by ordered group multiplication. Health and chord inequalities preserve the exact flag set, so exponent differences only involve supports containing the modified source. The resulting finite subset expansion is exact. Only the explicitly stated finite-shelf approx- imation is an error, and its allowance is selected after division by the node step has been included. Haar invariance gives the zero-integral identity for the same stock density used throughout the marker construction. Why the marker is a finite representative The equation insertion is constructed from a local change of density before the continuum quotient is taken. Its flags and replicated marks retain where that change is supported, and its expense calendar gives the root bounds needed by spectators. Thus the eventual equation-null class has an actual finite local representative on the chosen Wilson family. The surgery identity and the weighted estimates below establish this assertion together; a formally vanishing coefficient symbol would not supply the same conclusion. The lens itself has a finite, exact definition. For an oriented bond issuing from z, let Tν(j)U change its link by left multiplication by ν(j); for a negatively read bond the equivalent stored positive link is changed by the corresponding right inverse multiplication. Let SQ(U) denote the scalar stock of a support Q. Truncate the represented exponent by support-cover number: HM(U) = v| Site | + X covB(Q)≤M SQ(U). 449 ## Source page 452 A Wilson Yang–Mills construction J. Rodgers The lattice is finite, so this sum is finite. For the bounded local function J(U, j), the lens is L(U) = Z G h J(Tν(j)U, j)eHM(Tν(j)U)−HM(U) −J(U, j) i dνG(j). (TS10) The admissibility conditions on these data are concrete: J( · , j) reads the fixed support P; J(gU, gzj) = J(U, j); a nonzero value requires health at z with threshold θ/4; ν(aj) = aν(j)a−1; ν is measurable; and its chord is at most D, with 8D ≤θ. These conditions are preserved under the signed chamber copies used above. To justify expansion of (TS10), put V = Tν(j)U and suppose J(V, j) ̸= 0. Every link changes in chord distance by at most D. The chord of a plaquette changes by at most 4D, by telescoping its four unitary factors. A plaquette whose value changes touches z, so its chord at V is at most θ/4. Its chord at U is therefore at most θ/4+4D ≤3θ/4. Both are below θ. Unchanged plaquettes have identical threshold comparisons. Hence the flag sets at U, V agree exactly. The same reasoning after any signed motion shows that their relaxed kept-plaquette tests agree, since each relaxed threshold is at least θ. A stock whose support omits z is unchanged: the modified bond has an endpoint at z, all internal links of that support agree, and its flag data agree. Thus HM(V ) −HM(U) = X z∈Q covB(Q)≤M SQ(V ) −SQ(U) . For this finite set of supports, set tQ(U, j) = ( eSQ(V )−SQ(U) −1, J(V, j) ̸= 0, 0, J(V, j) = 0. The distributive law and exponential addition give the exact identity J(V, j)eHM(V )−HM(U) = X A⊆{Q:z∈Q, covB(Q)≤M} J(V, j) Y Q∈A tQ(U, j). If J(V, j) = 0, both sides are zero; this also explains the zero branch in the definition of tQ. It avoids any need to estimate exponent differences on configurations where the insertion is absent. This finite subset expansion is the local surgery expansion priced in the expense inequalities. Every term reads only the inspected union of P and the supports Q in that formula. The cover-extent inequality bounds its distance from z by pierSize = (l + M)(4B + 2) + 10B when covB(P) ≤l and 0, z ∈P. The resulting lens reads the origin-centered window of radius 2 pierSize. Its gauge invariance follows from the displayed covariance of J, ν, invariance of the stocks, and the change of Haar variable j 7→gzj. Measurability and boundedness follow from the finite stock expansion and its envelope. These facts provide the locality, gauge invariance and integrability used in the copied-lens and root estimates. For the equation marker used above, specialize (TS10) as follows. On a positive source link e let v = Adg ec be a unit color direction, let ν(g) = exp(tihv), and take J(U, g) = 1{healthy at the source with threshold θ/4}⟨Y (U)e, v⟩. For a reversed link use the equivalent reversed multiplication in (TS10). The current is gauge covariant at the source, so the covariance of J and ν is exactly the one required there. The step 450 ## Source page 453 A Wilson Yang–Mills construction J. Rodgers h = θx−1/24, with its fixed node multiples, eventually satisfies the stronger chart condition 64D ≤θ; thus it also satisfies the 8D ≤θ flag condition of (TS10). Let Le,tih(U) be this lens with the finite stock exponent, and sum over the active links, color basis indices and nodes: M(U) = 1 h X e,c X i 0 in the selected phase, with the same scalar diagnostic under the phase identifications established above. The mass-shell asymptotic argument excludes the finite comparison class specified in the 452 ## Source page 455 A Wilson Yang–Mills construction J. Rodgers theorem; the coherent-state argument gives the additional Maxwell countertests described in the trace chapter. Every witness in these conclusions has been constructed on the one chosen Wilson family and retained on the one composite refinement. Passing to the public fields, operators, correlations and channel coefficients is therefore their identification on that common table. This proves the theorem. 18 Conclusion The construction connects a finite Wilson probability law to a common continuum hierarchy through explicit kernels, nonlinear integrations, calibrated histories and physical coefficient spaces. The full mesh limit at each suitable fixed period precedes the outer volume refinement. Reflection positivity, the phase algebra and the tested boundary construction then give the fields on their common cyclic domain. The operator-product and contact constructions retain their finite horizons and chosen devices on that same hierarchy. The stress calculation returns to actual finite local functions, controls their equation residuals and fixes the centered trace amplitude. Its logarithmic short-distance asymptotic yields the finite mass-shell and Maxwell comparison conclusions with the hypotheses stated above. No positive mass gap is asserted. This paper provides the full natural-language mathematical exposition of the expanded source. The flagship in the same Zenodo release [2] develops Rodgers’s source–readout realization and recovery of this construction. Both papers retain the definitions and derivations required for their own arguments. A Editorial amendments and conventions The mathematical source is the expanded Full_pre_source_translation.md, included in the re- lease’s sources/ directory. Its 15,788 lines are counted with universal newlines. The coverage record locates every source heading in this paper. The following records the mathematical amend- ments and interpreted typography. A1: principal predictor (source line 1017). The extracted expression is m(q0)−1Ta/ga−H/U; the reconciled expression is m(q0)−1(Ta/ga −H/U). The direct subtraction in Section 5.22 shows E0 −r0 = SG0,a in the declared multiplier convention. The singular factor S−1 is cancelled before Fourier extraction, and the nonprincipal and dual branches retain their displayed definitions. The amended parenthesization implements the stated cancellation. A2: Gaussian prefactor (source line 1317). The extracted exponent −rh+dv is replaced by −(rh + dv). The active, dormant and frozen-link counts, and the substitution mg = c( √ 2πg)r, are calculated in Section 6.2. The dormant Gaussian integrates to one, and its normalization remains present. The determinant factor is unchanged. This calculation fixes the normalization used by the chart integral. Interpreted typography (E2). At source line 1575 use CK = (na+nb)AJ: the common row and column count multiplies their common estimate. At line 3218 use 4 pcap(4)( pdirecttop+pathstop)t; the common factor t comes from both base-row estimates. At line 3428 use HC(QRprice)n, consis- tent with the repeated bound at source line 4020. The source’s damaged comma in (CAL21) and split rm free in (GL0) are restored from their displayed mathematical context. 453 ## Source page 456 A Wilson Yang–Mills construction J. Rodgers Domains and interpretation. The combined startup allowances in Section 8.14 hold on x/2 ≤ β ≤2x. The forced estimate alone uses only the lower bound. The Gaussian comparison distin- guishes C0 and CV , with their determinant and inverse factors (Sections 8.23–8.25). The primitive relation retains ordered group multiplication (Section 7.22). Exact radius matching is geometric, while moment comparisons keep their error bounds (Section 16.2). The full idempotent-kernel proof is included in Section 11.5. These conventions also govern the corresponding recovery maps in the source–readout paper; the constructions used in this exposition are defined and derived locally. Scope of the results. The group and representation are given; no connectedness hypothesis is added. One Wilson family precedes the requested volume sequence. Its inner fixed-period mesh limit is full, and its outer volume limit is subsequential. Field assertions use the common cyclic invariant domain. Every OPE/contact statement retains its finite horizon, geometry and chosen subtraction device. The full-distance trace argument retains the same table and phase; the moving-mass estimates retain their positive-mass choice order; the real fourth-order remainder is distinguished from a formal jet and from leading transfer to weaker frames. The two comparison exclusions retain their specified hypotheses. No positive mass gap or Millennium-problem solution is asserted by this presentation. Transcription typography in the companion. Short formulae written directly in source prose are set in mathematical mode; subscripts, superscripts, inequalities and function names are joined into coherent expressions. The Unicode aggregate signs Σ and indexed Π are printed as P and Q where their context is a sum or product. The standalone heat-ground projection remains Π. These typography choices introduce no new operator or estimate. The following local interpretations make explicit the source’s intended application or grouping: Source line(s) Typeset expression Reason from the local argument 642 fξ(p) −fξ(y) The immediately preceding sentence ap- plies Cauchy’s estimate to the function fξ at p and y. 815 (d2B1)TF(d2B1) The transpose acts on the whole product: V = B1MB1, M = dT 2 Fd2, and B1 is sym- metric. This clarifies the extraction’s mis- placed transpose. 989 L4(L−3)2 The Gram row squares the L−3 coefficient bound and sums over L4 residues. 1431 [(A|Wj)−1]ij This is the ij entry of the inverse of the principal restriction, zero-padded outside Wj. 1699 Hp0 = 1 The distinguished patch point is p0; the value of its frame matrix is fixed to the identity. 3121 mg(β) = c( √ 2π β−1/2)r The damaged source exponent β−1/2 is the Gaussian scale g = β−1/2 already used in the active/dormant normalization. 454 ## Source page 457 A Wilson Yang–Mills construction J. Rodgers 4620, 4627 logReadi(1) The supplied name is an operator label; its receipt and reset identities define its use locally. 5195 Q i∈p\keys(M) ti The text explicitly removes the marked team’s keys before multiplying the remain- ing primitive-body tariffs. 9736 (ap(k),m)m This is a point in the countable product in- dexed by the retained level m, not a second subscript on one scalar. 9738 (Bm −bβm)/tm The preceding inequality bounds the nor- malized difference between the actual lim- iting coupling and its reference clock. 10044, 10050 Π The standalone symbol denotes the heat- ground projection; it is distinct from in- dexed product notation. In the reconstruction chapter, ASCII pairing brackets and postfix stars are set as Hilbert pairings ⟨·, ·⟩and adjoints on their declared domains. The zero-energy event is written {E = 0} and the low- energy function as ρ(E). These are the same pairings, spectral event and scalar function defined in the local proofs. Local internal source aliases identify provenance rather than supplying additional hypotheses or replacing a derivation. B Source, attribution and reproducibility The underlying Yang–Mills Lean work is credited to OpenAI. The source actually presented here is the expanded Wilson Yang–Mills Construction: Mathematical Translation [1], preserved in the release. Jeremy Rodgers is the author of this exposition, prepared with AI assistance. The related source–readout reconstruction develops the programme that Rodgers originated and directed [2]. The complete editable manuscript consists of the root LaTeX document, sections/*.tex, and references.tex. All mathematical content is contained in those files. The release includes the expanded source, coverage and amendment records, build instructions, and verification records. Source-unit comments connect the editable text to the original heading intervals. The verifica- tion record distinguishes the current build and focused calculations from retained numerical and structural checks; no new Lean/Lake execution is reported. References [1] Wilson Yang–Mills Construction: Mathematical Translation. Project source document: expanded natural-language extraction prepared for Jeremy Rodgers’s Yang–Mills project with AI assistance and supplied by Jeremy Rodgers. Included unchanged as sources/Full_pre_source_translation.md. OpenAI is credited separately for the underlying Lean work; the supplied materials do not identify an exact upstream repository or release. [2] J. Rodgers. Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction. Flagship paper in this shared Zenodo record. doi:10.5281/zenodo.24173681. [3] J. Rodgers. A Source-to-Readout Architecture for a Theory of Everything: Sector Physics as Coupled Projections from a Common Realization Carrier, v1.0. Zenodo. DOI 10.5281/zenodo.21366204. 455 ## Source page 458 A Wilson Yang–Mills construction J. Rodgers [4] K. G. Wilson. Confinement of quarks. Phys. Rev. D 10 (1974), 2445–2459. doi:10.1103/PhysRevD.10.2445. [5] A. Haar. Der Massbegriffin der Theorie der kontinuierlichen Gruppen. Ann. of Math. (2) 34 (1933), 147–169. doi:10.2307/1968346. [6] G. C. Wick. The evaluation of the collision matrix. Phys. Rev. 80 (1950), 268–272. doi:10.1103/PhysRev.80.268. [7] I. Gelfand and M. Neumark. On the imbedding of normed rings into the ring of operators in Hilbert space. Rec. Math. [Mat. Sbornik] N.S. 12(54), no. 2 (1943), 197–217. https://www.mathnet.ru/eng/sm6155. [8] I. E. Segal. Irreducible representations of operator algebras. Bull. Amer. Math. Soc. 53 (1947), 73–88. doi:10.1090/S0002-9904-1947-08742-5. [9] M. H. Stone. The generalized Weierstrass approximation theorem. Math. Mag. 21, no. 4 (1948), 167–184; continued, no. 5 (1948), 237–254. doi:10.2307/3029750; doi:10.2307/3029337. [10] K. Osterwalder and R. Schrader. Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31 (1973), 83–112. doi:10.1007/BF01645738. [11] K. Osterwalder and R. Schrader. Axioms for Euclidean Green’s functions II. Commun. Math. Phys. 42 (1975), 281–305. [12] K. Osterwalder and E. Seiler. Gauge field theories on a lattice. Ann. Phys. 110 (1978), 440–471. [13] J. Fröhlich, R. Israel, E. H. Lieb and B. Simon. Phase transitions and reflection positivity. I. General theory and long range lattice models. Commun. Math. Phys. 62 (1978), 1–34. [14] J. Glimm and A. Jaffe. Quantum Physics: A Functional Integral Point of View, 2nd ed. Springer, 1987. [15] R. F. Streater and A. S. Wightman. PCT, Spin and Statistics, and All That. Benjamin, 1964. [16] T. Bałaban. Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions. Commun. Math. Phys. 109 (1987), 249–301. [17] D. C. Brydges. A short course on cluster expansions. In K. Osterwalder and R. Stora (eds.), Critical Phenomena, Random Systems, Gauge Theories, Part I, Les Houches Session XLIII (1984), North-Holland, Amsterdam, 1986, 129–183. https://www.math.ubc.ca/~db5d/Seminars/les_houches_84.pdf. [18] E. Seiler. Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics 159, Springer, 1982. [19] D. J. Gross and F. Wilczek. Ultraviolet behavior of non-abelian gauge theories. Phys. Rev. Lett. 30 (1973), 1343–1346. [20] H. D. Politzer. Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30 (1973), 1346–1349. [21] A. Jaffe and E. Witten. Quantum Yang–Mills theory. In J. Carlson, A. Jaffe and A. Wiles (eds.), The Millennium Prize Problems, Clay Mathematics Institute and American Mathematical Society, 2006, 129–152. https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf. 456