Notation and analytic conventions

Notation and analytic conventions

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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, and the periodic lattice of side is (. A positive link is (), joining to; reading it in reverse gives the inverse group element. A plaquette in the positive plane 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

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⟨X, Y ⟩g = −1
m Re tr(XY ).

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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.

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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

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X
s = 11CA
24π2 ,
CA = −
1
dim g
α
tr(ad Tα)2.

The simple nonabelian assumption gives 0. Intermediate represented couplings are distinct from this reference coupling and are compared with it explicitly. For a finite support, its radius- halo is the union of the lattice balls of radius about its points. The cover number cov) is the minimum number of such balls whose union is connected

by unit lattice steps and contains. The centers need not lie in. 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 put log (log temp(

An envelope has rate if for every

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|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( temp()). 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 has rate and has rate, choose slightly larger exponents for their separate bounds; their product is then bounded by every exponent strictly greater than. 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 exponential 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.

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 Λ (, let be the configuration space and g its real space of left-link directions. A function is local on a site set if its value depends only on links with both endpoints in. Gauge invariance refers to For a plaquette, write) for its four visited vertices and for its holonomy. At threshold

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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; its boundary is not differentiated in any regular budget.

A coefficient which also reads the final flags in is required to agree whenever both the links internal to and) agree. Its actual link read set is bounded by

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Xins = X ∪
[
p:V (p)∩X̸=∅
V (p).
(AB2)

Agreement on internal links of this enlarged set fixes every plaquette incident to, 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, set

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(inc(U, tA))e = exp(tAe)Ue,
∇AF(U) = d
dtF(inc(U, tA))

t=0
.

For an ordered list), define recursively

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∇∅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 is a globally Borel, gauge-invariant local function such that(inc( near zero for every), and its recursively specified left derivatives have the derivative identities in (AB3) through order. 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 integers1, support exponent 0, and threshold 1. We denote it by p). For anchor, its allowed directions satisfy

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|Ae| ≤θ(1 + dist(e−, o))p
for every link e.
(AB4)

Hereis the initial vertex of the stored positive link, and the distance is the periodic maximum-coordinate distance. A nonnegative number is a derivative envelope for

whenever and everysatisfies (AB4) (AB5)

Norms replace absolute values for vector-valued coefficients. The support price is

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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 () with. A regular family has anchored budget if it has envelopes0 satisfying (AB5) and

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X
i:oi=o
es covB(Xi)bi ≤δ.
(AB7)
sup
o∈Λn

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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]

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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, a nonempty inventory, and a fixed extra predicate). Its coefficient is globally Borel and gauge invariant and vanishes unless

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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, hence have the physical read enlargement (AB2). A forced budget consists of numbers 0 such that on every configuration and

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X
i:x∈Yi
es covB(Yi)ci ≤κ.
(AB10)
sup
x∈Λn

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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
P
most Q
i ci. Restricting to compatible subsets only decreases it. If there are no flags,
i(1 + ci) ≤e
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 + 3, rather than, in (AB7). This is the pre-reset reserve. The three additional supportexponent units pay the local splitting and conversion operations explicitly described in the reset and signed-expansion chapters. Reducing the exponent from decreases every nonnegative summand. The rotation of derivative directions costs the finite factor

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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 +) costs 2 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 2 ordered lists at order; each has length and direction inflation at most 2. Multilinearity of the iterated derivatives bounds their sum by (2 2, which is at most (AB11) times for. 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

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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

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bPi = x−11/10
i
,
kPi = exp(−M1/5
i
).
(AB12)

Both are strictly positive. Thus means exactly that its regular and forced components obey (AB7) and (AB10) with budgets and, respectively, in their fixed address layout. The retained physical states have8. 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.