Free shell charges and separation of the Ward kernel · Part 2

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 +1. The polynomial ring on jets through order 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 bound their engineering weights. They generate every invariant polynomial as an algebra: for an invariant polynomial of length 0, express it in the ideal, take its length- component, and average the coefficients over the compact group. Averaging is a projection onto invariant polynomials and satisfies

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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.

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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 are absent by hypothesis. The limit is exactly the two-point Wick pairing B), where is the length- component. Ghost vanishing at every positive separated 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. Let be the first possibly nonzero length component and reflect its complex . The shelf, invariance, engineering cap and conjugate across the time hyperplane, giving length are preserved by this operation. The hosting result puts in the allowed span. At points its reflected Wick pairing with is strictly positive if= 0, by the polynomial Gram calculation above. Coalescence has made the same pairing zero, a contradiction. Thus Every polynomial has only finitely many length components, so this induction proves that a ghost lying on this fixed shelf is zero. Define

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d = max{r, 4 + j + L}.
(FS12)

Induct on the engineering weight of the candidate ghost. If its highest jet order is, cluster a pair of jet tests of orders and. Its weight is 4 +, and. Apply (HC2)–(HC4) with highest shelf order: the finite cutoffis 8 + + 1, and the extracted pole degree is 4 +. 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, where the hosting tests separate it. The weight-zero case is the constant, detected by the empty word. This proves

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∃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( 4); lower-cap conclusions then follow by the filtered

inclusions. Arbitrarily high output weight in (FS13) is permitted, but arbitrarily high individual spectator degree has not been silently introduced.

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. 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

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[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 [

Every polynomial is a finite sum of engineering-weight components, and map commuting with preserves each such eigenspace: projecting its image onto weight gives () times that projection equal to zero. Total inversion acts on a weight- monomial , because a derivative-order- curvature variable has sign ( But the vector covariance law gives. On a preserved weight component these two rules give simultaneously and. Hence = 0, and 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.

Obtaining a detector on the physical coefficient space

The detector used below is finite dimensional. At fixed degree let be the coefficient space, its equation ideal, and the dimension of the free-field quotient. The foundation identifies the kernel of the physical projection with an invertible image of. Rank-nullity therefore gives dim ran, uniformly in the cutoffs. In the notation of the detector trials = dim + 1. This dimension statement does not assert that every bare coefficient survives the equations.

Choose trained trial vectors, uniformly bounded by, 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

stable under sufficiently small entrywise perturbations. Denote the allowed perturbation by and the resulting left-inverse bound by. Train with tolerance4. The actual tested trial vector differs from its limiting response by at most two errors of size: the finite training error and the short-distance probe error.

There is a third error when is replaced by. Indeed has exactly zero projection. The equation-cancellation bound makes its probe tend to zero, hence eventually smaller than, for all finitely many simultaneously. Thus the projected trial response matrix is within 3 of the full-rank reference matrix. Its columns are independent. Since ran has dimension at most, the projected trials span that range.

The projection has a uniform operator bound, so each spanning vector has norm at most . Apply the stable left inverse to the detector response of any vector in the range and expand in these spanning vectors. Summing their norms gives

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∥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 for the reference response matrix, with sup norms, and choose 0 such that. Set If every column of has norm at most, then

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∥(H −G)u∥≤rδ∥u∥,
∥u∥≤K∥Hu∥+ Krδ∥u∥≤K∥Hu∥+ 1
2∥u∥.

Hence; one can take. The injectivity of the reference matrix follows directly from separation of the free quotient by the chosen correlation functionals: a vector annihilated by all rows is zero. Expressing these functionals in a basis identifies their matrix with This explains both the rank lower bound and its stability on precisely the same varying projected trial columns used in the rank upper bound.

Bounding physical coefficients before taking limits

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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,

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∥Xi∥≤Z∥LiXi∥.

An insertion already bounded in the Wilson laws, together with a vanishing error in the tested insertion seminorms, gives eventually. This alone does not bound, because the equation part need not have small coefficient norm. Instead put

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fi = (1 + ∥Xi∥)−1,
yi = fiXi,
zi = fi(xi −Xi).

Then1, and belongs to the same admissible coefficient class: that class is closed under bounded scalar multiplication and subtraction of bounded coefficient families.

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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,

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Z∥Li(Xi −xi)∥≤1
2(1 + ∥Xi∥).

The detector inequality and the triangle inequality now give

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∥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 acts only on the-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.

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, let be the free equation ideal, and let be its marked lift into local lattice insertions. The marking identity says that an insertion evaluated on, 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 : 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.

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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
Y
Eβ
fa = 0.
chambers a

The distinguished copy is exactly the original local equation insertion, including its scale normalization. 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 for the terminal chart at cutoff and for its calibrated invertible coefficient change. The physical projection has kernel equal to the image of under the composite marked chart. For a coefficient family in this kernel define

Then belongs to the marked equation carrier. The calibrated inverse estimate bounds the normalized coefficient size of by twice that of. Hence a coefficient family with the permitted subexponential growth retains that growth after this inverse change. The trained-chart comparison is

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upper(ZiPiyi) −hallmarks(yi) = Oroot(flight−1/32(i)).
(IC3)

Its finite-dimensional derivation is explicit. Decompose using the fixed finite signed-orbit tally of frame basis vectors. For each tally component the trained local comparison has root bound flight Its scalar coefficient is bounded by the coefficient norm times the fixed orbitcoordinate constant. Sum over the finite tally. The sum of these scalar bounds is subexponential, so it is eventually at most flight flight). This proves (IC3). The algebraic recombination is exact: the trained basis expansion is, whereas the original frame expansion recombines to. 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 , we obtain upper((flight whenker (IC4)

This is the banked equation-cancellation estimate. In the detector proof apply it to, using . 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, and the free observable map has kernel precisely. 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 byexp(, with all other links fixed, and denote this derivative by . The change preserves product Haar measure. The Wilson action derivative is

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∇e,AS(U) = ⟨A, Ee(U)⟩.

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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),

and at a link of axis the negative of the analogous difference in direction. 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() mat()) = repDim curvature(

This identity uses skew-adjointness of the color matrix and the orthogonal projection defining curvature. Let and be continuous, and let be bounded measurable and unchanged by this link variation. Haar invariance gives

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Eβ
∇e,AW −β⟨A, Ee⟩W

S
 = 0.
(IC5)

To justify differentiation, translate the derivative formula from zero to any by the one-parameter group law. On the compact configuration space the continuous derivative of the Wilson-weighted is bounded; multiply its bound by the bounded envelope of. 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= 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, on every torus of side at least a fixed, at every nonzero real 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

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p 7−→(Qp, fp),

where is a gauge-invariant local polynomial insertion with leading jet. At every graded piece, lies strictly above that piece in the finite lexicographic weight ordering. The construction retains the exact marking identity on the equation ideal. Triangularity makes invertible: a nonzero vector has a first nonzero graded piece, which leaves unchanged, so = 0 is impossible. In finite dimensions injectivity implies surjectivity. The same first-piece argument shows that

preserves each descending weight filtration.

Define a linear local realization by. It has leading jet and retains the required power filtration. Real conjugation and signed coordinate symmetry are imposed on the paired construction, so, its inverse, and 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 on the basis proves equality everywhere. The box contains zero and is invariant under signed coordinate permutations.

With, the resulting fixed section satisfies

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T(coords(Rp)) = fp.

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 are chosen once at grade, 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, let be its homogeneous label space and. The ordinary polynomial section supplies a linear lift on with the correct leading form and power bound. On, choose a finite basis and lift each basis vector by the positioned equation construction. Its output is a pair () with 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; all these properties are linear, including membership in the marking class.

Choose an algebraic projection. Define the lift on by

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Fm(p) = F eq
m (Lmp) + F ord
m (p −Lmp).

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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 act on labels by a signed coordinate transformation, and let pull

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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. Reindexing by left multiplication in proves signed equivariance. Finally let c), with the prescribed semilinear real structure on labels. Set

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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 orbit, while retaining a finite polynomial realization. If is a powered local polynomial, its orbit (gauge) spans a finite-dimensional space: each factor uses a fixed finite matrix representation and finite polynomial degree, and conjugation acts linearly on its finitely many coefficient tensors. The marking property is preserved by: in its expectation change variables by the constant 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.

The paired vectors (1) therefore lie in a finite-dimensional space. Their first coordinate and all configuration-evaluation coordinates are continuous functions of. These functionals separate that space, so their span is the whole dual; hence every linear coordinate is continuous. Integrate the coordinates of (1) against probability Haar measure to get a vector in their span. Its first coordinate is one and its second coordinate is exactly

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 ̄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. 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 ̄. 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- jet space let be a linear projection onto the range of the injective free gauge-jet map. Substituting in every jet variable defines an algebra homomorphism. A label polynomial belongs to the equation ideal precisely when = 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, split

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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. 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 this telescoping writes as a finite linear combination of such terms. When 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, the source first represents the equation functional 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-factor monomial modulo field-count degree at least + 1, by the finite product expansion. Multiply by the force factor, truncate to the retained jet