Free shell charges and separation of the Ward kernel · Part 3

degree, and the error has field-count degree at least + 2, strictly above the target count The force’s retained linear functional may have higher jet weights than; 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.

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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
X


F (l)
ab (x) −F (l)
incidenceν,a,b
ab (x −ei)
.
a<b

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l=0

The transport path drops out at this order because the curvature vanishes at the identity. The incidence signs are the same positive-link signs as in (IC5).

The leading homogeneous content of a transverse difference is its ordinary derivative. More precisely the difference of the two face averages equals a transverse interval integral of the derivative of their integrands. Together with the two face integrals, it is an integral over three unit intervals. This is the fundamental theorem of calculus, before any limiting argument. Rescale its base point and mesh by; for a homogeneous jet of orderthe rescaled transverse expression istimes its unit-mesh expression, after multiplication by the extra transverse factor

Suppose all transported force linearizations vanish on a finite jet array. For each source address choose a path to it avoiding its outgoing source edge. The preceding formula gives vanishing of the sum of transverse differences at every integer address. Multiply the order-terms byand use the rescaling identity. At points index), which tend to any prescribed the resulting continuous expression vanishes. Let0. All terms except the top order vanish, and the three interval averages shrink to a point. The remaining equation is

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X
i
∂iF (j)
iν (x) = 0
for every x, ν.

Its Bianchi identity already holds because these are curvature jets of radial potentials. Thus is a free solution jet. If a functional on that top jet annihilates the free gauge-jet range, it annihilates the common kernel of all force probes. In finite dimensions the annihilator of that common kernel is the span of the probes. This proves the force-span assertion and yields an actual finite linear combination, not an infinite sum over source points.

The edge-avoiding ordinary probes span the whole finite jet dual. Fix one forbidden edge. Except at a finite set of curvature-sampling addresses, one can reach the sampling plaquette along a path that avoids this edge and whose plaquette boundary avoids it as well. If every such safe sample vanishes on a jet array, all its curvature polynomials vanish at every integer point outside that finite exceptional set. The same scaling argument isolates its top homogeneous part: for nonzero, the sampled addresses index) eventually leave every fixed finite set, since their physical rescalings tend to= 0, while fixed addresses rescale to zero. Continuity gives vanishing at = 0 too. Hence the top homogeneous curvature jet is zero. Descending induction gives zero for every jet component. Radial reconstruction is injective, so the original jet array is zero. The safe samples therefore have zero common kernel; finite-dimensional duality proves that every desired ordinary jet functional is a finite linear combination of them.

Finally a transported force times an edge-avoiding multiplier has the exact marking identity claimed above. Move its root color test through the carrier by adjoint invariance:

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⟨X, Adg Ee⟩= ⟨Adg−1 X, Ee⟩.

The carrier avoids, so its transported color vector is unchanged under variation of. Expand that vector in a fixed real color basis. In each term its scalar coordinate, the avoiding multiplier, and the remote spectator are all unchanged by the varied edge. Apply the = 1 case of (IC5) to each such product and sum over the finite basis. The spectator locality radius contains the fixed force edge; a sufficiently large side prevents its periodic identification with a forbidden remote edge. This proves the exact local marking property of each force term, completing the positioned equation-lift argument.

Uniform placement and testing of detector trials

Detecting a coefficient with actual correlations A detector is a finite list of linear readings of a distinguished insertion against fixed spectator insertions and tested positions. To use its inverse on the whole relevant coefficient space, the same placement and continuity bounds must work for every column in a fixed coefficient ball. The proof below obtains that uniformity from one separated compact neighborhood and a finite coordinate telescoping argument. Its conclusion concerns the actual whole lattice cylinders, including their corrections, rather than only their leading Taylor polynomials.

The detector probes are actual lattice correlations, linear in the tested first insertion. Fix the trained spectator insertions. At cutoff

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"
#
n
Y
Mi(f; x) = Eβi
f(x0)
ν=1
Fν(xν)
.

Linearity follows by expanding the first factor under the finite Wilson probability integral; all local polynomial insertions are continuous on the compact configuration space. A forward lattice difference in the first insertion equals the corresponding difference in its first position. Iterating proves this identity for every finite direction list. The detector applies the corresponding finite averaging operator to at its prescribed placement. It is therefore a complex linear map on the first coefficient vector.

The trained placements lie a positive distance inside a common physical box and are generic: every pair of points differs in every coordinate used by the separation convention. For each fixed row and training index choose a small closed neighborhood of the placement inside this generic set and inside 1. Its pair-coordinate gaps have a positive minimum by compactness. These gaps persist for every allowed torus volume. Indeed all physical periods are at least four; a difference of absolute value less than two is no closer to a nonzero period translate than it is to zero. Thus the same positive gap works for every volume, including the smallest admitted torus.

The unextracted insertion estimates consequently give a common local continuity bound for all coefficient columns in a fixed norm ball. To derive it, connect two nearby lattice configurations of positions by replacing their 4(+1) coordinates one at a time. Every intermediate replacement stays within their original physical distance and within the chosen neighborhood. The one-coordinate telescoping estimate gives, if the physical distance is, a correlation change at most for each replacement, where 0 is the common separation reserve and the insertion allowance. Summing gives (IC6)

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 0 samples points displaced by at most 2 per direction in its fixed direction list. Equation (IC6) makes its difference from the unaveraged monitor arbitrarily small by taking0, 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 count(. 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 below4, then take so that the averaging error is below4, and finally use (IC4) so that projecting each trial changes its response by less than4. 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, 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 commuting square-zero markers. Let

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F(ε) =
X
A⊂{1,...,n}
Sfree(xA; qA)εA,
D = log F,

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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

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H(ε, U) = e−D(ε) Y
a
(1 + εaPa(U)).

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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 times gap. 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, and for two distinct slots the pair coefficient is

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

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 therefore has all these properties; no orthonormality at a probe separation is required. For any, let) 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 through the hyperplane perpendicular to; call the resulting quotient field. The reflected Wick positivity estimate gives

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Bx,y(p, p♯) > 0
(p ̸= 0).

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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 ofwould 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 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, write, and let be reflection perpendicular to. A real solution-tensor probe of homogeneous order has the Kelvin response

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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 byinv() and decompose it into homogeneous degrees 0. Multiply its order- component by; denote these lower-order probes by. The diagonal component is The sum of their responses is consequently

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r(y)2R⊗2
y sjP(inv(y) −inv(z)).

Along, the inversion difference is), where is smooth near zero and . Homogeneity shows that this response has a factor. Thus all its normalized Taylor jets

at below order vanish, and its order- jet is

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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 (. Hence the transformed reflected Gram form, on a tower of orders 0

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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 for this positive form and let 0 be its squared norms. Every finite Wick polynomial is a polynomial in such a finite tower. The cross-insertion Wick rule gives

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M
Y
i=1
αi! dαi
i .
⟨eα, eβ⟩refl= 1α=β

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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- tensor

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DP f(x) =
X
∂ν1 · · · ∂νjf(x)
j!
P(eν1, . . . , eνj).
ν1,...,νj

A solution tensor obeys the trace constraints corresponding to harmonicity. With normal normal (IC7)

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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

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∂iVm −∂mVi = (j + 2)Sim(x, . . . , x).

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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

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a
∂aDSaiG = −2(j + 1)(−2)j normal rj+2Vi.

The contraction of the two-curvature kernel with is twice the curl of this last vector, with the order dictated by the opposite-side derivative. Differentiating once more yields

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sjrj+2 [Sim −2r(x)(xiVm −xmVi)] .

The bracket equals reflection of both antisymmetric form indices through; its term quadratic in the two radial contractions vanishes by antisymmetry. Since is homogeneous of degree replacing its argument by inv( converts the prefactor into. This is precisely the Kelvin response used above, with normal. The sign ( in an order- probe at the second point comes from differentiating there; it is included once. All these identities hold offthe origin, where the Green function is smooth.

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 representatives, and its positive comparison width. At arbitrarily late comparison indices, 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, these data are available for every volume parameter. The representation and the readout may depend on; 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. 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, with its zeroth member equal to the distinguished unit. The coefficient bound for this inverse gives

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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 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. Thus the same metering bounds apply throughout. 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), 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) for the starting width and 0 for the testing scale. Along admissible indices,

The first assertion follows from the depth property of the scale calendar: if an admissible width stayed below a fixed, its plateau identity would contradict the eventual strict increase of the

calendar beyond the depth at. Given integers and 0, intersect the arbitrarily late supply set with the eventual conditions

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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 from it with

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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 configurations. 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, and every, 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 ), uniformly in. 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.