Exact stress correction and local quantitative estimates · Part 2

Indeed their respective ratios areΩ) andΩ), both tending to zero. These inequalities 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

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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

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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 already denote coefficient vectors; are scalar configuration functions until altoD is applied. The sign identity is, so the loop contribution is 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. The current here is the localized version of (TF0a), transported through the stationary descent, not an independently chosen continuum vector.

The term is a concrete finite-difference remainder. If 0 are the distinct nodes and their Lagrange polynomials on nodes, use

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wi = li(0)/ti,
Dm,hf =
X
i<m
wi{f(tih) −f(0)}.

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Their exact moment identities are P
= 1{j=0} for 0 ≤j < m, by interpolation of tj at
i<m witj+1
i
zero. Therefore Dm,hf −hf′(0) starts in order hm+1, with its integral Taylor remainder. Apply this
to the changed current pairing times the exponential action ratio on each active link, and average
the color directions over the Haar adjoint orbit of a unit color basis vector. Sum over the finitely
many active links. This is precisely Tm,h, including its current derivative and density derivative.
Its bounds after division by h are (LCE2).

The low-jet estimate for the first term in (LCE6) uses the quantitative stationary error as well as the identity (TF0c). Here is that step in detail. Write = boom , and for the stationary error. Equality of the analytic jets of and through order four gives

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Dj(F −R′)(0) = DjE(0),
0 ≤j ≤4.
(TF12)

The issued stationary chart bounds the analytic coefficient norm of by shock( on radius = east(). Converting the coefficient bound to ordinary derivatives through order four costs the previously defined analytic derivative price 4), called gable in the source. Localizing the tensor to Meld costs at most 4), where

There are two further replacements: the original current by its localized current, and the full principal action by its finite retained action. Use the exact identity

(TF13)

The current difference has weighted price; the localized current has price at most trophy(. The full action derivative has bound

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U(x) = bankCost(B(x))K momentPriceP (D/2)[GeoSize +2 paths],
T(x) = bankCost(B(x))Kx−100 + e−D rim(x)/4U(x).
(TF14)

Here 0 is the issued path width; depend on 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. The weighted product rule for contracting a current with an action derivative bounds (TF13) by ) = vocation) + vocation (TF15)

The prices in (TF15) have the following explicit meanings. Set

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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)

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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 2= 16. Summing ingives 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), and

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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 2 thinM(4, give the order-five rescaled remainder, with(1 + high) Summing the Taylor terms gives the stated envelope.

The quiet derivative 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. The term 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; its entire weighted sum is bounded in (LCE3). Thus the four small terms are exactly altoD, 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 of plaquette types and pair of color indices, define

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ca,b
γ
= 1{types of γ=a} coeffγ(b),
zγ = offsetγ,1 −offsetγ,0 .
(TF17)

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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)

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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)

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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 momentPrice after a sufficiently large cut and radius, for width 0 and full loop price. This proves the small appended-tail bound used for. 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. 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.

Source–readout formulation. The six-term split (LCE6) is an exact decomposition of one configuration function into retained source columns, each with its own reader, support and bill. Four are small after division by , altoD); two are not small and are computed ( the Haar Jacobian with its centred constant removed, and, which has zero scalar contraction exactly). The lift keeps all six, and keeps the distinction between the exact statement ((TF6b) for the augmented family) and the estimate ((TF19) for the appended part). The quiet moments used for the appended part are those of Proposition 14.10, obtained from the full family and not from a truncation of it.

25.6 Invariant scalar reading and the six-term local payment

For the bare divergence coordinates the scalar functional is

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ρ(bareVector w) = −4(w0 + 4w1).
(TR52)

The coordinate map is an isomorphism for simple g. The equation vector (1 1) is annihilated; more generally ideal membership gives 2(2 = 0, forcing the same. Composing

with the degree-five isomorphism gives, including the correction-column readings zero and For real regular local vector , its degree-five card reader obeys

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M
θ−2(altoD Va)a

 = 1
12 Re Holt(V ).
(TR53)

The actual positive threshold cancels through altoD DCard exactly. The card contraction equals the local polynomial contraction by reality and nonwrapping, without periodic aliases. This is an exact coefficient identity, not an error estimate. The principal vector splits

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V = W + altoD B −altoD K + altoD D + E + F.
(TR54)

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The regular split y + D = K and additivity of cards prove exactness. The central K has zero Holt
contraction by local cancellation. The B contraction differs from midPay /12 by at most calorie /12.
The four other terms W, altoD D, E, F, after θ−2, each have coefficient norm at most Wave(−1).
Thus
M(θ−2V ) −midPay
 ≤4∥M∥Wave(−1) + calorie
12
.
(TR55)
12

The total coefficient-size envelope is Wave(16), henceWave(16), since coefficient size dominates norm. Finally midPaylandPayshock and landPay supply TR51’s remaining errors and limit. This reduces the continuum scalar reading to the local cancellation, four paid envelopes and payment comparison on the same threshold. The lifted scalar source retains the invariant coordinate isomorphism and exact regular split. Its card reader commutes with the threshold cancellation by TR53; TR54–TR55 recover the principal payment and its residual with all six contributions retained.

Principal scalar functional and local payment recovery. Retain the fixed orbit map, bare functional and their composition. Transport of the finite coordinate spaces preserves this composition and its operator norm in transported norms. The issued-to-core error and retained norm therefore recover separately the first TR51 error and the principal norm bill. Closed norm inequalities and continuous linear maps give the same bounds along the original extraction. The remaining scalar payments stay attached to their finite-level calculation, rather than being replaced by an assumed continuum value.

For invariant coordinates the retained row is explicitlybareVector = ( equation ideal vector, and4 give

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−4(w0 + 4w1) = −4(w2 −w2) = 0.

In particular its values on Ω= (0 1) and ∆= (0 0) are exactly 0 and4. The coordinate isomorphism and horizontal readout preserve the full row, its kernel annihilation and these two column values. The local card source retains the actual positive, regular support and nonwrapping margins. Before any estimate,

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θ−2 altoD V = DCard V,
M(DCard V ) = 1
12 Re Holt(V ).

Horizontal transport of the cards leaves this literal coefficient identity unchanged. For TR54 write the four residual contributions as = altoD. Additivity and the central cancellation give exactly

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4
X
a=1
M(θ−2Ra) + Re Holt(B) −midPay
12
.
M(θ−2V ) −midPay
12
=

The omitted central scalar is exactly zero by TR53 applied to; its minus sign remains in the full vector decomposition. Each of the four retained residual norms is bounded by Wave(1). Thus this equality recovers TR55 with precisely four operator-norm payments and the calorie12 term. Adding the actual comparison to landPay yields explicitly

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|M(θ−2V ) + g| ≤4∥M∥Wave(−1) + calorie + shock
12
.

The same retained threshold cancels in the total norm envelope. Together with issued-to-core error and the source’s finite payments, these identities recover all TR51 error terms and their limiting value, keeping the subsequent massive calculation and TS near-kernel estimates as the specified dependencies.

25.7 Massive payment comparison and exact quiet-loop cancellation

Keep mass strictly positive through the volume estimates. The scalar resolvent primitive and differentiated shell satisfy

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d
dx Re Ri(ex) = −Re Si(ex),
| Re Ri(ex) + Lx| ≤C (x ≤0),
L = −12g.
(TR56)

Each fixed has an eventual estimate with the same; the massive matrix is strictly positive and its finite-volume resolvent derivative converges. For fixed0, the mean value theorem and the two endpoint estimates give] with

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| Re Si(μi) −L| ≤2C/(b −a).
(TR57)

The mass can depend on volume; no uniform mass convergence is assumed. The uncorrected local payment obeys, for every moving mass in a fixed [

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Xi −Si(μi) −μ2
i Ki(μi) →0,
|μ2
i Ki(μi)| ≤C1μi.
(TR58)

Given tolerance first choose negative with4, then large with 2 that the right side of (TR57) is at most4. The resulting positive is fixed before the volume limit, so TR58 applies and Re. Finite local resolvent-jet bounds give the remainder after summing finite indices and multiplying by. The same bound makes the local second-derivative correction vanish: choose small fixed first for its bound, then large volume for the comparison error. Uniform analytic jets identify the local difference quotient with the second derivative. Therefore

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

2
X
cresti −1
−→−12g.
(TR59)
midPayi = Re
d,a
floftd,a,i

The crest has the limit and every correction vanishes. For a finite gauge-invariant loop roster its periodization is

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P(U) = creset
X
X
seats s
pα(s · U).
(TR60)
α

It is analytic and gauge invariant. For analytic gauge-covariant in a nonwrapping neighborhood, is local, analytic, gauge invariant and zero at identity because is zero there. Covariance follows by differentiating the invariant identity with conjugated direction; locality by differentiating

supported links. The quiet roster Hessian pairs to zero between a supported direction and the prescribed distant polynomial directions through degree two. With loop length bound prevents wrapping. The derivative of remains supported there, so differentiating twice gives two zero Hessian pairings. In the third difference entering Holt, constant-direction terms vanish by gauge invariance and the local Ward identity; all others are these Hessian pairings. The finite difference is zero, and the coefficient identity summing those differences gives Holt(( 0. Add the finite boundary compensation to the original truncated sum, giving quiet Then, precisely in TR54. The compensation has the small calorie envelope; the original derivative retains its larger local envelope. Gauge invariance and loop-length bounds persist, accounting for the exact zero term and its separately paid boundary correction. The lifted massive source retains positive mass, endpoint estimates, mean-value choice and local jet errors. Its payment reader recovers TR59 in the required tolerance–mass–volume order. The quiet-loop source retains the compensation roster and nonwrapping supports; its exact directional reader recovers the zero central contraction while preserving the separate boundary residual.

Positive-mass choice and quiet-roster recovery. Retain the finite-volume primitive, shell quantity, local correction and their actual positive mass in the source record. Their scalar readouts are identical after horizontal transport. Put. At fixed0, the two eventual bounds give a mean-value point) with

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h′
i(ci) = L −Re Si(eci),
|h′
i(ci)| ≤
2C
b −a.

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This recovers TR57 from only two endpoint bounds. The same points and masses are retained in
the lifted scalar reader. Writing the TR58 comparison error as ei = Xi −Si(μi)−μ2
i Ki(μi), one has
explicitly

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| Re Xi −L| ≤|ei| + C1eb + 2C
b −a.

Choose, then, then the volume threshold; the fixed lower mass 0 makes the movingmass comparison applicable. This retains the exact tolerance–mass–volume quantifiers rather than assuming a mass-uniform primitive limit. Every local jet correction retains its finite index list and bound. Multiplication by gives ; first fixing a sufficiently small positive and then the volume recovers its vanishing. The recorded analytic-jet estimate identifies the local difference quotient with the actual derivative in that comparison. Hence the finite sum of corrections in TR59 vanishes, and its crest reader recovers the same payment.

For the quiet roster keep the actual finite loop functions, seats, reset coefficient and compensation. Configuration realization gives . A horizontal lift of the supported covariant direction satisfies exactly

The finite local jet readers intertwine by the same directional chain rule. The roster length, support and separation inequality are retained, so the source’s zero supported/distant Hessian pairings and local Ward cancellation are identical scalar readings. Their finite third-difference sum therefore recovers Holt(() = 0 exactly. Finally the retained equality gives before estimating either term. Thus the central zero and the separate boundary compensation of size calorie recover the same split in TR54, with no loss of its sign or nonwrapping condition. The determinant and massive estimates below remain the dependencies for TR56–TR59.

25.8 Determinant differentiation and finite roster moments

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The local primitive contracts

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1
2 log det(A(U) + μ2I) −log det(B(U) + μ2I).
(TR61)

The first matrix includes physical quadratic and gauge-fixing terms; the second is the gauge comparator. At fixed 0 both are invertible near identity and

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d
dμ log det(F + μ2I) = 2μ tr(F + μ2I)−1.
(TR62)

Joint mass/chart analyticity commutes this derivative with the finite coefficient derivatives. Apply the normalized contraction yardsites cob and sum ordered Its derivative is times the resolvent contraction; supplies the other mass factor, proving TR56. The period-limit comparison has eventual error at most one from the massive integral satisfying log(1, uniformly in 01 and allowed regulator segment. Twelve ordered-axis contributions give Re log(1

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eventually for each fixed mass, the bound in TR56. The massive-symbol calculation below supplies
this uniform integral estimate. For each axis/color pair the quiet charges and relative offsets obey
X
α
cα = 0,
X
α
cα(δα)a = 0
for all a.
(TR63)

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Hence P
α cαF(δα) = 0 for every additive real offset function, by its four coordinate values. Each
two-leg loop Hessian is the sum of two cross-leg curvature products. Shift seat summation so the
first leg has common base. Its nonzero supported curvature lies in the one-link enlargement; the
other curvature is affine on the loop-enlarged neighborhood: curl T(x + δα) = C + F(x) + F(δα).
Charge summation gives zero by TR63 for both leg orders. Signed rotations preserve the argument,
with at most one-link support growth. Summing seats and rotations proves the exact Hessian zero
used in TR55, retaining the original nonwrapping margins. The derivative in (TR56) can also be

read directly from its determinant definition. The lifted determinant source retains positive mass, matrix factors and normalized ordered contractions. Its derivative reader recovers TR56 with the correct mass factors and twelvefold normalization. The roster reader retains offsets and both finite moments, making its zero Hessian contraction an exact source identity rather than a small-error replacement.

Massive determinant and finite-moment recovery. Retain the finite physical and gauge-comparison matrices with their actual positive mass. On their horizontal copies, use the transported orthonormal bases, so determinant and trace are the original scalar readouts. In particular

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∂μ
h
1
2 log det(A + μ2I) −log det(B + μ2I)
i
= μ tr(A + μ2I)−1
−2μ tr(B + μ2I)−1.

This is an identity on the same positive-mass chart, before local coefficient contraction. Joint analyticity allows the retained finite local derivatives to commute with. The site-normalized reader and ordered axis sum are unchanged by horizontal transport. The chain rule at therefore retains both mass factors and the sign in TR56.

The four-dimensional ordered off-diagonal axis list has twelve entries. Its fixed-mass comparison error at most one and the source’s uniform massive coefficient estimate give exactly

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| Re Ri(μ) + 12g log(1/μ)| ≤12(C0 + 1).

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With L = −12g and log(1/ex) = −x, this is the stated | Re Ri(ex)+Lx| bound. Its index threshold
may depend on the fixed mass; the same constant is retained for every permitted regulator segment.
The massive integral below remains its quantitative dependency.
For the roster, retain the finite signed charges and relative offset vectors. If F(δ) = P
a δaF(ea)
is additive, then
X
α
cαF(δα) =
X
a
F(ea)
X
α
cα(δα)a = 0.

After the source’s exact seat reindexing, the supported first-leg curvature is fixed. Multiplying it by the second affine curvature and summing charges gives that fixed factor times

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(C + F(x))
X
α
cα +
X
α
cαF(δα) = 0.

The other leg order, finite signed rotations and seats obey the same identity. Horizontal configuration readout retains their actual links, offsets and nonwrapping margins, so it also retains this zero before any limit or estimate. Consequently TR63 recovers the quiet Hessian cancellation used in TR55 and its full local third-difference reader, with the original compensation kept separate.

25.9 The local resolvent bound

We derive the local resolvent bound in (TR58). Use a periodic lattice of side, angular momentum spacing, and normalized momentum weight. Write lift() for its centered momenta. If, any subset of momenta with has at most

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(3R/h)4
(LR1)

members. Indeed their centered integer coordinates have absolute value at most. The lift is injective, and the integer coordinate box has at most (2 points. This upper bound remains true when the coordinate box wraps on the torus. For1, group momenta according to

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2jμ ≤|kx| + μ ≤2j+1μ,
j ≥0.

Assign boundary points to one of their adjacent groups. A group has at most (6 points by (LR1); its summand ( is at most (2. Therefore

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n−4 X
x
(|kx| + μ)−5 ≤
64
X
j≥0
2−j ≤2 · 64
(2π)4μ.
(LR2)
(2π)4μ

Only finitely many groups are nonempty at a given; extending their sum by zeros justifies the infinite geometric majorant.

Here is the integrand estimate to which (LR2) applies. At 0, scale an external momentum direction by The zero-field matrix, one-field vertex and two-field vertex have sizes 1 with bounded scaled derivatives through the finite orders in use. These follow from