Exact stress correction and local quantitative estimates · Part 3

their vanishing origin jets and the analytic symbol bounds proved in the massive-symbol section. Ellipticity gives the inverse size. Differentiating gives

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D(A−1) = −A−1(DA)A−1,

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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, with constants depending on the fixed directions and finite matrix dimensions but not on

For such an analytic integrand , let [! be its homogeneous Taylor coefficient. The scaled bound gives

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z3|[V ]3(p)| ≤Cz−2/3!,
hence
|[V ]3(p)| ≤C′z−5.

For the second coefficient it gives. The angular cube and1 give a fixed upper bound, so this too is bounded by. Summing the finite matrix entries and applying (LR2) bounds both contracted coefficients by a constant times

The local readings are explicit finite coefficient selectors. For a scalar function on integer arguments set

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Pan f = 1
2(f(1) −2f(0) + f(−1)),
Flop f = 1
12[8(f(1) −f(−1)) −(f(2) −f(−2))].

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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,

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∃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 and the moving mass family. Once is fixed, eventually, as required in (LR2). Sum (LR3) over the finitely many tensor indices. Multiplication by yields, the remainder bound in (TR58).

For the local second-derivative correction, its exact massive comparison has error tending to zero at each fixed 01, and its massive term is at most by the second-selector version of the same argument. Given 0, choose 01 with2, and only then take the volume large enough that the comparison and selector errors total less than2. This proves vanishing of that correction, including the actual second Taylor coefficient rather than merely its finite difference. Consequently

Source–readout formulation. In the lift the masses, the selectors Pan and Flop and the local lens are numerical readers of the realized determinant films of Section 14. The mass is an auxiliary spectator parameter with no target counterpart; it is removed by the two-step limit and is never set to zero inside an integral. The quantifier order in (LR3), with the constant chosen before the lower mass cutoffand the threshold after it, is part of the recovered statement: for the local second-derivative correction the massive term is at most a constant times, and one chooses first and the volume afterwards. The selectors are exact on polynomials of the stated degrees and differ from the corresponding derivative coefficients by terms vanishing with the momentum spacing, uniformly for masses in a fixed interval [ 1]; that discrepancy is retained until the volume limit is taken.

25.10 The logarithmic momentum coefficient and its uniform remainder

Use and. For the actual massive symbol’s half second external derivative

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Z
B
Re Fμ(k)d4k.
(TR64)
α(μ) = c4

The change from the unit-period torus has Jacobian (2. At fixed positive mass the symbol is analytic and its first two external derivatives are bounded on a compact external interval times the closed internal cube. Dominated differentiation moves them under the integral; the half coefficient and definition’s factor two give precisely TR64. Uniform symbol estimates are

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|Fμ(k)| ≤A(r + μ)−4,
|Fμ(k) −PG(k)| ≤Dr−3 + Dμ2r−6
(μ ≤r),
(TR65)

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with common A, D ≥0 for 0 < μ ≤1 and
PG(k) = 4Cadj

k2
1/r6 −4k2
1k2
0/r8 −r−4
.
(TR66)

The distinct axes are the external direction and polarization. Packaging the admissible symbol/mass families at scales and supplies the same-mass estimates without shifting to a preferred radius. On, polar integration gives

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Jμ =
Z
r−4d4k = 2π2 log(1/μ),
Z
r−3d4k = 2π2(1 −μ),
Z
r−6d4k = π2(μ−2 −1).
(TR67)

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Rotational invariance and P
a k2
a = r2 give
R k2
a/r6 = Jμ/4.
Rotate to (ea ± eb)/
√
2 and add
fourth-power identities; odd terms cancel, giving diagonal fourth moment three times the mixed
one. Summing over b gives
R k2
ak2
b/r8 = Jμ/24 for a ̸= b. Consequently

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Z
Aμ
PGd4k = c4 4Cadj(1
4 −4
24 −1)2π2 log(1/μ)
c4
= −11Cadj
24π2 log(1/μ) = −g log(1/μ).
(TR68)

Split the cube into, this annulus and 1. The inner integral is at most times the radius- ball volume, giving2. The outer integrand is at most, giving vol(). Annular

difference error integrates to at most 2. Thus absolute bounds before region summation yield

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|α(μ) + g log(1/μ)| ≤c4[Aπ2/2 + A vol(B) + 3Dπ2].
(TR69)

This is TR56’s mass-independent constant, with the factor eleven and normalization explicit. The lifted momentum source retains the actual symbol, angular measure, same-mass comparison and three regions. Its pole reader recovers TR68 by finite angular moments; its residual reader retains all absolute integral errors in TR69, so logarithmic recovery preserves the uniform mass bound and constants.

Angular pole and uniform remainder recovery. Retain the actual massive symbol, external derivative, angular cube and normalized measure. Rotational invariance determines the required angular moments without a sphere normalization convention. The horizontal matrix realization uses transported bases, so its scalar half second derivative is the same) at every fixed positive mass. Thus

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Z
αZ(μ) = c4
B
Re F Z
μ (k)d4k = α(μ),
P Z
G (k) = PG(k).

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The coordinate change k = 2πx retains its Jacobian, and the analytic external derivatives commute
with this same compact integral at fixed mass. The constants in TR65 retain their common families
and same mass, rather than a mass chosen separately at each radius.
For the rotational moment reader put A4 =
R
Aμ k4
0/r8 and B4 =
R
Aμ k2
0k2
1/r8. The retained
rotation identities give A4 = 3B4 and A4 + 3B4 = Jμ/4, hence B4 = Jμ/24. These are exact finite
linear readouts of the angular integral. Consequently the pole factor is exactly

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1
4 −4
24 −1 = −11
12,
c4 4Cadj

−11

2π2 = −11Cadj
24π2
= −g.
12

This recovers TR68 with its original logarithm, sign and color normalization. More explicitly, define the remainder reader by the same three measurable regions:

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α(μ) + g log(1/μ) = c4 Re
 Z
r≤μ
Fμ +
Z
Aμ
(Fμ −PG)
+
Z

.
B∩{r>1}
Fμ

Boundary choices have zero Lebesgue measure. The first integral is bounded by2 and the last by vol(). The middle absolute integral is bounded exactly by 2 Every scalar integrand, region and normalization is unchanged under readout, so the same mass-independent TR69 constant is recovered. The vector/ghost and weighted matrix estimates below remain the explicit dependencies establishing the pole and TR65 bounds used in this recovery.

25.11 Vector and ghost pole traces and massive scalar assembly

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For orthonormal adjoint color tr T 2
b = −Cadj. At external polarization one and momentum te0, the
vector vertex is (pI4 + Q(t)) ⊗Tb, the ghost vertex pTb, where

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p = −2ik1,
Q(t)ab = 2it(δb0δa1 −δa0δb1).

Free propagators are and shifted denominator . Spatial/color traces factor, with tr[(. Hence

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Hgh(t) = −Cadjp2
r2D(t) ,
Hvec(t) = −Cadj(4p2 −8t2)
r2D(t)
.

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For r > 0 denominators are locally nonzero. Twice differentiating with D′(0) = 2k0, D′′(0) = 2,
p2 = −4k2
1 gives
H′′
gh(0) −1
2H′′
vec(0) = 2PG(k).
(TR70)

This retains the vector determinant’s half weight and pole sign. In the massive log determinant let denote the upper and bubble second derivatives. Before bounding,

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Fμ = 1
2[ 1
2(Uv −Lv) −(Ug −Lg)].
(TR71)

Their common order- bounds at give TR65’s first bound. On use and the more precise estimates

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|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)

TR70–TR72 bound the difference from . Since in the angular cube, take = max() for TR65. Since multiplication by 12 decreases the absolute value, (TR71) yields the first bound in (TR65) upon taking. All constants are chosen over all admissible regulator symbols, masses and internal momenta before restriction to. The weighted matrix estimates establishing TR72 follow next. The lifted pole source retains the explicit tensor vertices, color trace and shifted denominator. Its second-coefficient reader recovers TR70 with both determinant weights. The massive reader TR71 is an exact assembly before absolute bounds; retaining the common symbol family transports TR72 into the same-mass comparison used by the logarithmic integral, with no regulator-dependent constant selection.

Weighted vector/ghost scalar recovery. The horizontal symbol source retains the spatial and adjoint-color bases, vertices and external momentum parameter. Matrix composition and traces are unchanged in those transported bases. For 0, the actual denominator obeys

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(D−1)′′(0) = 8k2
0
r6 −2
r4 .

With, direct substitution in the retained bubble readers gives

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H′′
gh(0) = 32Ck2
1k2
0
r8
−8Ck2
1
r6
,
H′′
vec(0) = 128Ck2
1k2
0
r8
−32Ck2
1
r6
+ 16C
r4 .

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Their difference with the exact half vector weight is 2PG. Thus the lift recovers TR70 and its pole
sign before angular integration, retaining the adjoint trace normalization and both vector and ghost
contributions.
For the massive scalar define the retained bubble discrepancies Ev = Lv −H′′
vec(0) and Eg =
Lg −H′′
gh(0). TR70–TR71 give the exact identity

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Fμ −PG = 1
4Uv −1
2Ug −1
4Ev + 1
2Eg.

All four terms and their coefficients are identical horizontal scalar readouts. Hence the common order- bounds recover the first TR65 bound at. On the retained family and the triangle inequality give, with its nonnegative constants,

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|Fμ −PG| ≤(u + x)T + A + B
T 3
+ (A + B + v + y)μ2
T 6
.

Since, the recorded common constant max(() recovers the second TR65 bound. The weights have only been enlarged for this estimate, not changed in the scalar identity. The mass remains the same in the symbol, its propagators and comparison. Uniform weighted matrix estimates below retain the necessary whole-family constants and supply the remaining TR72 dependency.

25.12 Weighted matrix calculus for massive symbols

Use fixed finite matrices and a submultiplicative norm with tr. Size at scale means (TR73)

Products have size, constant at most 4, retaining the two middle Leibniz terms. Products of several differing factors are compared by replacing one factor at a time; (TR73) controls every term. Sums add constants and bounded scalar multiplication changes size accordingly. If , then

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(A−1)′ = −A−1A′A−1,
(A−1)′′ = 2A−1A′A−1A′A−1 −A−1A′′A−1

give inverse size, with weighted constants . Analyticity follows from the determinant or local geometric inverse. The exact comparison

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A−1 −B−1 = −A−1(A −B)B−1
(TR74)

turns discrepancy into Telescoping compares finite products with all raw derivative coefficients retained. Along1, zero-field matrix, one-field vertices (including incoming shifts) and two-field vertex have sizes 1. Symbol products have two one-leg and four two-leg Leibniz placements; inversion preserves the bounds with inverse size. The zero-field error from has jets below order three zero; the one-field error from its continuum linear vertex has jets below order two zero. Common analytic-strip Cauchy bounds on the fixed ball, with chain-map norms included, give Taylor sizes and weighted derivatives through two. The two-field vertex has unscaled bounded derivatives. Integrating highest derivatives along segments proves for the vanishing-order- errors. For admissible 1, lattice magnitude4 gives

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s(k)2 + μ2 ≥T 2/64.
(TR75)

The ghost matrix is this scalar times identity; vector coercivity is at least min(, with common 0. Both inverses are uniformly), including the joint small-mass/momentum regime. The lower bubble) has size; its second derivative and trace are). For upper), massless is-independent and is unscaled bounded, giving). The regulator vertex has size; after the inverse and two derivatives it gives). These prove TR72’s upper bounds; full sizes and bounded give the rough bound. For bubbles at, set. The zero-field error is , so TR74 gives inverse error. Vertex error is Telescoping four factors gives bubble error and second trace error, proving the last TR72 bounds and TR65.

Weighted matrix and propagator recovery. For each finite symbol index space use horizontal maps with and transported norm. This completes the matrix power counting underlying (TR65); the origin-jet identities and common strip bounds invoked here are the corresponding symbol inputs. If, then on its horizontal image

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(AZ)(j) = JA(j)Q,
(AB)Z = AZBZ,
(AZ)−1 = JA−1Q.

Thus all weighted derivative norms and traces in TR73 agree exactly. In particular the inverse derivatives retain their two ordered terms and their minus term; no factorial convention is changed. The inverse bound is recovered with common constant + at size. These sizes follow from precise Taylor cancellations at the origin.

For two invertible families the horizontal realization of TR74 is again the exact difference of inverses. For ordered factors the comparison reader is the finite identity

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m
X
A1 · · · Am −B1 · · · Bm =
j=1
B1 · · · Bj−1(Aj −Bj)Aj+1 · · · Am.

Differentiating twice retains every product placement and binomial coefficient. Hence the relativeerror transfer and bubble comparison are recovered before their norm estimates.

The source’s scale assumptions and coercive inequalities are unchanged by the transported inner product. Explicitly gives ), while16; these imply the stated weaker bound64. With the same positive ellipticity constant, both massive inverse readers therefore retain uniform size. The positive mass controls the zero momentum branch. The lower bubble has size and its second derivative has trace size the upper reader the unshifted inverse remains independent of the external parameter: the massless raw second vertex derivative costs no additional scale, giving; the regulator piece instead has

size) and its second derivative gives. All these are the original ordered products on the horizontal image. On set. The exact relative zero-field discrepancy is; the inverse identity gives size and the vertex discrepancy has size. Each term of the four-factor telescope therefore has size, and its second derivative trace has size, namely . This recovers TR72 on the same-mass family and retains the origin-jet/strip estimates below as its symbol inputs.

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For analytic functions, ray bounds f(tx) = O(tu) imply vanishing jets below u: a first nonzero
homogeneous polynomial would be nonzero on a real diagonal by polarization. Conversely bounded
order-u derivatives give uniform Taylor bounds. Ray constants may vary; final derivative constants
are common. For qa(k) = eika −1, the difference from ika is second order. Product expansion
therefore makes P
a qa(−k)qa(k) −|k|2 vanish to order three, proving the ghost cancellation. The
actual vector matrix is Avec = d∗
1M(k)d1 + d0d∗
0, with M(0) = I.
Subtracting M = I gives
two first-order exterior factors and a first-order metric error, hence third order. The free exterior

identity makes the matrix the lattice scalar times identity. Thus vector cancellation has the same order with longitudinal gauge fixing retained. This uses the actual quadratic symbol normalization, including its longitudinal gauge-fixing term. The origin cancellations used in this calculus can be stated independently of the family bounds. The lifted matrix source retains raw derivatives, placement order, coercive inverse and analytic jets. TR73–TR74 transport its products and inverse readers exactly before bounding; TR75 keeps the zero-mode branch controlled by the actual positive mass. The resulting bubble and upper readers recover the quantitative powers in TR72 on the same admissible symbol family.

25.13 One-field vertices and determining position Ward columns

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The exact ghost vertex is

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[−e−i(ka+pa)qa(k) + qa(−k −p)eika](Tb)xy.

Scale both momenta by and divide by: the limit is. Analyticity makes its error second order. The gauge-fixing contribution has linear part [ Adding Wilson curvature gives exactly

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[δij(−i(2ka + pa)) + 2i(pjδia −piδja)](Tb)xy.

The curvature sign follows by second differentiation of the group commutator and third-order chain rule: its Hessian is the two first-derivative products minus their reversed products, and zero first-jet group factors contribute zero. Prescribed soft order three makes a permitted perturbation’s one-field vertex second order, so the same continuum linear vertex survives. Common analytic-strip bounds supply the uniform derivatives for this entire symbol family.

Origin jets and one-field vertex recovery. The symbol source retains its actual analytic lattice matrices and coefficient maps. On horizontal index spaces the intertwining gives , so every zero origin jet and every bounded derivative used in the Taylor estimate is preserved. For a matrix-valued analytic function the ray argument applies to each fixed coordinate, and finite coordinate reconstruction preserves the common norm bound. Thus ray-dependent constants are not silently promoted to uniform constants; the latter remain supplied by the retained strip derivative estimates.

Write) with the corresponding zero jets. The exact ghost discrepancy decomposes as

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qa(−k)qa(k) −k2
a = Ra(−k)qa(k) + (−ika)Ra(k).

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Each term has third-order zero jet.
For the vector matrix the exact additional discrepancy is
d∗
1(M −I)d1, again a product of three first-order factors. The free exterior identity and longitudinal
term are retained in its horizontal matrix, recovering both zero-field cancellations used in the
weighted propagator comparison.
The actual one-field ghost expression has the recovered linear jet

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−[1 + O(t)] [itka + O(t2)] + [−it(ka + pa) + O(t2)] [1 + O(t)] = −it(2ka + pa) + O(t2)

along each fixed ray, with its same adjoint matrix factor. The retained gauge-fixing and Wilson curvature jets give the displayed continuum vector vertex, including its antisymmetric curvature sign. For clarity the mixed Hessian of the commutator along

is. This is the same ordered local derivative readout after horizontal transport; the finite third-order chain rule therefore keeps the source’s sign and all zero-first-jet cancellations.

The permitted soft perturbation retains zero one-field jets below order two. Adding it leaves both displayed linear vertices unchanged. The common strip bounds are also retained on the entire admissible family, so these exact jet identities yield the same weighted Taylor errors and the same continuum pole. This closes the symbol input dependency in TR73–TR75 without replacing a finite lattice vertex by its limit before differentiation.

For real degree-four frame with the stated symmetry and conjugation compatibilities, exact decomposition gives simultaneous real coefficients

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P −1
s
(s4Tab) = κsT 0
ab + dsδabO + esδab1,
O = Oscale.
(TR76)

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The physical result gives ds/u2
s →d2 for the same leading coupling normalization, including Wilson
charge; thus ds →0. At (e0, 0) with exterior O, let Es be the first-point divergence of the moment
with PsT 0, and Bs that with Ps(δO), the first time derivative of the scalar pair. Derivatives in
unscaled positions supply chain factor s. Let Rs, Qs be the translation shell-flux tests with stress
PsT 0 and Ps(δO), inserted and exterior PsO. Conservation and Ward give exactly

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κsEs + dsBs = 0,
κsRs + dsQs = Bs.
(TR77)

The flux integral scales by (volume, cutoffderivative); the translated-field derivative adds . Both sides therefore have the same factor, which cancels. Unit divergence and flux vanish exactly. These are separated tests, with no contact identity used. Compact separated frame-jet comparison yields (TR78)

Free Ward gives the second limit; the free scalar moment = 3 dim g, gives

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B0 = −8cG ̸= 0.
(TR79)

The second TR77 equation and0 imply, hence1. The first then gives

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Es/u2
s = −ds/u2
s
κs
Bs −→−d2B0.
(TR80)

Thus the conservation column is determined at its first even order; the other three columns need only order zero. The lifted Ward source retains the exact tensor decomposition, scaled shell window and both chain factors. Its finite-column reader recovers TR77 before any limit. The nonzero scalar derivative then recovers the stress amplitude and TR80 on the same position frame, preserving the distinction between formal even coefficients and later real error estimates.