Gaussian short-distance comparison · Part 2

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This is an integrable common bound in the finite-dimensional configuration space. The transported
step function is literally the same function, so dominated convergence commutes with its readout.
For the actual rounding cell Cm,o the constant reading and its volume give the exact finite
identity
Z

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Cm,o
ψ(ρmx)KZ
m(ρmx) dx = h4n
m ψ(hmo)Km(o).

Summing the cells intersecting the fixed support recovers PT16 with its full 4-dimensional normalization. Eventual nonwrapping uses the unchanged. In arity zero this identity is the one-point integral with weight one. Through the same field-address identity the reduced order-zero polynomials represent the fixed free labels, so this profile recovery gives PT14 once the corrected covariance and moving Wick-row limit below are supplied. Those remaining dependencies are retained rather than replaced by an independent choice of continuum profile data.

Reduce the row limit to covariances by indexing every occurrence of every generator separately. Write) for its insertion slot and for its generator label. The order-zero Gaussian product moment on a finite occurrence set satisfies integration by parts:

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|A|Mm(A) =
X
X
b∈A\{a}
Cm(a, b)Mm(A \ {a, b}).
(PT17)
a∈A

The empty moment is one by probability normalization. Continuum Wick moments obey the same recursion with. Induction on, the covariance limit and finite summation prove convergence after division by the positive integer Distinguishable occurrences preserve multiplicities of repeated generators. Finite polynomial expansion then proves the limit for all allowed sections; the field-address identity makes the limit exactly their free Wick moments, without a new fit. Applying this along every refining subsequence gives PT16’s moving-limit hypothesis. The corrected covariance limit, including its finite-regulator subtraction, is supplied next. For because the Gaussian law is a probability measure. The lifted row source retains occurrence indices, reduced polynomial addresses, grid cells and their actual volume. Its Gaussian recursion reader preserves multiplicities and normalized empty moments; its profile reader is PT16. Joint moving convergence supplies the common bound needed by the integral reader, recovering PT14 without a pointwise-only convergence substitution.

22.6 Private Gaussian marks and corrected bilinear covariance

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Fix the maximal jet grade d in the row and its finite coordinate keys a.
The regulated real
potential Z has precision P and covariance P −1; its symmetrically averaged linear jet coordinate at
x is kT
h,x,aZ, using the perturbative integral’s same density. For sine/cosine waves, axes and color
labels c, define the infinite lattice subtraction matrix by their responses lh,a,c(p):

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Sh(a, b) = (2π)−4
Z
[−π,π]4 δ(p)−1 X
c
lh,a,c(p)lh,b,c(p) dp.
(PT18)

Responses are bounded by continuity on the cube and is integrable in four dimensions. Entries exist, symmetry holds, and

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X
!2
dp ≥0.
vT Shv = (2π)−4
Z
δ(p)−1 X
a
valh,a,c(p)
c

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Thus Sh is the covariance of a possibly degenerate real centered Gaussian vector, requiring no inverse.
Give each insertion i an independent such vector Yi, independent of Z, and set Xi,a = kT
h,xi,aZ+iYi,a.

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Their bilinear covariance is

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Ch((i, a), (j, b)) = kT
h,xi,aP −1kh,xj,b −1i=jSh(a, b).
(PT19)

Mixed terms vanish by independence/centering; private marks are independent across insertions and their imaginary coefficients give the minus sign within one insertion. No conjugation occurs. Same-slot entries need not vanish at finite regulator. The normalized perturbative constant term is exactly the product of reduced insertion polynomials evaluated at these and then averaged. Weight and vacuum denominator have constant coefficient one. Each linear insertion’s constant term is its physical coordinate plus imaginary private mark; finite polynomial expansion commutes with constant extraction. Private integration gives the prescribed within-insertion subtraction; independence combines it with physical integration. Polynomial Gaussian integrability, from and finite Gaussian moments, justifies expansion and Fubini. For a real centered linear Gaussian probe of variance, its moment generating function exp(2) gives Apply to arbitrary real linear combinations and compare coefficients to obtain by polarization

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|A|E
Y
α∈A
Xα =
X
X
β∈A\{α}
E(XαXβ)E
Y
γ∈A\{α,β}
Xγ.
(PT20)
α∈A

Complex multilinear expansion extends this from real probes, including degenerate laws. Repeated generators retain distinct occurrences; the empty product is one. This is precisely PT17’s finite recursion.

Private-mark source and exact polynomial recovery. The source retains the real potential law and a separately named private mark for each insertion. A concrete realization of a possibly degenerate mark is , with independent standard real Gaussian vectors, independent also of. Its pushforward covariance is exactly; the positive square root requires no inverse. On the resulting product Gaussian source define the complex linear readout

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Xi,a(Z, η1, . . . , ηn) = kT
h,xi,aZ + i(S1/2
h
ηi)a.

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Private coordinates are indexed by insertion, not by distinct physical position or by generator name.
Thus repeated occurrences in one slot share its mark, while different slots retain independent marks.
The same readout on the horizontal potential copy uses the transported Gaussian law and coefficient
maps, giving precisely the original Xi,a. Its bilinear second moment is PT19, with i2 = −1 and no
conjugation. This preserves finite-regulator same-slot terms rather than setting them to zero.
For an insertion polynomial Qi and a fixed physical coordinate vector zi = (kT
h,xi,aZ)a, the
private integral has the exact finite polynomial form

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



−1
X
exp
Qi
(zi).
EYiQi(zi + iYi) =
a,b
Sh(a, b)∂a∂b
2

The exponential terminates on each polynomial. This follows by finite Taylor expansion and Gaussian moments, or by comparing polynomial coefficients of the Gaussian generating function. Independence of the marks gives, conditional on the same physical

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EY1,...,Yn
Y
i
Qi(zi + iYi) =
Y
i
EYiQi(zi + iYi).

Integrating the physical law recovers exactly the normalized perturbative constant term. The weight and vacuum constant terms are one, so no additional denominator appears at this order.

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Polynomial integrability justifies each finite expansion and this product-law integration, even for a
degenerate mark law.
For occurrence indices α ∈A, set G(t) = E exp(P
α∈A tαXα). Its Gaussian coefficient identity
is



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X
1
.
G(t) = exp
α,β∈A
tαCh(α, β)tβ
2

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The identity ∂tαG = P
Only its finite derivatives at zero are used.
β Ch(α, β)tβG, followed by
extraction of the multilinear coefficient and summation over α, recovers PT20 and its factor |A|.
Distinct occurrence variables preserve all repeated-generator multiplicities. The empty moment
remains one. Induction over |A| then transports PT17 exactly and, once the corrected covariance
limit is supplied, recovers every finite polynomial moving-row limit with the original field addresses.

The corrected covariance limit has two roles: for distinct limiting positions it tends to the Maxwell kernel) on the corresponding free solution-jet functionals; for the same slot it tends to zero: (PT21)

This vanishing follows after regulator removal. For the finite signed coordinate group the averaging identity beginning its proof is

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A(wideLinearh,x) = cS
X
s∈S
A ◦turns ◦radialMap(charts,h,x).
(PT22)

Write for the inner functional. The next sampling/reconstruction identity identifies its free limit. The lifted Gaussian source retains and the separate private, with the same finite density and explicit subtraction matrix. The polynomial reader is expectation after ; PT19 proves compatibility with the prescribed subtraction. Keeping bilinear rather than Hermitian covariance recovers the sign and same-slot remainder exactly. PT20 supplies the finite occurrence reader without a covariance inverse, so degeneracy introduces no lost branch. The limit reader retains both the cross-slot kernel and the corrected diagonal tending to zero rather than imposing Wick subtraction prematurely.

22.7 Averaged reconstruction and corrected covariance limits

Radial reconstruction inverts sampling on free solution jets, so

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plains(As) = A ◦gaugeJet .

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Independently transformed stencils with inverse turns therefore give the same free two-point form
for all s, t, with fixed relative endpoints. The finite covariance is c2
S
P
s,t ordinarys,t(h; x, y; As, Bt).
Expanding both wave factors in PT18 gives the same finite double average of infinite-lattice shell
forms; integrability justifies interchange with the integral. The separated chart limit gives the same
free pairing for each summand. At equal endpoints the return chart limit makes each ordinary form
minus its shell form tend to zero. Finite averaging proves both PT21 and the separated limit. The
reporting schedule is

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hm = L−m,
Nm = 4Lm2+m+1,
hmNm = 4Lm2+1,
L > 1 integer.
(PT23)

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Thus Nmhq
m →∞for every fixed integer q, and 4 | Nm. Center positions at the quarter-period
point. Convergent centered physical coordinates eventually have integer coordinates smaller than

4, so fixed stencil shifts of differences do not wrap and have the required physical limit. Sums of positions are twice the quarter center plus bounded physical displacement; their lifts have a coordinate at least4 in absolute value. Their physical distance tends to infinity, including at coincident observed positions, supplying the inversion-image return condition. PT20 induction now gives moments with zero within-insertion pairings and Maxwell cross-insertion pairings, exactly the reduced free Wick rule. Finite polynomial expansion and PT15–PT16 give every moving row and profile limit. This completes the Gaussian constant-coefficient step needed in the leading perturbative comparison.

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An elementary stencil is a finite signed shifted-edge sum, with axis, offsets and ordered deriva-
tive list Da. Its two-point form consequently has four terms, with signs +, −, −, +. Its Fourier
polynomial factors into sign, translation character and difference multipliers. Multiplication gives
the exact identity
X
Y
atbsχp(z + ut ± vs) = σa,b,±χp(z + ua,± + ub,±)
qν(p).
(PT24)
t,s
ν∈DaDb

Multiplying by the inverse-Laplacian weight and summing identifies covariance with the differentiated scalar torus kernel, including the same sign and factor. On the infinite cover, sum sine/cosine channels using cos cos+sin sin = cos(). Unequal axes or colors give zero; matching ones give the offset-difference cosine. Integration against (2 is the cover scalar Green function, justified by cos1 and integrability. Curvature coordinates are differences of two stencils, giving the same four signs + + in torus covariance and cover subtraction. PT24 reduces separated terms to scalar differentiated limits and inversion-image terms to the zero far-distance limit. At coincidence subtract the cover value before rescaling: the return estimate vanishes after any fixed inverse power of by PT23. Finite array coordinate functionals span each raw jet-response dual; bilinear expansion therefore extends these elementary limits to arbitrary chart functionals in PT22; the responses are arrays of color components, vector-index pairs and derivative-index words, so coordinate projections span the dual of each finite array space. The lifted stencil source retains the exact Fourier factors, turns, relative endpoints and common torus/cover subtraction. Its finite average reader recovers the free solution-jet functional, and PT24 transports covariance to scalar kernel readers without replacing any stencil by its limit. The quarter-period placement and PT23 preserve the far-image and return hypotheses needed for the corrected diagonal.

Finite averaged-covariance recovery. Record the actual signed group and every ordered pair ) in its double average. Denote the corresponding torus and infinite-cover forms by and . The exact same-slot identity is

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Ch((i, a), (i, b)) = |S|−2 X
s,t∈S
(Oh;s,t −Hh;s,t).

Horizontal transport of each stencil functional, sampling map, turn and radial reconstruction leaves these scalar forms unchanged. On free solution jets the equality plaingaugeJet recovers the same output for each signed record. In particular

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|Ch((i, a), (i, b))| ≤max
s,t |Oh;s,t −Hh;s,t| −→0.

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The subtraction is inside each summand before this limit. For separated slots no private subtraction
occurs; if K is their common free pairing, the corresponding estimate is

|S|−2 X
s,t
Oh;s,t −K

≤max
s,t |Oh;s,t −K| −→0.

Finite group size makes these implications exact and preserves the independent transformations of both stencils. The subsequent scalar return, separated and far-image estimates supply these entrywise limits with their stated rescalings. The schedule is retained literally. Its explicit growth is

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Nmhq
m = 4Lm2+(1−q)m+1 −→∞
for every fixed integer q.

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For a bounded physical displacement and any fixed finite stencil offset, their ratio to hmNm tends
to zero. Hence differences have the stated nonwrapping lifts; a sum at twice the quarter-period
center has lifted distance at least hmNm/4 eventually, after allowing the bounded displacement
and offsets. The same geometry holds when the observed endpoints coincide. No observation point,
stencil shift or period is reselected in the horizontal lift.
For explicit scalar-stencil recovery set Pa(p) = P
t atχp(ut).
Then the left side of PT24 is
χp(z)Pa(p)Pb(±p). Multiplication of the actual finite factorizations of Pa, Pb gives exactly its sign,
offset and ordered difference factors. The horizontal reader retains these coefficients, so both the
Fourier sum and the infinite-cover integral have their original normalization and h−|Da|−|Db|−2

price. If and are curvature readings, their bilinear covariance, and separately its cover term, is exactly

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B(La, Lb) = B(La,1, Lb,1) −B(La,1, Lb,2) −B(La,2, Lb,1) + B(La,2, Lb,2).

Thus the common four-term subtraction is recovered before scaling. Finite coordinate expansions then recover arbitrary chart functionals. The occurrence recursion uses these very covariance entries, so its limit retains zero within-insertion pairings and the specified Maxwell cross-insertion pairings. This closes the moving-row/profile recovery conditional on precisely the scalar estimates developed next.

22.8 Scalar return with every fixed jet rescaling

On put) and cos(2) for fixed integer displacement, assigning(0) = 0 as in the inverse-Laplacian zero-mode convention. The integral is unaffected and the symbol bound gives. Outside null cell boundaries, quadrature representatives obey and when nonzero. Thus and, including the zero cell. For 0 and, the representative norm is at least2. Bounded derivatives of cosine and symbol give

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by the exact quotient subtraction

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(cosa(y) −cosa(x))d(y)−1 + (d(x) −d(y))d(y)−1d(x)−1 cosa(x).

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Using r/2 gives the outer error AaN−1r−4. The inner singularity is integrable in four dimensions,
with
Z
∥x∥<2r
∥x∥−2dx = r2
Z
∥u∥<2
∥u∥−2du.

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Split the error at r to obtain

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| Re GN(a) −G∞(a)| ≤Car2 + AaN−1r−4.
(PT25)

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The torus sum is exactly
R
Q fa(vN(x))dx since cells have volume N−4, and the cover integral is
R
Q fa by p = 2πx. Even N makes symmetric representatives tile this cube. For fixed integer d ≥0
choose r = hd+1; PT23 eventually ensures r ≤1, 8/N ≤r. Then

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h−d| Re GN(a) −G∞(a)| ≤Cahd+2 + Aa(Nh5d+4)−1 −→0.
(PT26)

This retains every fixed inverse mesh power needed by a finite jet. The lifted return source keeps the zero-mode convention, representative cells, exact quadrature normalization and the singular/regular split. Its scalar reader is the difference of torus and cover values before rescaling; PT25 transports that difference with an explicit bound and PT26 recovers its vanishing at the jet reader’s actual scale.

Quadrature subtraction and fixed-power recovery. The scalar return source retains the actual representative cells, zero-mode value and fixed integer displacement. Its two scalar readers and their exact discrepancy are

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RN(a) = N−4 X
u
fa(u) =
Z
Q
fa(vN(x)) dx,
R∞(a) =
Z
Q
fa(x) dx,
EN(a) =
Z
Q
[fa(vN(x)) −fa(x)] dx.

Thus) before multiplication by any inverse mesh power. The normalization (2 under and cell volume are retained entries of these readers. Pulling the source record to its horizontal section changes neither reader nor their difference.

For a nonzero representative the inequalitiesgive the integrable bound 26 for the discrepancy; in the zero cell)) = 0 gives the same bound directly. All these statements are up to the stated null boundaries and zero point. Splitting at gives explicitly

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|EN(a)| ≤26
Z
Q∩{∥x∥≤r}
∥x∥−2 dx + AaN−1r−4 vol(Q).

The first term is at most a fixed constant times in dimension four and vol() = 1. This recovers PT25 using the combined subtraction rather than separate singular estimates after rescaling. For fixed0 the retained choice yields exactly

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h−dr2 = hd+2,
h−dN−1r−4 = (Nh5d+4)−1.

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Along PT23, Nhd+1 →∞makes 8/N ≤r eventually, and Nh5d+4 →∞gives the second vanishing
term. Consequently the same reader recovers PT26 for each fixed power required by a finite jet. If
a fixed finite stencil calculation uses P
t λtEN(at) with constant coefficients and bounded rescaling
degree, the finite sum of these bounds proves its return limit with the same schedule, after taking
the maximum of finitely many thresholds. This transports the actual torus-minus-cover cancellation
used for arbitrary finite chart functionals.

22.9 Whole scalar kernels from annular readers

Suppose) for every 0, and with constants independent of eventually for each fixed

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|Am(s)| ≤Csp,
|Fm −Am(s)| ≤B/s,
|A(s)| ≤Esp,
|F −A(s)| ≤D/s,
p ≥1.
(PT27)

Then. Choose small1 with (3, then integer with ( Finite annular telescoping gives). Take late enough for its error below3. Add large-cutofftails and small-cutoffsizes to bound. The order is; cutoff-uniform convergence is not assumed. For differentiated torus kernels at nonzero physical separation use + 2. Smooth Fourier cutoffs have bound and complement away from zero. Their mesh conditions (22 hold eventually for fixed by0. Annular Riemann convergence and PT27 give the separated scalar limit. Its continuum candidate is. With Fourier character integrable heat regulator 0 gives inverse transform

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∂DG(z) −∂D(G(z)e−π2∥z∥2/b),
z ̸= 0.

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Set b = v−2; the second term is v|D|+2∂D(Ge−π2∥·∥2)(vz), tending to zero by rapid derivative decay
at infinity. For a physical displacement whose norm tends to infinity, the same comparison principle
applies with F = 0, A(s) = 0. At fixed Fourier cutoff, dominated convergence removes heat inside
the integral. Annular tail bounds persist and sum as P
j C/(2js) ≤2C/s, supplying the continuum
estimates in PT27. For diverging physical separation take F = A = 0. Summation by parts gives

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2π
∥h lift(x)∥
|D|+3
,
| annulush,s,D(x)| ≤CD
s

which vanishes for fixed. Low-frequency and tail bounds remain valid once distance is at least one. PT27 proves the far-image limit.

Whole-kernel recovery from annular differences. Retain both the annular reader and its small-cutoffanchor in the source. For fixed 0 and finite integer0, put. The exact telescoping discrepancy is

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q−1
X
Am(2j+1s) −Am(2js) −A(2j+1s) + A(2js)
 −→0.
Em;s,q =
j=0

An empty sum at = 0 is zero. The full recovered difference satisfies

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Fm −F = [Fm −Am(S)] + Em;s,q + [Am(s) −A(s)] + [A(S) −F],

and therefore, eventually for the fixed cutoffs,

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|Fm −F| ≤B + D
S
+ |Em;s,q| + (C + E)sp.

The choice of small, then finite, then sufficiently large is preserved by the lift. These identities expose the anchoring term) which controls the additive constant left undetermined by annular differences alone. Every scalar cutoffand torus reader has its identical horizontal readout, so these finite identities and bounds recover the whole kernel, not merely its annular increments. For the separated torus application, the retained mesh inequalities are exactly

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2π(|D| + 4)
Nh
≤s,
hs ≤1
2.

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For each fixed cutoffthey follow from the same reporting schedule; no cutoff-dependent change of
continuum witness is introduced. The limiting inverse transform uses the stated character conven-
tion and the actual function G(z) = (4π2∥z∥2)−1. For z ̸= 0, the heat correction retains the exact
scaling
∂D
G(z)e−π2v2∥z∥2
 = v|D|+2[∂D(Ge−π2∥·∥2)](vz).

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Choose a decay power greater than |D| + 2 to recover its vanishing at each nonzero z as v →∞.
Heat removal at a fixed Fourier cutoffand the retained dyadic tail P
j≥0 C/(2js) = 2C/s supply the
continuum comparison without asserting absolute integrability of an uncut differentiated symbol.
These operations use the same scalar readouts before and after horizontal transport.

For the far-image application retain = 0 and) = 0. The source’s annulus bound gives annular convergence to zero at each fixed as physical distance diverges. The preceding exact comparison then recovers0 with the same small-cutoffand tail constants once distance is at least one. This is the far-image reader used in the quarter-period covariance construction, including the coincident-observation return geometry.

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For the nonoscillatory low-frequency bound set t = hs, q = p/(hs).
The denominator is
dt(q) = P3
ν=0 q2
ν sinc(tqν/2)2. For ∥q∥≤1, 0 ≤t ≤2, the sinc bound 2/π gives dt(q) ≥(2/π)2∥q∥2.
Each normalized difference numerator is at most |qν| ≤1, so the normalized cutoffsymbol is
bounded by (π/2)2∥q∥−2 regardless of list length.
For centered integer momentum lifts in the
cube Bm, |Bm| ≤(2m + 1)4. On Bm \ B⌊m/2⌋inverse squared norm is at most 4/m2. Strong
induction gives P
p∈Bm ∥lift(p)∥−2 ≤432m2, with zero term assigned zero: the inner bound is
108m2 and outer bound (3m)4(4/m2) = 324m2, starting at m = 0. Writing η = 2π/N, T = η/t,
rescaling bounds the unit-ball inverse-square sum by 432T −4. Multiply by physical normalization
(Nh)−4s|D|−2 and use Ts = 2π/(Nh):

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| pieceh,s,D(x)| ≤432(π/2)2
(2π)4
s|D|+2.
(PT28)

This low-frequency estimate also holds at coincidence. The lifted kernel source retains annular differences together with both small-cutoffsize and large-cutofftail data. Its telescoping reader recovers the whole kernel and fixes the constant not determined by differences alone. Heat removal, the zero momentum convention and physical normalization remain attached to the same reader, preserving the separated and far-image limits used in the corrected covariance.

22.10 Discrete oscillation bounds and moving annular quadrature

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Translation of a finite torus momentum sum gives exact integration by parts for every integer v:
X

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p
χp(x)∆v
−eνf(p) = (χeν(x) −1)v X
p
χp(x)f(p).
(PT29)

The normalized annular symbol is smooth, supported at2, with derivatives of every fixed finite order uniformly bounded for 01, by cutoffaway from zero and smooth sinc extension at = 0. Momentum differences have normalized step, so repeated fundamental theorem of calculus bounds order by1, support expands only to3, with at most 8 representatives since1. Boundary crossings cause no loss: the periodic uncut symbol agrees on both lifts and the annular cutoffvanishes on both if2. For the high-frequency complement at2 use uniform derivatives on the rescaled Fourier box; all representatives contribute but, giving the same bound. Thus

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|χeν(x) −1|v
≤CD,vT v−4.
X
p
χp(x)f(p)

Choose a coordinate at least half the physical norm; the centered-lift circle chord inequality gives lift(). With + 3, physical normalization and) give

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2π
∥h lift(x)∥
v
.
(PT30)
| annulush,s,D(x)| ≤CD,v
(2π)4s