Gaussian short-distance comparison

Gaussian short-distance comparison

The comparison retains its sequence and normalization. The ultraviolet comparator is a free Maxwell polynomial hierarchy on one weak scaling sequence. Simultaneous finite-degree frames

transport labels and derivatives to that comparator, and finite determining position tests fix the prescribed calibration columns. An exact normalization of these finite columns is a statement about those tests; it is not equality of every interacting moment with its Gaussian value at a finite scale.

Private Gaussian marks, corrected bilinear covariances and the occurrence recursion recover the whole separated kernel and its fixed derivatives. The scalar return and annular quadrature estimates retain their Fourier measures, powers of the scale and normalization. The Gaussian prefactor amendment of Appendix I.1 fixes the earlier finite integration convention; it does not authorize a new normalization of these later kernels. Only their stated errors tend to zero, and the weak-sequence comparison by itself supplies no interpolation to every positive small radius. The later trace argument provides the distinct all-distance passage for its specified selected observable.

22.1 Common weak-sequence Gaussian comparison and stress normalization

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The mean Euclidean hierarchy uses the same weak sequence rk →0, with inverse ultraviolet scale
r−1
k ; this statement does not interpolate to every positive small scale. For free Maxwell field spaces
E0
d and simultaneous frames Uk,d : E0
d →Ed, unit, conjugation, signed rotation and filtration
compatibilities hold eventually in each fixed degree, and

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rkU−1
k,d+1∂aUk,d −→∂0
a.
(PT1)

Transported infinitesimal rotations also converge; the real conserved stress space is one-dimensional. The free translation-normalized comparator is the negative of the original Wick Maxwell tensor. To fix the sign, if the same and nonzero derivative family satisfy translation shell identities with both, subtraction and separated exterior testing give ( by field separation. The explicit Maxwell tensor has convention; compare with the constructed tensor to force constant1. The curvature-label calculation identifies its underlying Maxwell tensor. For

any finite free homogeneous list, any stress-slot subset and indices, use in regular slots and in stress slots. At positions the mean moments and every fixed position-derivative order converge uniformly on compact distinct sets to Wick moments, with in stress slots. Pull every slot back by: regular labels are fixed All use the same sequence and frames. and stress labels converge by physical comparison and Ward normalization. Fixed-basis expansion reduces to finitely many compact jet comparisons, with converging coefficients and bounded free derivatives. This proves the mixed claim for any number of stress slots on the same table as the selected/reconstructed hierarchy.

Common-sequence frame and mixed-jet recovery. Use the horizontal maps on the interacting degree spaces and keep the free spaces as the common comparison coordinates. Put

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UZ
k,d = JdUk,d,
(UZ
k,d)−1 = U−1
k,dQd,
∂Z
a = Jd+1∂aQd.

The inverse is on the horizontal image. The exact finite-level identity

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rk(UZ
k,d+1)−1∂Z
a UZ
k,d = rkU−1
k,d+1∂aUk,d

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recovers PT1 before taking its limit. The unit, filtration, conjugation, signed rotations and infinites-
imal rotations intertwine by the same cancellation QdJd = 1.
For a stress slot the pulled-back label is U−1
k,4(r4
kTab), identical to the lifted pulled-back label
(UZ
k,4)−1(r4
kJ4Tab). Its comparator is the same T 0,tr
ab . The nonzero derivative family in the subtracted
Ward identities is transported injectively, so (c −c′)D = 0 and (c −c′)JD = 0 are equivalent. This

preserves the source’s normalization and its one-dimensional real conserved stress space.

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For completeness expand the finitely many pulled-back labels in fixed free bases. Write ck,A for
the finite product of their coordinates and Wk,A for the corresponding mean basis moment at scaled
positions; let c0,A, W0,A denote their limits. For every fixed position multi-index α and compact
distinct set K, the exact finite expansion gives





∂α X
(ck,AWk,A −c0,AW0,A)
A

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K
≤
X
A
|ck,A| ∥∂α(Wk,A −W0,A)∥K
+
X
A
|ck,A −c0,A| ∥∂αW0,A∥K −→0.

Coefficients are position independent, bounded and convergent; the free derivatives are bounded on. Every lifted term has the identical scalar readout, so this also recovers arbitrary mixed stress-slot compact jets. The sequence, frames and continuum witness remain the same for all the fixed finite choices in the source; no interpolation in the scale parameter is used.

22.2 Finite position tests and exact frame normalization

At finite horizon let be the real space of small endomorphisms annihilating the unit, preserving each lower filtration space and commuting with conjugation, signed symmetries and multiplication by. Set (PT2)

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It is complex linear and near zero invertible by geometric series, fixing the unit and the stated
structures. The determining tests choose bk →0 and frame Uk,dMbk. At degree four all lower
fields are fixed exactly: for simple gauge algebra the free spaces below four consist of unit multiples,
whose degree inclusions remain the unit, annihilated by b. A test i has finite arity ni, labels vij ∈E0
d
and smooth compact complex profile ψi supported on distinct configurations. Let Λ be a finite real
linear combination of real readings. In a complex basis ea, let w contain every profile-integrated
basis moment for each test and tuple. Multilinearity gives

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Di(w; v1, . . . , vni) =
X
Y
j
(vj)ajwi,a1,...,ani,
a1,...,ani

and the finite real polynomial map is

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CQ(w, b) = Λ
Re Di(w; Mbvi1, . . . , Mbvini)


i.
(PT3)

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For the free integrated array w0
Q, a square determining collection has invertible derivative

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∂bCQ(w0
Q, 0) : Bd −→Rdim Bd.
(PT4)

At degree four its existence follows from the rigidity calculation below. A test derivative in direction is the real integrated value of

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Jb(x; v) =
X
j
⟨v1(x1) · · · (bvj)(xj) · · · vn(xn)⟩0.
(PT5)

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If all real derivatives vanish, using iψ also kills imaginary parts. Smooth compact test separation
and continuity give pointwise vanishing on distinct configurations. Rigidity forces b = 0, so these
real functionals separate B4. Choose finitely many spanning their dual span; their combined map
is injective and has a linear left inverse. Use it and coordinates on B4 as Λ. Then PT3’s derivative
is that coordinate isomorphism, proving PT4 with all profiles away from collisions. The finite-
dimensional implicit-function theorem gives a locally unique smooth φ near w0
Q such that

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φ(w0
Q) = 0,
CQ(w, φ(w)) = CQ(w0
Q, 0).
(PT6)

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For nearby (w, b) equality holds exactly when b = φ(w). Polynomial smoothness and PT4 supply
its hypotheses. The actual mean array wQ(k) uses Uk,dea at rkx. Compact separated convergence
gives wQ(k) →w0
Q: each integrated error is bounded by its supremum on the fixed compact profile
support times ∥ψi∥L1. The compact-profile convergence needed in (PT7) requires no distribution
extension through collisions. The lifted comparison source retains the common sequence, frames
and Ward-normalized tensor, rather than separately selecting stress and regular limits. Its finite-
test reader is the polynomial PT3 with the complete basis moment array. PT4–PT6 recover the
unique small frame correction from that array; preserving the profile and coordinate data makes
this calibration explicit. The rigidity dependency is retained for the ensuing degree-four calculation.

22.3 Exact frame correction and finite numerical jets

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Set
bk = φ(wQ(k)).
(PT7)

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Then bk →0, Mbk is eventually invertible, and PT6 gives exact determining equations in the inter-
acting mean hierarchy. This identifies the correction chosen by the local map with the correction
appearing in the endpoint. Finite-basis expansion and interchange of finite sums with integrable
profiles identify these equations with PT3. For any other finite separated test collection R, conti-
nuity gives
CR(wR(k), bk) −→CR(w0
R, 0).
(PT8)

Thus exact normalization preserves every other leading position test.

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Finite-test calibration and frame recovery.
The calibration source records the finite basis,
every basis-tuple moment, the actual profiles, input labels and real map Λ.
The lifted mean
moments have identical scalar readouts, so their profile arrays satisfy wZ
Q(k) = wQ(k) exactly in
these recorded coordinates. Hence the full polynomial, not only its derivative, is recovered:

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CZ
Q(w, b) = CQ(w, b),
∂bCZ
Q(w0
Q, 0) = ∂bCQ(w0
Q, 0).

Here the free calibration coordinates are kept unchanged; the interacting frame carries the horizontal lift. The real derivative in a direction ̇ is explicitly

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

ni
X
∂bCQ(w, 0)[ ̇b] = Λ
j=1
Di(w; vi1, . . . ,  ̇bvij, . . . , vini)
Re

i
.

Thus PT5 is recovered by the same finite multilinear expansion and profile integral. Replacing a profile by recovers its imaginary functional as well. The source’s ensuing rigidity result makes their common kernel zero at degree four. Thus the degree-four adjustment is compatible with the

already chosen frames in degrees zero through three. Choosing an injective finite combined derivative and a left inverse, the prescribed real coordinate map can be written Λ = coord; then Λ = coord. The horizontal translation retains and this coordinate isomorphism, recovering PT4 with the same profiles and the same rigidity dependency.

Since the entire polynomial and base point agree, local uniqueness in PT6 gives the identical correction function in these coordinates. In particular

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bZ
k = φ(wZ
Q(k)) = bk,
bUZ
k,d = JdUk,d(I + bk),
Qd bUZ
k,d = Uk,d(I + bk).

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For ∥bk∥< 1 its inverse on the horizontal image is (I + bk)−1U−1
k,dQd, with

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(I + bk)−1 −
m
X
l=0
(−bk)l





 ≤∥bk∥m+1
1 −∥bk∥.
∞
X
(I + bk)−1 =
l=0
(−bk)l,

The finite filtration, unit, conjugation and signed symmetry laws are retained by this inverse and the correction. At horizon four all included degree-zero through degree-three fields are fixed exactly, since their unit multiples are annihilated by

For each profile the unchanged integral error estimate is times the corresponding compact moment error. It proves , hence0, while the determining equations hold exactly for all sufficiently large. Any other fixed finite profile collection has its same transported array ) and therefore the same PT8 continuity limit. Thus calibration preserves the other leading position tests on the original common sequence and adjusts the frame by the exact local solution, with no independent selection of moments or stress data.

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For a fixed finite torus of side greater than one, let wQ,l(m) be order-lcoefficients of its regulated
profile series; order zero tends to w0
Q as m →∞. At fixed finite formal horizon p, the polynomial
cQ,p,m(t) = Pp
l=0 tlwQ,l(m) is used only for numerical derivatives, without convergence of the full
series. For sufficiently large m its base point, which tends to w0
Q, is in the PT6 neighborhood,
so CQ(c(t), φ(c(t))) is identically the prescribed constant near zero. All positive derivatives vanish
exactly and the zeroth one has that value. Another query has order-zero numerical reading

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CR(wR,0(m), φ(wQ,0(m))) −→CR(w0
R, 0).
(PT9)

Subtracting at from PT8 gives difference tending to zero on the actual-theory sequence. Order zero is independent of. This calibrates each finite determining jet and compares other profiles at order zero; it does not compare every higher coefficient to the interacting theory. PT6 local uniqueness makes every other small correction solving the same equations eventually equal to PT7. Together with PT2 this retains all frame symmetries and exactly fixes raised lower fields through degree three.

Finite numerical-jet recovery. Retain each regulated coefficient array as a separate numerical entry, with its finite formal horizon and mesh index. Equality of the scalar coefficient readouts gives

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cZ
Q,p,m(t) = cQ,p,m(t),
dj
dtj cQ,p,m(0) = j!wQ,j(m)
(0 ≤j ≤p).

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The polynomial curve starts at wQ,0(m), so for each sufficiently large m it lies in the local calibration
neighborhood for t near zero. Write HQ(w) = CQ(w, φ(w)) there. PT6 makes HQ the constant
CQ(w0
Q, 0), whence

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(
CQ(w0
Q, 0),
j = 0,
0,
j > 0.
dj
dtj HQ(cQ,p,m(t))

t=0 =

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The identical polynomial and correction function recover this equality in the lift at every retained
finite horizon. No infinite series is evaluated. For a different query put ak = CR(wR(k), bk) and
dk = CR(wR,0(k), φ(wQ,0(k))). Both converge to a0 = CR(w0
R, 0), so explicitly

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∥ak −dk∥≤∥ak −a0∥+ ∥dk −a0∥−→0.

The second reading uses only order-zero arrays and is independent of. The same local uniqueness recovers the eventual equality of any other small correction satisfying the determining equations. This retains exactly the finite calibration and order-zero comparison scope of the source.

22.4 Degree-four rigidity of the position tests

The free degree-four basis is the unit together with unordered plane-pair curvature labels

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ec =
X
α
F α
abF α
uv.
(PT10)

Each plane has increasing-axis orientation; reversing one changes sign and swapping the two planes preserves the product. The free quotient calculation supplies this basis and dual coordinates At separation ( 0) curvature pairings are diagonal on six planes with nonzero. Wick contraction gives (PT11)

Repeated planes have two equal pairings; distinct planes have their single matching pairing. The coefficient is the product of nonzero curvature factors, dim g and its nonzero multiplicity, absorbed in the dual normalization. Expectation is the unit coordinate. For zero unit coordinate the reversed pairing has the same formula. Basis expansion proves these identities without requiring this bilinear form to be positive definite. For) define parity. Axis reflection gives sign minus precisely on axes in that set. Commutation with reflections forces for different parities. Equal parities admit an involutive axis permutation exchanging the plane pairs up to orientation and swap. If parity is empty both repeat a plane and a transposition or double disjoint transposition exchanges the planes. If it has two elements, each pair consists of two planes sharing one of the other two axes; exchange those common axes when they differ. For size four, a transposition exchanges the different complementary-plane partitions. Identical pairs use the identity. These lie among identity, six transpositions and three double disjoint transpositions. The induced exchange signs agree, since squaring the involution is identity on nonzero basis vectors. Commutation gives. Also within parity: multiplicities agree, absolute curvature factors agree and the product of curvature signs at the axis-zero separation depends only on whether zero is in parity. If PT5 vanishes for every tuple, arity one gives) = 0; arity two gives (PT12)

Taking yields = 0. Different-parity entries vanish already; equal-parity symmetry gives 2 = 0, so all vanish. The unit column is zero since1 = 0 and its row by arity one. Thus = 0, closing PT4’s determining-test existence proof through degree four.

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Degree-four basis and rigidity readout.
Transport the source’s actual free quotient basis
and its dual by eZ
c = J4ec, 1Z = J41, and lZ
c = lcQ4, lZ
∗= l∗Q4. For the allowable matrix use
bZ = J4bQ4. Then

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lZ
c (bZeZ
d ) = bcd,
⟨(J4v)(e0)eZ
c (0)⟩Z
0 = aclc(v),
J Z
bZ = Jb.

The horizontal free reader here is the transported free Wick reader. The oriented-plane signs, unordered products and fixed multiplicities are retained, so no new positivity convention enters PT11. Reflection characters and the finite signed axis permutations are conjugated by They therefore give the same parity blocks, zero off-block matrix entries and symmetric withinblock entries.

On the curvature-label block let = diag(). The arity-two identity is exactly

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Db + bTD = 0.

Within each parity block with= 0, and the source’s involutions give on that block. Thus the recovered identity is 2 = 0 there. Every off-block entry already vanishes; the unit column is zero from1 = 0 and the unit row from arity one. The same full matrix is recovered by the dual coordinates after horizontal transport, so = 0 on its horizontal domain if and only if = 0. This recovers the precise degree-four rigidity used for the determining derivative, with its quotient-basis and free contraction dependencies intact.

For the physical stress limit let the nonzero raw centered conserved tensor satisfy center(= 0, by the one-dimensional divergence kernel. The centered physical tensor is nonzero because its translation charge differentiates a nonzero field. Shell identities and field separation give= 0 is the preliminary comparison scale, weak derivative comparison gives= 0. Rescaling the raw limit therefore yields

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U−1
k,4(r4
k center(T)) −→κ−1
0 T 0.
(PT13)

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The difference from T is a fixed unit tensor whose inverse-scaled contribution is r4
k times that
unit and tends to zero, with eventual exact unit identifications. Ward normalization above gives
κ0 = −1. The same factor (c/ξk)−1 normalizes the preliminary angular-generator limit, giving
the final rotation limit. For compact separated profiles no contact extension is used: if fk →f0
uniformly on support and δ = ε/(1 + ∥ψ∥1), eventually

Z
ψ(fk −f0)
 ≤∥ψ∥1 sup
supp ψ
|fk −f0| ≤∥ψ∥1δ < ε.

The lifted calibration source retains finite formal curves separately from actual hierarchy arrays. PT7 reads their unique small frame correction, and PT9 recovers only the asserted numerical order-zero comparison. The plane-pair basis, parity action and Wick coefficients supply an explicit injective derivative reader, recovering from the determining tests. PT13 retains the charge normalization and exact unit removal, recovering the same physical stress sign used in all mixed-slot limits.

Physical stress and compact-profile recovery. With the horizontal maps on degree four, the raw/physical relation and Ward derivative relation recover exactly as

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J4Jraw = cJ4 center(T),
JDraw = cJD.

The same nonzero and comparison scale are retained. Before any limit the normalization identity in free coordinates is

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U−1
k,4(r4
k center(T)) = (c/ξk)−1 ξ−1
k U−1
k,4(r4
kJraw).

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Using (UZ
k,4)−1 = U−1
k,4Q4 makes this identical for lifted tensors. Since c/ξk →κ0 ̸= 0, it recovers
PT13 and the same normalization of the angular generator. Write T −center(T) = λ1 componen-
twise, with fixed λ. Exact unit identification gives

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(UZ
k,4)−1r4
kJ4(T −center(T))

 = r4
kλ10 −→0.

Thus horizontal recovery retains the Ward-normalized sign1 and the full physical tensor, including the vanishing center correction. For each separated compact profile the finite moment readout is unchanged; its Bochner integral error remains bounded by times the uniform moment error on its support. This recovers the actual arrays without any contact extension.

22.5 Moving Gaussian rows and normalized profile sums

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Smooth separated moments make the preceding profile integrals ordinary Bochner integrals, proving
every component of wQ(k) →w0
Q. For numerical regulator arrays, order-zero Wilson coefficients
are Gaussian Wick contractions of reduced local polynomials. The common field-address identity
identifies these with the free labels. With fixed refinement base L > 1, denominator Lm, spacing
L−m and periods growing faster than each fixed polynomial margin, the Gaussian row limit holds
for every refining subsequence and moving grid tuple tending to distinct continuum positions. The
profile sum then gives
wQ,0(m) −→w0
Q.
(PT14)

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Here are its normalization and convergence details. Let hm = D−1
m →0, Dm integer, and torus side
Nm satisfy hmNm →∞. Coordinate rounding in arity n gives

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|ρmx −x| ≤4nhm.
(PT15)

Every bounded set eventually embeds without wrapping. Assume the joint moving limit: for every subsequence, allowed, and grid tuple embedded at positions tending to This implies an eventual uniform bound on rounded points of any compact allowed set. Otherwise choose values exceeding at; compactness gives a convergent position subsequence, and PT15 plus nonwrapping contradicts the moving limit. For a continuous compact profile supported in the allowed region put). It converges pointwise to); outside closed support the rounded positions are eventually outside. If support lies in a radius- ball,

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these step functions lie in radius R + 4n for hm ≤1 and have one common bound there. They are
integrable, and dominated convergence applies. Every grid cell has volume h4n
m , so the integral is
exactly
X
h4n
ψ(hmo)Km(o).
(PT16)
m
o∈(Z/NmZ)4n

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No wrapping occurs on the support. Arity zero has a one-point grid and space with cell volume
h0
m = 1, giving the same conclusion.

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Moving-row and grid-integral recovery.
Retain the finite regulator row, mesh, period and
rounding cells in the source record. Write KZ
m for its lifted scalar reading; then KZ
m(o) = Km(o)
for every retained grid tuple. Consequently its joint moving limit and every selected violation of a
uniform compact bound are the identical scalar statements after recovery. A fixed compact profile
support lies inside a compact neighborhood of the allowed open region. For sufficiently fine meshes,
all rounded points relevant to its nonzero values lie in that neighborhood. The moving-limit bound
therefore gives some finite M with

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|ψ(ρmx)KZ
m(ρmx)| ≤M∥ψ∥∞1{|x|≤R+4n}(x).