The Gaussian short-distance comparison

The Gaussian short-distance comparison

The short-distance comparison has two normalizations. A weak scaling sequence compares the actual hierarchy with the free Gaussian hierarchy. A finite determining collection of position tests then adjusts the coefficient frame by a small invertible map. Those finitely prescribed columns are matched exactly. General interacting moments still satisfy the purchased error bounds; they are not all equal to their Gaussian counterparts at a finite station.

Leading perturbative comparison on a common weak scaling sequence

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There is a further conclusion about the mean Euclidean hierarchy.
Its scale parameter is the
same weak sequence rk →0, with ultraviolet inverse scale r−1
k ; it does not assert interpolation to
every small positive real scale. Let E0
d be the free Maxwell field spaces and Uk,d : E0
d →Ed the
simultaneous scaling frames. Their unit, conjugation, signed rotation and filtration compatibilities
hold eventually in each fixed degree. The transported first derivatives satisfy

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rkU−1
k,d+1∂aUk,d −→∂0
a,
(PT1)

and the transported infinitesimal rotations converge to the free ones. The real conserved stress space has dimension one.

The free stress comparator is translation-normalized: it is the negative of the source’s original Wick Maxwell tensor. This sign is fixed by a Ward identity, rather than by conservation alone. If a nonzero first-derivative family and one tensor satisfy the translation shell identities both with and with, subtract those identities, test against arbitrary separated exterior fields and use

field separation. The resulting equality ( = 0 gives. Apply this to the constructed free tensor and the explicit Maxwell tensor, whose translation convention is. It forces their comparison constant to be1. The explicit curvature-label calculation identifies the underlying tensor with the original Maxwell tensor, giving the claimed negative comparator, which we denote

Fix any finite list of free homogeneous labels, any subset of its slots to be replaced by stress components, and a pair of tensor indices in each such slot. In a regular slot use; in a stress slot use . Then, for every compact set of jointly distinct positions and every positionderivative order, their mean moments at converge uniformly on, with those derivatives, to the free Wick moments with stress slots . All arities, finite label choices and derivative orders use the same sequence and frames.

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To prove this passage, pull every slot back by U−1
k,di. Regular slots become constant labels; stress
slots converge to T 0,tr
aibi by the physical stress comparison and its Ward normalization. Expand the
resulting finite-dimensional labels in a fixed basis. The common compact jet comparison handles
the finitely many basis moments, while coordinate convergence and compact boundedness of the
free derivatives handle the varying coefficients. This proves the mixed statement with any number
of stress slots. The construction of the sequence supplies the mean hierarchy, derivatives, rotations
and physical stress on the same table as the selected and reconstructed theory; no change of
continuum witness is made here.

The position normalization through degree four is a second finite-dimensional construction. At a general finite degree horizon, let be the real vector space of allowable small endomorphisms of the free field space . They annihilate the unit, preserve every lower filtration subspace, commute with conjugation and signed symmetries, and commute with multiplication by. The last property makes complex linear. Put

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Mb = I + b.
(PT2)

For near zero it is invertible; the inverse follows from a geometric series in the endomorphism norm. It fixes the unit and preserves the listed structures. The determining finite collection of position tests will choose0, and the adjusted frame is

At horizon four the adjustment preserves all lower fields exactly. The free degree- field space for 4 consists of unit multiples for a simple gauge algebra. The degree-raising maps take that unit to the unit. Since annihilates the latter, fixes every raised lower field. Thus the degree-four adjustment is compatible with the already chosen frames in degrees zero through three.

A determining position test consists of a finite arity, labels , and a smooth compactly supported complex profile whose support lies in the open set of distinct configurations. A finite real linear map Λ combines the real parts of these readings. Choose a complex basis of . A finite array records the profile-integrated basis moments, for each test and every basis tuple. Multilinearity reconstructs the reading of any tuple:

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

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Define the finite position map
CQ(w, b) = Λ
Re Di(w; Mbvi1, . . . , Mbvini)


i.
(PT3)

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It is a polynomial in its finite real coordinates. Let w0
Q be the array obtained by integrating the
free Wick moments against the profiles. A square determining collection means exactly that

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∂bCQ(w0
Q, 0) : Bd −→Rdim Bd
is invertible.
(PT4)

This condition is proved for a suitable collection at degree four. For this existence proof, a single test’s derivative in direction is the real part of the integral of

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

If all these real integrated derivatives vanish, replace each profile by to see that their imaginary parts vanish as well. Continuity on distinct configurations and smooth compact test separation then show that (PT5) vanishes pointwise there. The degree-four rigidity calculation below proves that this forces The resulting real linear functionals therefore separate In a finite-dimensional space finitely many suffice: choose a finite spanning subset of their span in the dual. Their combined map has zero kernel, hence a linear left inverse. Compose that left inverse with coordinates on to choose Λ. By differentiation of (PT3), its-derivative is then precisely that coordinate isomorphism. This proves (PT4), with profiles supported away from every collision.

The finite-dimensional implicit-function theorem now provides a function on a neighborhood of , with (PT6)

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and local uniqueness: for (w, b) near (w0
Q, 0), the equality in (PT6) holds if and only if b = φ(w).
The map is smooth to every finite order at the base point. The proof uses only the polynomial
map and the invertible derivative (PT4).
Let wQ(k) be the same array measured in the actual mean hierarchy with labels Uk,dea at
positions rkx. Compact separated convergence implies wQ(k) →w0
Q. To justify the integrals, the
smooth profile has a fixed compact separated support, so the uniform moment error is bounded by
its supremum there times ∥ψi∥L1. Set

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bk = φ(wQ(k)).
(PT7)

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Then bk →0, Mbk is invertible eventually, and (PT6) gives the exact determining equations in the
interacting mean hierarchy. Equality of (PT3) with those measured equations follows by expanding
the inserted labels in the finite basis and exchanging a finite sum with the integrable profile. For any
other finite collection R of such separated profiles, continuity of CR and convergence (wR(k), bk) →
(w0
R, 0) give
CR(wR(k), bk) −→CR(w0
R, 0).
(PT8)

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Thus the exact normalization does not spoil any other leading position test.
The endpoint also compares this construction with numerical jets of finite Wilson data. For
a fixed finite torus with side greater than one, let wQ,l(m) denote the coefficient of order lof its
regulated profile series at mesh parameter m. Its order-zero coefficients tend to w0
Q as m →∞.
For a fixed truncation order p, form the finite polynomial curve cQ,p,m(t) = Pp
l=0 tlwQ,l(m). This
curve is used only to compute numerical derivatives. It requires no convergence of the full formal
series. Equation (PT6) holds along the curve near t = 0, for all sufficiently large m, because its
base point tends to w0
Q. Therefore every positive derivative at zero of CQ(cQ,p,m(t), φ(cQ,p,m(t)))
vanishes exactly, and its zeroth derivative equals the prescribed constant.
For another query R, the order-zero numerical reading is

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

Subtract (PT9), at, from (PT8). Their difference tends to zero on the same actual-theory sequence. The order-zero numerical reading is independent of the chosen polynomial truncation

. These are exactly the numerical-jet calibration and leading-comparison assertions: calibration of the determining equations at each finite formal horizon, and order-zero comparison for other profiles. They do not assert comparison of every higher-order coefficient with the actual interacting hierarchy.

Any other small correction satisfying the same exact determining equations equals (PT7) eventually, by the local uniqueness in (PT6). This identifies the correction chosen by the local map with the correction appearing in the endpoint. Combined with (PT2), the unit and symmetry identities and the exact preservation of raised fields of degree below four, it proves all the frame properties claimed for the degree-four position normalization.

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Here is the degree-four rigidity used in (PT5). Its free field basis consists of the unit and the
color-contracted curvature products
X
F α
abF α
ec =
uv,
(PT10)
α

indexed by unordered pairs of oriented coordinate planes, each plane put in its increasing-axis orientation. Reversing a plane changes the sign and interchanging the planes preserves the product. The free quotient calculation proves that these labels, together with the unit, form a basis; its dual coordinates will be denotedand

At the separation ( 0), the free curvature two-point form is diagonal on the six coordinate planes, with nonzero coefficients. Wick contraction therefore gives

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⟨v(e0)ec(0)⟩0 = aclc(v),
ac ̸= 0.
(PT11)

For a repeated plane the contraction has its two equal pairings; for two different planes it has the single matching pairing. Accordingly is the product of the two nonzero curvature coefficients and dim g, multiplied by the corresponding nonzero Wick multiplicity. The dual-coordinate normalization absorbs that multiplicity. The unit coordinate is expectation. If) = 0, the reversed pairing has the same formula. These facts follow by expanding in the basis and applying the diagonal contractions, so no positive-definiteness of this particular bilinear form is required.

For each plane pair), let its parity set be the symmetric difference of their axis sets. Reflection in axis multiplies by1 exactly when belongs to this parity set. If commutes with all such reflections, its matrix coefficient) vanishes whenever have different parity sets.

For equal parity sets there is an involutive permutation of the four axes exchanging the two plane pairs, up to reversing individual plane orientations and swapping the planes. This finite assertion has three cases. If the parity set is empty, both pairs repeat a plane, and a transposition or a product of two disjoint transpositions exchanges the two 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. If it has four elements, each pair is a partition into two complementary planes; a transposition exchanges any two different such partitions. The identical-pair case uses the identity. These permutations belong to the list consisting of the identity, six transpositions and three products of disjoint transpositions.

The induced action sends to and to, with the same sign: applying the involution twice gives product of signs one and the basis vectors are nonzero. Commutation of with this action gives. Moreover for equal parity sets. Their Wick multiplicities agree, and the product of their two curvature signs at the axis-zero separation depends only on whether zero belongs to the parity set; their absolute curvature factors are the same. This proves the equality of the coefficients in (PT11).

Now suppose (PT5) vanishes for every separated configuration and tuple. Arity one gives ) = 0 for every. Arity two at ( 0) gives

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⟨(bv)(e0)w(0)⟩0 + ⟨v(e0)(bw)(0)⟩0 = 0.
(PT12)

Take Equation (PT11) and the zero unit coordinate give For different parity sets both coefficients already vanish. For equal parity sets, symmetry and = 0 give 2 = 0, hence again both vanish. The unit column vanishes because the unit row vanishes by arity one. Every coefficient of is zero, proving the rigidity assertion. This closes the existence proof of the square position tests through degree four.

The physical stress comparison used before (PT2) can be proved directly on this weak sequence. Let= 0 be its constructed raw conserved centered tensor, and let= 0 be the scale used in its preliminary stress comparison. The one-dimensional divergence kernel implies

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Jraw = c center(T),
c ̸= 0.

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The centered tensor is nonzero because its translation Ward charge differentiates a nonzero field.
Compare its charges with those of Jraw. Field separation applied to every translation shell identity
gives equality of their derivative families, Draw = cD. The weak derivative comparison then yields
c/ξk →κ0, where κ0 ̸= 0 is the free stress Ward amplitude. Multiplying the preliminary raw stress
limit by (c/ξk)−1 gives
U−1
k,4(r4
k center(T)) −→κ−1
0 T 0.
(PT13)

The difference from is a fixed constant unit tensor. Its inverse-scaled contribution is times that unit, tending to zero; the unit identifications are exact eventually. The Ward argument preceding (PT2) gives1, proving the stress-slot limit used there. Similarly, the preliminary angulargenerator limit is normalized by the factor (, giving the transported rotation limit in the final physical normalization.

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The compact-profile convergence needed in (PT7) requires no distribution extension through
collisions. If ψ is one of the allowed compact separated profiles and fk →f0 uniformly on its
support, take δ = ε/(1 + ∥ψ∥L1). Eventually

 ≤∥ψ∥L1 sup
Z
ψ(fk −f0)
|fk −f0| ≤∥ψ∥L1δ < ε.
supp ψ

Smoothness of the separated moments makes both integrals ordinary Bochner integrals. This supplies every component of

For the regulator arrays in (PT9), the order-zero Wilson coefficient is the Gaussian Wick contraction of the reduced local field polynomials. The common field-address identity identifies these polynomials with the free field labels. At fixed lattice refinement base 1, the denominator is, the mesh spacing is, and the chosen period tends to infinity fast enough for every fixed polynomial margin. The Gaussian row limit holds for every refining subsequence and every sequence of grid tuples tending to a distinct continuum tuple. The grid-sum limit for a compact separated profile then gives (PT14)

The Gaussian kernel, Wick row and grid-sum arguments underlying this assertion are developed next, including the finite-regulator subtraction and its limiting cancellation.

15.1.1 From moving Gaussian rows to profile integrals

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Here is the grid-sum passage in (PT14). Let hm > 0 tend to zero, hm = D−1
m with integer Dm, and
let the torus side Nm satisfy hmNm →∞. For an arity n, round every coordinate of a continuum
configuration x to the grid. Denote the rounded point by ρmx, so

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|ρmx −x| ≤4nhm.
(PT15)

Every fixed bounded set eventually embeds in the torus without wrapping, because

Suppose lattice kernels have the following joint moving limit: for every subsequence , every in the allowed open region, and every grid tuple with embedded positions tending to , one has). This is stronger than convergence only at a fixed rounded point and is exactly what the Gaussian row calculation supplies. It implies a uniform eventual bound for on rounded points in any fixed compact allowed set. Otherwise choose and points in that compact set with. A convergent subsequence, (PT15) and eventual nonwrapping give a sequence of embedded grid tuples tending to, contradicting the moving limit.

For a continuous compactly supported profile with support in the allowed region, define the step function

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It converges pointwise to ψ(x)K0(x). On the support use the moving limit and continuity of ψ;
outside its closed support the rounded points are eventually outside it, so the expression vanishes.
If ψ is supported in the radius-R ball, the step functions, for hm ≤1, are supported in the radius-
R + 4n ball and are bounded by a common constant there. They are integrable step functions.
Dominated convergence gives convergence of their integrals.
Each grid cell has exactly volume h4n
m .
Summing the cells, with no wrapping on the fixed
support, identifies the integral exactly with

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h4n
m
X
o∈(Z/NmZ)4n
ψ(hmo)Km(o).
(PT16)

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The cell partition and its volume hold also at arity zero, when both the configuration space and the
grid consist of one point and h0
m = 1. This proves convergence of (PT16) to the desired continuum
integral with its correct normalization.
The Gaussian row limit is reduced to a two-point covariance limit by an exact finite recursion.
Give every occurrence of every generator in the finitely many local polynomial labels a distinguish-
able index a. Let τ(a) be its insertion slot and let ζa be its field-generator label. At order zero, the
row is the Gaussian moment Mm(A) of the product of the corresponding centered linear variables,
for each finite set A of occurrences. Gaussian integration by parts gives

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

For) = 1 because the Gaussian law is a probability measure. The continuum Wick moments obey the same recursion with covariance). Induct on: the covariance limit and the finitely many already proved smaller-product limits give convergence of the right side, and division by the positive integer gives convergence of

This occurrence indexing accounts for repeated powers without identifying identical generators prematurely. An arbitrary monomial is the product over its occurrence set; arbitrary finite polynomial labels are finite sums of such monomials with their coefficients. Multilinearity and finite summation extend the preceding limit to every tuple of the allowed local polynomial sections. The

field-address identity sends the reduced polynomial section of a free label to that same label in the free Wick hierarchy. Therefore the limiting row is precisely the continuum free moment, not an independently fitted polynomial. Applying the result along every refining subsequence supplies the moving-limit hypothesis used in (PT16). This proves (PT14), once the corrected covariance limit used in (PT17) is established.

15.1.2 Private Gaussian marks and the subtraction covariance

Private Gaussian marks distinguish contraction occurrences without changing the physical covariance. Their subtraction covariance must be computed with the stated Fourier and opposite-momentum convention. Only after the corrected bilinear covariance is identified can the occurrence recursion and scalar return estimate be used for higher products.

The covariance used in (PT17) has a finite-regulator subtraction that must be retained before taking the limit. Fix the largest jet grade occurring in a polynomial row. Its finitely many linear coordinates are indexed by keys. Let be the centered real Gaussian coordinate vector of the regulated potential, with precision matrix and covariance. Write for the coefficient vector of the linearized, symmetrically averaged jet coordinate at. Thus its value on is These coordinates use the same Gaussian density as the regulated perturbative integral.

The subtraction matrix is defined on the infinite lattice. If ranges over sine and cosine waves, vector axes and color indices, and) is the jet coordinate evaluated on that wave, set

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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)

Here is the lattice Laplacian symbol. The wave responses are continuous on the cube, hence bounded; is integrable in four dimensions, so every entry exists. Symmetry is immediate. For any real vector

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

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Consequently Sh is the covariance of a possibly degenerate centered real Gaussian vector.
No
inverse of this matrix is needed.
For insertion indices i = 1, . . . , n, take independent copies Yi of this Gaussian vector, indepen-
dent also of Z, and put
Xi,a = kT
h,xi,aZ + i Yi,a.

All these variables are complex linear functions of one finite-dimensional real Gaussian vector. Their means vanish and 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)

Indeed the mixed terms vanish by independence and centering; the two private marks have covariance zero for different insertions, and multiplication of their imaginary coefficients gives the minus sign for equal insertions. This is a bilinear covariance, with no complex conjugation. At finite regulator its same-insertion entries need not vanish.

The constant term of the normalized perturbative integral is exactly the expectation of the product of the reduced insertion polynomials evaluated at these. To see this directly, the weight and vacuum denominator both have constant coefficient one. The constant coefficient of each linear insertion generator is its physical Gaussian coordinate plus its imaginary private mark.

Expanding a polynomial is a finite operation, so taking constant coefficients commutes with that expansion. Integration over each private mark produces the prescribed subtraction within its own insertion. The product law then combines these independent integrals with the physical Gaussian integral. Every integrand is a polynomial in a finite Gaussian vector and is integrable: a polynomial is bounded by, and all Gaussian moments are finite. This also justifies each finite expansion and use of Fubini.

For completeness the moment recursion does not require a nonsingular Gaussian law. A centered real linear Gaussian variable, of variance, has moment generating function exp( Differentiating at zero gives

Apply this identity to arbitrary real linear combinations of a finite family of centered real linear probes and compare coefficients. Polarization gives

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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

Initially the probes can be taken real. Expanding each complex probe into its real and imaginary parts extends the identity complex multilinearly. Occurrences, including repeated occurrences of the same generator, remain distinct elements of. The empty product has expectation one. These facts prove the finite Gaussian recursion used above.

15.1.3 The limit of the corrected covariance

Same-insertion subtraction The corrected covariance removes contractions within one prescribed insertion while retaining contractions between distinct insertion positions. This is an occurrence-level subtraction: repeated generator names do not merge their occurrence indices. Its two limits have different purposes. At different positions it recovers the Maxwell two-point kernel; at the same insertion it tends to zero. The latter is a regulator-removal conclusion, not an identity imposed on every finite torus covariance.

The subtraction in (PT19) has the following two different limiting roles. For distinct physical limiting insertion positions

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Ch((i, a), (j, b)) −→K(ka, kb; zi −zj),

where is the corresponding coordinate functional on free solution jets and is their Maxwell two-point kernel. For (PT21)

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Thus zero same-insertion covariance is a conclusion after removing the regulator.
It is not an
identity for a finite torus.
Here is the averaging step that identifies both limits.
Let S be the finite group of signed
coordinate transformations and cS = |S|−1. For a jet functional A, its averaged lattice stencil has
the exact form
A(wideLinearh,x) = cS
X
s∈S
A ◦turns ◦radialMap(charts,h,x).
(PT22)

Denote the functional inside this sum by. On free equation solution jets the radial reconstruction is an inverse to sampling; equivariance therefore gives

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plains(As) = A ◦gaugeJet .