Apply the universal selection statement to this particular sequence. Equation (TLim1) retains this inequality at every
, whereas the preceding sandwich proves that its left side tends to zero. This contradiction proves (TR2) for the full small-scale limit. No density or ratio condition on a single favorable sequence of radii is being assumed.
The positive measure in (TR3) is unique and is the trace spectral measure. With the Fourier sign and normalization used in the reconstruction, it satisfies
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∥Θ(f)Ω∥2 = Z | bf+(p)|2 dμΘ(p), ⟨Θ(f)Ω, Θ(g)Ω⟩= Z bf+(p)bg+(p) dμΘ(p). (TR4)
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Its support lies in the forward cone, it is Lorentz invariant and of temperate growth, and every positive time damping has finite total mass. The full two-point Wightman distribution is its relative Fourier transform, also on joint Schwartz tests. Existence follows from the reconstructed joint momentum measure on the dense family of smeared trace vectors; these quadratic forms patch to a single positive measure. Uniqueness follows by testing all | bf+|2, polarizing, and using compactly supported test functions to identify the measure. The regularized point-vector construction (TR5) and the measure argument (TR10)–(TR17) below give the patching and ultraviolet limit explicitly.
The spectral construction can be made directly from the regularized point vectors. Let
be the single trace shot with positive imaginary-time height
, centered at the origin. The shot identities already proved give
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vs+t = e−tHvs, W2 at heights (−s, s) = ⟨vs, U(x1 −x0)vs⟩.
Let
be the diagonal joint-momentum measure of
. Functional calculus gives
Define 
(TR5)
The semigroup identity shows equivalently
for every
0: compare both heights through their sum. Hence
is finite, and its characteristic function is the regularized two-point function. This proves (TR3), with
2 in the shot identity. Smearing the same identity and taking the previously established boundary limit yields (TR4) and the full joint-test Fourier representation. The momentum cone support is inherited from the shot measures, and scalar Lorentz covariance identifies each Lorentz pushforward with the same unique measure. The polynomial growth estimate for these measures follows from the common polynomial shot-height bound; this estimate will be detailed below.
The free contraction constant in (TR2) has a direct finite calculation. For
= 0, put
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Rai(x) = δai −2xaxi/|x|2.
This is an orthogonal reflection, since multiplying its two rows and summing the common index gives
. The free curvature covariance has the form
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Curlabij(x) = D(x) RaiRbj −RajRbi , D(x) = 4 normal |x|−4.
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Orthogonality yields X a,b,i,j Curlabij(x)2 = 24D(x)2.
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Indeed summing first over j, then i, gives D2(2 −2δ2 ab); summing the a, b indices gives 24D2. The scalar curvature-square field has coefficient 1/4 in each sum, two Wick contractions between its two quadratic factors, and dim g color choices. Thus its two-point function is (1/16) 2 dim g 24D(x)2 =
3 dim g
) with the source’s Green-function normalization. This supplies the free constant used above. We next derive the ultraviolet measure limit from (TR2). Write
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Z(ρ) = ρ−8 log2 ρ, νρ = Z(ρ) (p 7→p/ρ)∗μΘ, ρ > 1.
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These are positive Lorentz-invariant cone-supported measures. For s > 0, Z e−sp0 dνρ = Z(ρ)E(s/ρ) −→AGs−8, (TR6)
because
. Lorentz invariance extends (TR6) to every strictly future timelike damping vector. The comparison measure is
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dν∞(p) =
AG
1536π (p2
0 −|p|2)21{p0≥|p|} d4p.
(TR7)Its time-axis Laplace integral is
, as can also be checked directly: integrate spatial radius 
], obtaining
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4π Z p0 0 (p2 0 −r2)2r2 dr = 32π 105 p7 0,
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and then
R ∞
0 e−sp0p7
0 dp0 = 7!s−8. The product is 1536πs−8. Positivity, cone support and finite
timelike Laplace transforms allow the Laplace convergence theorem to identify the compact-test
limit with (TR7); its proof is unfolded next.
There is a uniform polynomial bound strong enough for Schwartz tests. Since e−p0/R ≥e−1 on
p0 ≤R,
νρ{p0 ≤R} ≤e
Z
e−p0/R dνρ ≤CR8,
R ≥1,
(TR8)uniformly for sufficiently large
. For the last inequality use (TR2) at (
, and
The same bound holds for
. On the forward cone, Euclidean norm is comparable to
; dyadic annuli and Schwartz decay therefore make the tails of every Schwartz test uniformly small. Combining this with compact-test convergence gives
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Z(ρ) Z φ(p/ρ) dμΘ(p) −→ Z φ dν∞ (φ ∈S(R4)). (TR9)
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For joint two-point tests use their relative wave transform. The normalized position dilation fρ(x) = ρdf(ρx) satisfies bfρ,+(p) = bf+(p/ρ) by change of variables. The same identity for the joint wave transform carries (TR9) to the source’s relative and joint Wightman distribution limits. We now prove the polynomial growth and Laplace continuity assertions used in (TR6)–(TR9), then the finite mass-shell obstruction. The subsequent stress correction and Ward calculation derive the trace coefficient and its Wilson normalization.
16.6.1 Polynomial bounds and the Laplace continuity argument
Which topology the spectral passage controls The spectral reader is first defined at positive time, where exponential damping makes its integral finite. Passing from these Laplace readings to a tempered distribution requires uniform polynomial growth bounds on the rescaled positive measures and a uniqueness argument on the allowed tests. The proof below supplies those bounds before taking the Schwartz limit. Convergence at a few positive-time values would not by itself justify arbitrary Schwartz smearing or the later spectral comparison tests.
Here are the measure arguments used above. They apply to positive measures; no assertion about convergence of arbitrary signed measures is needed. Suppose
is supported in the closed future cone and 
(TR10)
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The damped point-vector estimate supplies this inequality directly. Indeed, on the energy slab the weight e−2p0/R is at least e−2. If the squared norm of the vector at height R−1 is at most A0Rk, its spectral representation gives (TR10) with A = e2A0. A Euclidean ball of radius R is contained in that energy slab. On the dyadic sets 2j ≤1+∥p∥≤2j+1, the mass is at most A2k(j+1). Consequently Z (1 + ∥p∥)−k−1 dμ(p) ≤2k+1A. (TR11)
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The sets may overlap on their boundaries, which only enlarges this upper bound. This proves the polynomial temperateness needed for Schwartz testing. For completeness, let μi, ν be supported in the future cone C, with finite Laplace transforms at every u in its open dual cone C◦, and suppose Z e−u·p dμi(p) −→ Z e−u·p dν(p) (u ∈C◦). (TR12)
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The pairing here is the real Euclidean pairing used for the damping parameter. On the one-point
compactification C ∪{∞}, the functions su(p) = e−u·p, extended by zero at infinity, are continuous:
for p ∈C,
u · p ≥1
2(u0 −∥u∥)∥p∥.Their products satisfy
. Together with the constant function they separate points. They distinguish infinity from finite points by positivity; if two finite points have equal pairings with every
, perturbing the time unit vector by a sufficiently small multiple of any real vector shows that the two points coincide. The real Stone–Weierstrass theorem therefore makes their linear span uniformly dense in the continuous functions on this compactification. Fix
and damp all the measures by
Their total masses converge by (TR12). Integrals of the constant and of every
converge, since the latter are precisely the Laplace trans-
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forms at u + v. Uniform approximation and the bounded total masses give convergence against every continuous function on the compactification. If f is continuous and compactly supported in the ambient space, eu·pf(p), restricted to C and extended by zero at infinity, is such a function. R f dμi. Its damped integral equals This proves convergence against real compactly supported continuous tests, and then against complex tests by taking real and imaginary parts.
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If (TR10) holds with common A, k for all these measures and the limit, this convergence extends to every Schwartz function. To see this explicitly, put q = 2k+1A and choose B with |f(p)| ≤ B(1 + ∥p∥)−k−2. Let χR be a smooth cutoffequal to one on the radius-R ball, supported in the radius-R + 1 ball, with 0 ≤χR ≤1. For R ≥1, Z Z |f −χRf| dμi ≤Bq/R, |f −χRf| dν ≤Bq/R.
Choose
to make both tails smaller than a prescribed third of the error, then apply compact-test convergence to
. This proves (TR9) from (TR6)–(TR8), with all tail estimates uniform. The same density argument also proves uniqueness of the positive cone-supported measure from its all-future Laplace transform.
16.6.2 The finite polynomial comparison class and its obstruction
The mass-shell exclusion is a finite comparison statement. The allowed functions are the finite complex linear combinations of finite products of the specified mass-shell integrals, with the stated integer and homogeneity restrictions. Their radial expansions have the prescribed power and logarithmic form, incompatible with the inverse square logarithm of the trace. The separate Maxwell argument retains its own locality, sector and comparison hypotheses.
The polynomial-field exclusion has no bound on the number of colors or on the degree of the two chosen polynomials. Each elementary contraction, however, has a finite sum of fixed mass-shell kernels with finite polynomial momentum numerators. More generally, the proof permits a real continuous homogeneous spatial numerator
of nonnegative integer degree
, an energy power 
0, and a mass
0. Write
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ωm(p) = q m2 + |p|2, a(p) = ωm(p)kP(p) 2ωm(p) , p+ = (ωm(p), p). (TR13)
Values at the isolated massless zero momentum do not affect the integrals. A contraction track with
elementary factors acts on a total-momentum test by
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n X ! n Y L(φ) = Z n Y α=1 aα(pα) φ α=1 (pα)+ α=1 dpα. (TR14) (R3)n
The empty track is evaluation at zero. Allowed comparison kernels are finite complex linear combinations of these tracks. This includes units, arbitrary constant contraction coefficients and statistics signs; positivity of the comparison kernel is unnecessary.
Indeed, expand each polynomial into finitely many monomials. A cross-vertex pairing is a bijection between the occurrences at the two vertices. For each arc, expand its finite sum of elementary shell kernels, and then expand each polynomial numerator into monomials in energy and spatial coordinates. Each resulting term is (TR14); all arc coefficients, monomial amplitudes and pairing coefficients multiply into its constant coefficient. Summing the finite choices proves the asserted inclusion for any two finite polynomial fields. An infinite set of available generator labels causes no difficulty, since only finitely many occur in the two fields being compared.
At positive imaginary time
, a track is absolutely damped and its value factors:
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Z n Y R3 aα(p)e−sωmα(p) dp. (TR15) L(e−sp0) = α=1
Polar coordinates and homogeneity reduce an elementary factor to
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Z S2 P(θ) dθ Z ∞ 0 rj+2(m2 + r2)(k−1)/2e−s √ 1 2 m2+r2 dr. (TR16)
These formulas also give absolute integrability for compact total-momentum tests: such a test is bounded by a constant times
on its support.
The small-
radial expansion used here has the following exact meaning. For every integer
the radial factor differs by
) from a finite sum of terms
(log
, where
and
Finite sums and products preserve this property. For the product statement, first bound each factor by some integer power of
, then take the other factor’s expansion far enough to pay that bound. Thus the real part of every finite combination of (TR15) admits, in particular, such a finite expansion with bounded remainder. The radial expansion itself will be derived below from the massive integral; it is not an assumption on the comparison fields.
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No real function with such a bounded-remainder expansion can have a nonzero finite limit after multiplication by s8 log2(1/s). To prove this, the bounded remainder contributes zero. Combine equal terms in the finite expansion and set s = 2−n. A nonzero term becomes a constant times 2−(8+a)n nb+2. (TR17)
Order the finitely many terms lexicographically by their exponential base and then their polynomial degree. If the largest base is below one, every term tends to zero. If it is at least one, its term diverges in magnitude and every lower term divided by it tends to zero. Dividing a supposed finite limit by this largest term would force its nonzero coefficient to vanish. These two cases prove the assertion.
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It remains to justify that equality only on the product tests stated in the theorem would imply equality of these imaginary-time values. Every smooth compactly supported momentum test v can be represented as the relative wave transform of a tensor product of two Schwartz position tests. Choose the first Fourier test equal to one on a ball containing supp v; choose the second by Fourier surjectivity so that the relative product is v. Equivalently one may multiply v by the first Fourier test and its conjugate, whose squared modulus is one on this support. Hence equality on R v dμ and the finite track combination for every such v. the indicated tensor tests gives equality of
Choose smooth compact cutoffs
tending pointwise to one, with 0
1, and apply this equality to
. On the trace side, finite damped mass permits dominated convergence. On each track side, the absolute damped integrability in (TR15) permits the same passage. Therefore equality of the tested two-point functions would imply
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M X l=1 clLl(e−sp0) (s > 0). E(s) =
Taking real parts and applying (TR17) contradicts the positive limit (TR2). This proves the stated exclusion for every finite polynomial comparison of this class. The promised radial expansion follows from a convergent-integral argument, so it does not require summing a formal asymptotic series. Put
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Mj,k(s) = Z ∞ 0 rj+2 √ m2 + r2 (m2 + r2)k/2e−s √ m2+r2 dr.
For
0 these integrals are finite. The cases
1 are dominated respectively by
and 
; higher cases follow from the exact recurrence
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Mj,k+2 = m2Mj,k + Mj+2,k. (TR18)
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The gamma integral gives Mj,0(s) ≤Γ(j + 2)s−j−2. Differentiation under the integral is justified locally at s > 0 by the integrable bound with damping s/2, and gives M′ j,k = −Mj,k+1. Integration
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of the derivative of rj+3e−s √ m2+r2 gives
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sMj+2,0 = (j + 3)Mj,1, d ds sj+3Mj,1 = −m2sj+3Mj,0. (TR19)
The radial boundary terms vanish at zero and infinity. The derivative is integrable by the preceding bounds. Here is the finite-order integration rule used with (TR19). Say that
has expansion to order 
if
is a finite linear combination of integer powers of
times nonnegative powers of log
up to
has expansion to order
, and
1, then
has expansion to order 
+ 1. Every finite expansion has an antiderivative of the same kind: integrate
(log
directly, and use integration by parts for all other powers. After subtracting this antiderivative it remains to treat
1, integrate from
to a fixed positive point to obtain
1, the derivative is integrable at zero, so
has a finite limit and differs from that constant by 
). The arbitrary integration constant is itself an allowed expansion term. This proves the rule.
Start with the bound at order
2 for
. If it has expansion to some integer order 
, multiply by
, apply the integration rule to the second identity in (TR19), and divide back by that power. Since
1, the exceptional integration exponent is avoided. This yields an expansion of
to order
+ 1. Its negative is the derivative of
; integrating improves the latter to at least order
1, first weaken the derivative remainder to order 
2 and integrate that remainder, obtaining the required order
1. Otherwise the integration rule gives order
+ 2, which can be weakened to
+ 1. Induction supplies expansions of
to every integer order. The preceding deduction then does the same for
, and (TR18) proves it for every 
. This proves the expansion asserted after (TR16), and closes the polynomial-field exclusion.
16.6.3 The Maxwell comparison and the reduction to local sector estimates
The Maxwell conclusion concerns a specified comparison of representations. For
copies use the photon measure
), with its irrelevant zero-momentum value assigned arbitrarily, and the Hilbert direct sum
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Famb =
M
M
s:{1,...,n}→{1,...,m}×{1,2,3}
L2((R3)n, λ⊗n).
(TR20)
n≥0
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The physical subspace consists of waves symmetric under simultaneous permutation of momenta and labels and transverse in each slot. Transversality means invariance under applying Πab(k) = δab −kakb/|k|2 to that slot’s spatial component. Translation is multiplication by the plane wave at total momentum P(k) = P i(|ki|, ki). The vacuum is the unit wave in sector zero.
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The comparison hypothesis is a complex linear isometry U : H →Famb whose image is physical, which takes the selected vacuum to this vacuum and intertwines translations. It requires neither surjectivity nor an identification of all fields. For a list h of real Maxwell electric test functions, its one-particle wave is X Πab(k)|k| bhc,b,+(|k|, k). zh(c, a; k) = b
Its coherent wave has sector values
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n Y E(h)n,s(k) = e−∥zh∥2/2 in √ α=1 zh(sα; kα). (TR21) n!
Schwartz decay and
make
square integrable. The sector norm sum is the exponential series, so (TR21) defines a physical Hilbert vector. Put
)Ω. The excluded condition is the vacuum matrix identity
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⟨E(−h), σ(f)⟩= ⟨σ( ̄f), E(h)⟩ (TR22)
for every compact smooth
and compact smooth real list
whose supports are spacelike separated from that of
. The proof actually needs this only for
supported in one fixed ball of any positive radius. Thus it yields the precise small-region variant as well.
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Here is the initial spectral reduction, which uses only the comparison hypothesis. Let |Uv|2 dΛ denote the squared-wave measure on the direct sum (TR20). Its pushforward under P is the translation spectral measure of v: their characteristic functions are both ⟨v, T(b)v⟩. For v = Θ(f)Ω, this measure is | bf+|2 dμ. Polarization, also after bounded momentum multipliers, therefore gives
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bg+(P) σ(f) = bf+(P) σ(g) almost everywhere on the wave space. (TR23)
For clarity, the squared norm of the difference equals the corresponding norm in
), where the two products are identical. This proves (TR23) without pointwise evaluation of a Hilbert vector at exceptional momenta.
The remaining sector argument constructs common amplitudes
from (TR23), proves from (TR22) that they satisfy the local position identities, and uses their finite dilation spectrum. Its numerical end is particularly useful to state now. Write
, and let 
. Positivity gives
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νqn,s(Bn,j) ≤μ{p : p0 ≤(n + 1)Rj}.
(TR24)
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The trace asymptotic implies that the right side, multiplied by R−8 j , tends to zero. The local sector estimate proves that every sector with n ≥4 vanishes and that each of the remaining finitely many labeled sectors has νqn,s(Bn,j) ≤Cn,sR6 j. (TR25)
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Summing gives μ{p : p0 ≤Rj} ≤CR6
j, since an energy slab is contained in each sector’s correspond-
ing barrel. This contradicts the R8/ log2 R high-energy law already proved. The common-amplitude
construction and the local sector estimate are unfolded next.
Choose a countable sequence of Schwartz functions vj whose positive Fourier transforms have
modulus one on the radius-j ball. At each wave-space point choose the least j with bvj,+(P) ̸= 0,
and define
q = σ(vj)/bvj,+(P).
(TR26)
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This is measurable, since the selection is made from a countable measurable partition. For any fixed f, intersect the countably many full-measure sets in (TR23) for g = vj. On that intersection, σ(f) = bf+(P)q. Thus the construction gives one amplitude for all f, with the usual separate almost-everywhere interpretation for each test. Weighted spectral equality then implies
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Z
μ = P∗(|q|2dΛ),
μ(B) =
X
{P(k)∈B}
|qn,s(k)|2 dλ⊗n(k).
(TR27)
n,s
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To remove the Fourier weights, use the same vj: their squared weights equal one on an exhausting sequence of momentum balls. Equality on each ball and countable exhaustion prove equality of the measures. This proves (TR24), since P 0 = P i |ki| ≤nRj on Bn,j. The harmless n + 1 in that bound also covers the vacuum sector. We record explicitly the algebraic and measure-theoretic part of the local sector estimate. Treat amplitudes as measurable functions modulo almost-everywhere equality. For a slot i, let
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(Diq)(k1, . . . , kn) = q(k1, . . . , 2ki, . . . , kn).
The operators
commute. The photon measure scales with degree two: the pushforward under 
is
. For a parcel
this gives
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νDiq(B(d)) = 1 4νq B(d1, . . . , 2di, . . . , dn) . (TR28)
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The space of amplitudes with finite mass on every dyadic barrel and R−8 j νq(Bn,j) →0 is a real vector space, stable under every Di. Addition uses |q + r|2 ≤2(|q|2 + |r|2); stability follows from (TR28) and containment in the next barrel, with the fixed loss 26. Thus all finite polynomials in the Di preserve this growth restriction. The local distribution argument gives an integer K, common to the finitely many slots, such that K−1 Y
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a=0 (Di −2a−1)q = 0 (1 ≤i ≤n). (TR29)
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Here the exponent a−1 includes the electric-test factor |ki|. To split these identities while preserving the growth restriction, work entirely in the preceding invariant real vector space. A linear operator annihilated on a vector by a polynomial with distinct real roots decomposes that vector into its finitely many eigencomponents. One direct induction takes v = Q a<t(Di −2a−1)q, which is an eigenvector of eigenvalue 2t−1; subtract
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Y !−1 v a<t (2t−1 −2a−1)
from
and reduce the number of factors by one. All denominators are positive. Since the slot operators commute, this splitting can be repeated in every slot inside the eigenspaces already selected and inside the kernels of the other annihilating polynomials. Hence
is a finite real sum of simultaneous eigenwaves, each satisfying the growth restriction. Let
for such an eigenwave. Equation (TR28) yields
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νw(B(d1, . . . , 2di, . . . , dn)) = 22aiνw(B(d)), νw(Bn,j) = R 2P i ai j νw(Bn,0). (TR30)
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If ai = 0, shrink only its radius successively by halves. The parcel masses remain equal; continuity of a finite measure from above identifies their limit with the mass of the set ki = 0, which is zero. Starting with any barrel proves w = 0. Thus every nonzero eigenwave has ai ≥1. If P i ai ≥4, the growth restriction and (TR30) again force its base barrel mass to vanish, so w = 0. The surviving waves therefore have n ≤P i ai ≤3, and (TR30) bounds their barrel masses by a constant times R6 j. Finite summation using the squared triangle inequality proves (TR25), including sector zero.
The weaker projected and averaged versions require no new sector estimate. A self-adjoint bounded map
which is the identity on
can be moved across each inner product in (TR22), removing it from the coherent vector. For the open-region version, compact spacelike-separated supports have open neighborhoods with spacelike-separated closures, by compactness and openness of the spacelike relation. The neighborhood of supp
can also have closure inside the prescribed ball. Thus its asserted region identity implies the ball-local identity used above.
More generally, suppose a probability space indexes maps
preserving inner products between physical waves and fixing
pointwise. Since coherent vectors are physical,
, and likewise with the arguments reversed. The scalar integrands in the proposed averaged identity are therefore constant in
. Integration removes the averaging and gives the excluded local identity. This proves the averaged exclusion with exactly these hypotheses, without requiring boundedness, linearity, or a projection property of the individual
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We now prove the distribution step leading to (TR29). The polynomial slab bound for μ and (TR27) imply, for each fixed sector amplitude, a weighted square-integrability bound Z |q(k)|2 Y (1 + |ki|)−2d dλ⊗n(k) < ∞ (TR31) i
Source: A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition. Both papers belong to one Zenodo record. No external funding.