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The same identity holds for its lifted shot. On the recorded dyadic sets the weighted contributions are bounded by 2−(k+1)jA2k(j+1) = A2k2−j, X j≥0 A2k2−j = 2k+1A.
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This recovers TR11 and its temperate bound with the original constant.
For the Laplace continuity step fix u ∈C◦and write λi = suμi, λ = suν on the one-point
compactification. Their total masses are eventually bounded by some M, and the limit mass is
bounded by M as well, increasing M if necessary. The functions sv and constants retain their exact
multiplication and separating properties after readout. If h is continuous there and a finite linear
combination a of this algebra satisfies ∥h −a∥∞≤δ, then
Z
h dλi −
Z
h dλ
≤2Mδ +
Z
a dλi −
Z
a dλ
.The last term tends to zero by the finite collection of Laplace readings at
and
. First choosing 
and then the measure index proves convergence for
. For a compact continuous ambient test take
, extended by zero at infinity; its damped readout is exactly the original
integral. Thus the compact-test recovery is quantitative and uses bounded positive masses, rather than an uncontrolled undamping of test functions at infinity. The same argument for two measures with identical Laplace readings gives their identical compact-test readings and uniqueness.
Finally set
. Outside radius
1 the Schwartz bound gives
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|f −χRf| ≤B R(1 + ∥p∥)−k−1,
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so TR11 supplies each tail bound Bq/R. In particular
Z
f dμi −
Z
f dν
≤2Bq
R
+
Z
χRf dμi −
Z
χRf dν
.Choose
before the index. All weights, cutoffs and constants are unchanged scalar readouts, so this also recovers the lifted Schwartz limit TR9. The argument retains positive measures, the common polynomial budget and the full future damping family throughout.
24.5 The finite mass-shell polynomial comparison class
There is no bound on colors or on the degrees of the two finite comparison polynomials. Their elementary contractions, however, are finite sums of fixed mass-shell kernels with finite polynomial momentum numerators. More generally allow a real continuous homogeneous spatial numerator
of integer degree
0, energy power
0 and mass
0, with
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ωm(p) = q m2 + |p|2, a(p) = ωm(p)kP(p) 2ωm(p) , p+ = (ωm(p), p). (TR13)
The isolated massless zero value does not affect integration. A track of
factors is
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X ! Y L(φ) = Z n Y α=1 aα(pα)φ α (pα)+ α dpα. (TR14) (R3)n
The empty track evaluates at zero. Finite complex linear combinations allow units, contraction coefficients and statistics signs without positivity. Expanding the two polynomials into monomials,
cross-vertex occurrence bijections, each arc’s finite shell sum and each numerator’s energy/spatial monomials gives exactly finite tracks. Each resulting term is (TR14); all arc coefficients, monomial amplitudes and pairing coefficients multiply into its constant coefficient. Only finitely many generator labels occur, even if infinitely many are available. At positive imaginary time damping makes a track absolutely integrable:
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Z n Y L(e−sp0) = R3 aα(p)e−sωmα(p)dp. (TR15) α=1
Polar coordinates give each factor as
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Z S2 P(θ)dθ Z ∞ 0 rj+2(m2 + r2)(k−1)/2e−s √ 1 2 m2+r2dr. (TR16)
Compact total-momentum tests are absolutely integrable too, being bounded by a constant times this damping on their support.
Mass-shell track source and damping recovery. For a fixed track retain its finite ordered arc list, all masses, energy powers, homogeneous numerators and constant contraction coefficients. Its source space is (
, with the realization and weighted readout
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n X α=1 (ωmα(pα), pα), L(φ) = Z (φ ◦Q) Y α aα(pα) dpα. Q(p1, . . . , pn) =
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The source also retains the absolute weight Q α |aα| for integrability estimates. Horizontal copying of each momentum coordinate with its transported Lebesgue measure leaves this realization and readout identical. The empty list has the one-point source, Q = 0 and weight one, recovering the unit track. Each total energy is the sum of nonnegative shell energies, so
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e−sQ0 = Y α e−sωmα(pα).
For one factor, continuity of
on the sphere bounds it by a constant times
= 0 the absolute radial integrand is at most a constant times
, integrable at zero and infinity. At 
0 it is locally bounded by a constant times
near zero and has polynomial times exponential decay at infinity. Thus the absolute damped track integral is finite. Product-law Fubini recovers TR15 with its actual signed or complex factors. If a compact momentum test is supported where 
, then on the source 
This proves the required compact-test absolute integrability and the locally finite total-momentum readout, without requiring its weight to be positive.
For a single arc,
) and
, while the shell denominator contributes 
). These three exact factors recover TR16, including its 1
2 and radial power
. Its angular integral may vanish or have either sign; it is retained as part of the track coefficient.
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Finally the finite monomial expansion, occurrence pairing bijections, arc-shell choices and nu- merator expansion are recorded as a finite index set I. For the two chosen polynomial fields their comparison reader is exactly P ι∈I cιLι, with every statistics sign and contraction coefficient in cι. The lift recovers each summand and its coefficient before finite addition. There is no uniform bound on field degree or color count across the comparison class; only the finite data of each chosen
comparison are used. The subsequent radial-expansion obstruction applies to this same retained finite class.
The radial expansion proved below has, for every integer
, a finite sum of
(log
, with remainder
). Finite sums and products preserve it: bound one factor by an integer power and expand the other deeply enough to pay it. In particular each real finite damped track combination has such an 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. No such real function has a nonzero finite limit after multiplication by
log
). The bounded remainder vanishes. Combine equal terms and take
; each remaining term is a nonzero constant times
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(2−(8+a))nnb+2. (TR17)
Order the finite terms by exponential base then polynomial degree. If the largest base is below one all vanish. Otherwise the leading term diverges and every lower term divided by it tends to zero; dividing any supposed finite limit forces its nonzero coefficient to vanish, a contradiction. To transfer the theorem’s product-test equality to these damped values, represent any compact smooth momentum test
as the relative wave transform of two Schwartz tests. Choose one Fourier test equal to one on a ball containing its support, and the other by Fourier surjectivity to give 
; equivalently multiply by that first test and its conjugate, whose squared modulus is one there. Equality on tensor tests therefore gives equality on all such
. Insert
with compact
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smooth 0 ≤χn ≤1 tending pointwise to one. Finite trace damped mass and each track’s absolute damping justify dominated convergence, giving E(s) = PM l=1 clLl(e−sp0). Real parts and TR17 contradict the positive TR2 limit. This excludes precisely these finite polynomial comparisons, without extending the claim to arbitrary free or generalized-free classes. The lifted comparison source retains occurrence pairings, finite shell choices, coefficients and damping. Its track reader is TR14 and its imaginary-time reader is TR15. Product-test recovery and dominated cutoffre- moval identify the same reader as the trace Laplace transform under any proposed equality. The asymptotic obstruction therefore preserves the exact comparison class and its stated limitation.
Finite asymptotic obstruction and tested recovery. The track source retains each radial function and its finite expansion at the requested order, including its remainder bound. To make the product rule explicit, suppose
) for nonnegative integers
. Choose finite approximants
with
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f −F = O(sN+B), g −G = O(sN+A).
The orders may be increased so that
) as well. Then
). Finite products and sums therefore retain a finite power/log expansion to any requested order. Every identity here concerns the same scalar radial readout after lifting; the existence of its expansions retains the convergent-integral argument below as a dependency. At bounded remainder order, write the real finite combination as
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h(s) = X (a,b)∈I ca,bsa(log s)b + O(1), a ∈Z, b ≥0,
with equal pairs combined and zero coefficients removed. At
the normalized term is exactly
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s8 log2(1/s)ca,bsa(log s)b = ca,b(−1)b(log 2)b+2(2−(8+a))nnb+2.
This retains TR17’s nonzero coefficient and sign. If the finite index set is empty the normalized remainder tends to zero. Otherwise the largest exponential base and then largest polynomial
degree determine the leading term. A base below one gives zero; a base at least one gives divergent magnitude, since
2. All lower terms divided by it vanish. The same exact scalar sequence is recovered by the lift, so none has TR2’s positive finite amplitude.
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For product-test recovery take η ∈C∞ equal to one on supp v, and use Fourier surjectivity to c choose Schwartz f, g with bf+ = η and bg+ = v in the retained convention. Then bf+bg+ = v on all momentum space. Thus the actual tensor-test reader recovers every compact smooth momentum test without a density limit in the pair of fields. For fixed s > 0, insert vn = χne−sp0. On the trace side its limiting error is controlled by the finite positive damped mass; on each finite track it is controlled by the absolute damped product weight. Dominated convergence therefore recovers
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M X l=1 clLl(e−sp0) E(s) =
from any proposed equality of the stated tested two-point functions. Taking real parts gives exactly the preceding finite obstruction. The source–readout translation thus preserves the exclusion of the specified finite polynomial mass-shell comparison class and every coefficient or statistics sign, without changing that class’s scope.
24.6 Convergent-integral derivation of the radial expansion
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For s > 0 define Mj,k(s) = Z ∞ 0 rj+2(m2 + r2)(k−1)/2e−s √ m2+r2dr.
For
1 the bounds are respectively
and
; all higher cases follow from
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Mj,k+2 = m2Mj,k + Mj+2,k. (TR18)
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The gamma integral gives Mj,0 ≤Γ(j + 2)s−j−2. Local differentiation is justified by damping s/2 and gives M′ j,k = −Mj,k+1. Integrate the radial derivative of rj+3e−s √ m2+r2, whose boundary terms vanish at both ends and whose derivative is integrable, to obtain
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sMj+2,0 = (j + 3)Mj,1, (sj+3Mj,1)′ = −m2sj+3Mj,0. (TR19)
Say expansion to integer order
means a finite sum of integer powers and nonnegative log powers If
has order
1, then
has order
with remainder
Indeed each finite term has an antiderivative of the same type:
(log
integrates directly; other powers use integration by parts. Subtract this primitive. For the remaining derivative bounded by
integrate to a fixed positive point if
1, giving
1, integrability at zero gives a finite constant plus that remainder. The integration constant is an allowed term. Start
at order
2. Given order
, multiply by
and apply the integration rule to TR19, then
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divide: j + 3 + a ≥1 avoids the exceptional exponent and gives Mj,1 to order a + 1. Integrating −Mj,1 = M′ j,0 improves Mj,0 to at least order a + 1. If a + 1 = −1, weaken its derivative remainder to order −2 before integration to obtain order −1; otherwise the rule gives a + 2, which can be weakened to a + 1. Induction gives all integer orders for Mj,0 and then Mj,1; TR18 gives every k. This proves the finite expansions used after TR16, without summing an infinite formal series, and closes that comparison exclusion. The lifted radial source retains its actual damped integrals, derivative identities and finite-order remainder. Integration is the reader transport, with constants retained as allowed terms. The induction recovers each requested finite asymptotic horizon and its explicit error, exactly the data required by the finite-track obstruction.
Finite radial-horizon recovery. Retain (
) and the actual radial Lebesgue integral in the source record. Its horizontal scalar readout is the identical
) for every
0. Multiplying the integrand by
recovers TR18 before integration. Differentiating on a compact positive 
interval uses the same damping bound and gives the identical derivative reader
particular 
This explicitly recovers the second TR19 identity from the first and the finite recurrence, retaining all boundary and domination conditions. The finite primitive reader for a term is explicit. For
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Z s−1(log s)bds = (log s)b+1 b + 1 ,
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and for an integer l̸= −1 it is
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Z sl(log s)bds = sl+1 b X (−1)rb! (b −r)!(l+ 1)r+1 (log s)b−r. r=0
Both formulas retain an arbitrary additive constant. For a derivative remainder
1, integration from zero gives
+ 1). For
1, integration to a fixed positive point gives
), with the finite constant absorbed because
diverges. These bounds recover the source’s precise primitive rule without integrating an infinite series or discarding an integration constant.
Starting with the zero approximant and remainder order
2, record each finite primitive and its error. If
has order
2, the derivative of
has remainder order
Its primitive therefore has order
+ 4; division by
gives order
+ 1 for
. A further primitive gives order
+ 2 for
unless the remainder exponent is
1. In that exceptional case weaken the remainder to order
2 and recover order
1, exactly as in the source. Thus each cycle raises the guaranteed order by at least one. TR18 then supplies every fixed
by finitely many linear operations. The horizontal reader retains every finite approximant, constant and error bound. This completes the required translation of the radial-expansion dependency in the finite polynomial comparison obstruction.
24.7 Maxwell comparison and common sector amplitudes
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For m copies use dλ(k) = d3k/(2|k|) and
Famb =
M
M
s:{1,...,n}→{1,...,m}×{1,2,3}
L2((R3)n, λ⊗n).
(TR20)
n≥0
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The isolated zero-momentum convention is irrelevant. Physical waves are symmetric under simul- taneous label/momentum permutations and transverse under Πab(k) = δab −kakb/|k|2 in each slot. Translations multiply by total-momentum plane waves with P(k) = P i(|ki|, ki); the vacuum is the unit zero-sector wave. The comparison assumes a complex-linear isometry U : H →Famb with phys- ical image, preserving the selected vacuum and intertwining translations. Neither onto-ness nor a full field identification is required. For real electric tests h set zh(c, a; k) = P b Πab(k)|k|bhc,b,+(|k|, k) and
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E(h)n,s(k) = e−∥zh∥2/2 in √ n Y α=1 zh(sα; kα). (TR21) n!
Schwartz decay and
give square integrability; the exponential sector norm series defines a physical vector. With
)Ω, the excluded condition is
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⟨E(−h), σ(f)⟩= ⟨σ( ̄f), E(h)⟩. (TR22)
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It is required for compact smooth f and real compact smooth h with mutually spacelike supports. The proof needs f only in one fixed ball of any positive radius, giving the small-region variant. The pushforward P∗(|Uv|2dΛ) is the translation measure of v by equality of characteristic functions. For trace vectors it is | bf+|2dμ. Polarization with bounded momentum multipliers gives
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bg+(P)σ(f) = bf+(P)σ(g) almost everywhere. (TR23)
The difference’s norm equals its
) counterpart, identically zero, so no exceptional-point Hilbert evaluation is used. For
and
, the resulting common amplitudes obey 
(TR24)
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TR2 makes this upper bound o(R8 j). The local sector argument below kills every n ≥4 and bounds the remaining finite sectors by νqn,s(Bn,j) ≤Cn,sR6 j. (TR25)
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An energy slab lies in its sector barrel, so summing gives CR6 j, contradicting the previously obtained R8/ log2 R high-energy law. For the common amplitude choose countably many Schwartz vj with |bvj,+| = 1 on radius-j balls. On the measurable partition where j is the first nonzero Fourier weight at P, put q = σ(vj)/bvj,+(P). (TR26)
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For each fixed f, intersect the countably many TR23 full-measure sets to get σ(f) = bf+(P)q, with a separate almost-everywhere interpretation for each test. Weighted measure equality and exhaustion by those balls remove the weights, proving
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Z
μ = P∗(|q|2dΛ),
μ(B) =
X
{P(k)∈B}
|qn,s(k)|2dλ⊗n.
(TR27)
n,sOn a barrel
, proving TR24;
+1 also covers vacuum. The lifted comparison source retains the isometry, physical sector constraints, coherent probes and measurable amplitude partition. TR23 is its compatible smeared reader; TR26–TR27 recover a single amplitude measure from all test-weighted spectral readings without claiming simultaneous pointwise identities for uncountably many tests. The following local sector reader supplies the remaining TR25 bound; its scope remains precisely the TR22 comparison hypothesis.
Comparison-isometry and common-amplitude recovery. Keep the Maxwell ambient space, photon measures, labels and momentum map unchanged. If
is the reconstruction unitary, the transported comparison isometry is
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UZ = UV −1, (UZ)∗UZ = IHZ, UZΘZ(f)ΩZ = UΘ(f)Ω= σ(f).
Its physical image, vacuum and translation intertwining are therefore identical. No surjectivity is added. The transverse projector and simultaneous permutations are the same operators on the ambient wave space. The coherent probes retain their normalization explicitly:
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X n,s ∥E(h)n,s∥2 = e−∥zh∥2 X ∥zh∥2n n! = 1. n≥0
Thus TR22 has exactly the same two scalar matrix elements after transport, for the same spacelike-separated tests and the same permitted fixed small ball. Let
denote bounded multiplication by
) on the ambient space. Polarized spectral equality for trace vectors gives
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⟨Maσ(f), Mbσ(g)⟩= Z a(p) bf+(p)b(p)bg+(p) dμ(p).
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Applying this to the two products in TR23 gives the exact squared norm of their difference as R |bg+ bf+ −bf+bg+|2dμ = 0. Both multiplication operators are bounded since Fourier transforms of Schwartz tests are bounded. This recovers the wave-space identity without evaluating vectors at exceptional points. Retain the countable test sequence vj and the measurable pieces
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Aj = {bvj,+(P) ̸= 0} ∩
\
l<j
{bvl,+(P) = 0}.
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They exhaust the wave space, since each finite total momentum lies in some ball where a chosen weight has modulus one. The ratio reader q|Aj = σ(vj)/bvj,+(P) is identical under transport. For each fixed f, the same countable intersection of full-measure sets recovers σ(f) = bf+(P)q. On any momentum ball use a vj with unit-modulus weight there; weighted measure equality then recovers TR27 on that ball, and countable exhaustion recovers it on all Borel sets. Every labeled sector retains this same amplitude and squared-wave measure. Since P 0 ≤nRj on Bn,j, its barrel mass obeys TR24 with the same harmless n + 1 allowance for the zero sector. Conversely P 0 ≤Rj forces every |ki| ≤Rj. Once the subsequent local sector result supplies vanishing for n ≥4 and TR25 for the remaining finite labeled sectors, summing those barrel bounds gives the same energy-slab estimate. Thus the final comparison obstruction retains precisely its original local-sector dependency and high-energy trace-law dependency in the lifted representation.
24.8 Sector dilation algebra and projected comparison variants
Amplitudes are measurable functions modulo almost-everywhere equality. Define the commuting slot dilations by 
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Photon measure scaling (k 7→rk)∗λ = r−2λ gives
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νDiq(B(d)) = 1 4νq(B(d1, . . . , 2di, . . . , dn)). (TR28)
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Amplitudes finite on every dyadic barrel with R−8 j νq(Bn,j) →0 form a real vector space, by |q+r|2 ≤ 2(|q|2 + |r|2), stable under Di by TR28 and containment in the next barrel with fixed loss 26. Thus every finite polynomial in the Di preserves this restriction. The local distribution argument below supplies a common finite K with
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K−1 Y a=0 (Di −2a−1)q = 0 (1 ≤i ≤n). (TR29)
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The shift a−1 includes the electric factor |ki|. Distinct real roots split q into finite eigencomponents within the invariant space. Explicitly, if v = Q a<t(Di −2a−1)q, it has eigenvalue 2t−1; subtract v/ Q a<t(2t−1 −2a−1) to remove that factor, with positive denominators. Commuting slot operators
permit successive splitting inside already chosen eigenspaces and the other polynomial kernels. Each simultaneous eigenwave retains the growth bound. For
, TR28 gives
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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, shrinking that radius by halves leaves masses equal; continuity from above for finite parcel mass gives the mass at ki = 0, which is zero. Exhaust barrels to get w = 0. If P i ai ≥4, the sub-R8 restriction forces zero base barrel mass and again zero wave. Nonzero waves therefore satisfy n ≤P i ai ≤3, and have mass at most CR6 j. Finite squared triangle bounds yield TR25, in- cluding sector zero. This completes its algebraic conclusion once TR29 is established. For projected variants, a bounded self-adjoint map acting identically on UH moves across TR22’s inner products and is removed from coherent vectors. Compact spacelike-separated supports have open neighbor-
hoods with spacelike-separated closures; the trace neighborhood can have closure inside the chosen ball. Thus the region identity implies the same ball-local identity. For probability-indexed maps 
preserving inner products between physical waves and fixing
pointwise, physical coherent vectors satisfy
and its reversed counterpart. Scalar integrands are constant in
, so averaging removes the maps and gives the excluded identity. This uses precisely these conditions, without adding boundedness, linearity or a projection condition for individual
. The weaker projected and averaged versions require no new sector estimate. The lifted sector source
retains the invariant growth space, all commuting dilation operators and finite polynomial kernels. Eigencomponent readout stays within that same growth restriction; TR30 recovers the parcel masses with exact photon-measure scaling. The zero-slot and large-degree branches are removed by finite-measure continuity and the retained ultraviolet bound, not by pointwise wave evaluation. Projection and averaging readers preserve exactly the matrix hypothesis specified above.
Dilation eigencomponents and comparison-variant recovery. The horizontal Maxwell comparison has the identical wave amplitudes on the same photon measure space. Thus every slot dilation
and almost-everywhere equivalence class is unchanged by readout. Since
change of variables in one slot recovers TR28 with factor 1
4. In particular
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R−8 j νDiq(Bn,j) ≤26R−8 j+1νq(Bn,j+1) −→0.
This proves that all finite polynomial readers in the commuting
remain within the same growth space. A fully explicit simultaneous splitting of TR29 uses its distinct roots
On the cyclic polynomial space generated by
, set
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Ei,a = Y Di −λb λa −λb , wa1,...,an = Y i Ei,aiq. 0≤b<K b̸=a
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The denominators are nonzero real constants. Polynomial interpolation modulo the annihilating polynomials gives q = P a1,...,an wa1,...,an and Diwa1,...,an = λaiwa1,...,an. These are finite algebraic operations, with no bounded-operator spectral theorem assumed for Di on this growth space. Read- out preserves every component and its growth restriction. For an eigenwave TR28 gives the exact parcel ratio 4|λai|2 = 22ai. If ai = 0, decreasing only its radius to zero leaves the parcel mass constant; finite-measure continuity identifies it with the zero mass of ki = 0. If P i ai ≥4, TR30 and the vanishing normalized barrel mass force the base
Source: Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction. Both papers belong to one Zenodo record. No external funding.