The centered trace and its non-freeness test · Part 5

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for some integer d.
Indeed a sector ball of radius R lies in the total-energy slab (n + 1)R, so
its squared-wave measure has polynomial ball growth; apply (TR11) and then dominate its in-
verse radial weight by the product weight with d sufficiently large. Cauchy–Schwarz and
R (1 +
|k|)−4dλ(k) < ∞show that |q(k)| Q
i(1 + |ki|)−d−2 is integrable. Every integral below is controlled
by this one after taking sufficiently high Schwartz seminorms. Slot dilation preserves (TR31), since
1 + 2|k| ≤2(1 + |k|).
Use bg−(p) = bg+(−p). After selecting a label sector, the consequence of (TR22) is
Z
q(k)
Y
j
|kj|bgj,−((kj)+) dλn =
Z
q(k)
Y
j
|kj|bgj,+((kj)+) dλn
(TR32)

whenever every is compactly supported in the spacelike region relative to zero. Notice that no separation between two different’s is required. To pass from the ball-local identity to zero, smear the trace with a real normalized bump). For largeits support is inside the chosen ball and spacelike to all the finitely many supports. Its Fourier transform tends pointwise to one and has modulus at most one. The sector coefficient identity extracted from (TR21) therefore tends to (TR32) by the integrable bound just established. Extraction of those coefficients, including the use of symmetry and transversality, is given below.

Keep all slots but tested, and define the two equal-time data functionals

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Ab(f) =
Z
q(k)|ki|b+1 bf−(ki)
Y
j̸=i
|kj|bgj,−((kj)+) dλn,
Bb(f) =
Z
q(k)|ki|b+1 bf+(ki)
Y
j̸=i
|kj|bgj,+((kj)+) dλn,
(TR33)
Tb(f) = Ab(f) −(−1)bBb(f).

Here is a spatial Schwartz test. These are continuous complex linear distributions. More precisely their norms are bounded by a constant, depending on and the other tests, times the fixed seminorm

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2J∥bf−∥0,0 + ∥bf−∥J,0 + ∥bf+∥0,0 + ∥bf+∥J,0

,
J = d + 2 + b + 1.
(TR34)

The order of this seminorm does not depend on the other tests. If has compact support avoiding zero, insert into slot the spacetime testx ). Its support is eventually spacelike to zero. Its two Fourier transforms in the time variable carry the factors ( and (, respectively. Applying (TR32), then taking, proves Dominated convergence follows from (TR31) and the modulus-one bound for the Fourier transform of the normalized bump. Thus is supported at the spatial origin.

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A distribution controlled by a fixed continuous Schwartz seminorm and supported at zero de-
pends on a uniformly finite jet at zero.
Here is the proof with the uniformity needed in this
application. Bound that seminorm by a finite maximum of Schwartz seminorms and let L be the
largest derivative order occurring. If all derivatives of f through order L vanish at zero, multiply
f by a fixed smooth cutoffχ(sx) equal to one near zero. Point support gives Tb(f) = Tb(χ(s ·)f).
On the cutoffsupport |x| ≤C/s, Taylor’s theorem bounds the derivative of f of order L −j by a
constant times s−j−1. Each order-j cutoffderivative costs sj; the Leibniz formula therefore bounds
all the finitely many required seminorms by Cf/s. The weights are bounded on this shrinking
support. Letting s →∞proves Tb(f) = 0. Hence some Lb, depending only on the seminorm in
(TR34), suffices for every choice of the other remote tests and every slot.
Let C2f(x) = 23f(2x). Change variables in the photon measure to obtain

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Tb[Diq](f) = 2−3−bTb[q](C2f).
(TR35)

The order- jet of times that of. For = 0, the induced factor on this jet is 2, the root 2 in (TR29); for, the root indexed by. Choose Applying the polynomial in (TR29) to transfers by (TR35) to a polynomial on tests that kills every jet through, for both 1. Therefore the resulting amplitude has on all spatial tests, for every choice of the other remote tests.

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These two zero data imply r = 0, as follows. Take a compact smooth momentum test w avoiding
zero. In the b = 0 identity use the inverse negative Fourier test of |k|w(k); in the b = 1 identity use
that of w(k). Multiplication by |k| preserves smoothness on this support. The negative-frequency
parts of the two identities agree, as do their positive-frequency parts, but the latter have opposite
signs. Adding the identities isolates the negative-frequency integral and proves
Z
Y
|kj|bgj,−((kj)+) dλn = 0.
r(k)|ki|2w(ki)
(TR36)
j̸=i

Translate each a sufficiently small amount; it remains remote by openness and compactness. Multiply by the corresponding shell plane wave in its slot, still a smooth test avoiding zero. Equation (TR36) then says that the Fourier transform, on a neighborhood of zero in all spacetime translation variables, of its integrable integrand vanishes.

An integrable function on a product of forward massless shells with this property vanishes almost everywhere. To prove it, restrict its Fourier transform to a line in a strictly future direction. Nonnegative shell energies give a bounded continuous function on the closed upper half-plane, holomorphic in its interior; the derivative is justified by the bound. A zero boundary interval extends by Schwarz reflection and the holomorphic identity theorem to the whole line. First continue along strictly future lines, then express an arbitrary displacement as the difference of two sufficiently large strictly future displacements. The Fourier transform vanishes everywhere. Restricting to spatial translations and using ordinary Fourier uniqueness of the resulting finite complex measure proves the assertion.

Finally choose a sequence of remote bumps whose shell Fourier transforms are eventually nonzero at every fixed momentum, obtained by shrinking a normalized bump around a fixed spacelike point. Choose also spatial cutoffs avoiding zero that are eventually one at every nonzero momentum. Intersect the countably many almost-everywhere conclusions for their integrands in (TR36). Almost every shell momentum is nonzero, so all the multiplicative factors are eventually nonzero. It follows that = 0 almost everywhere. Apply this to each slot’s polynomial amplitude to obtain (TR29). This completes the locality-to-dilation argument.

To finish the promised extraction in (TR32), define for a physical vector

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Z
Lx(g1, . . . , gn) =
X
xn,s(k)
Y
i
zgi(si; ki) dλn.
s

Cauchy–Schwarz makes every term integrable. Decomposition of the Hilbert inner product by orthogonal sectors gives the convergent coherent expansion

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⟨x, E(h)⟩= e−∥zh∥2/2 X
in
√
n!
Lx(h, . . . , h).
(TR37)
n≥0

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For ⟨E(−h), x⟩, the corresponding coefficients are inLx(h, . . . , h)/
√
n!.
Apply (TR22) to th for
every positive real t. The identical positive Gaussian factors cancel. The resulting power series is
zero for every t > 0, and each coefficient is zero. One can justify coefficient uniqueness without an
analytic-series theorem: after the first N coefficients vanish, divide by tN; the tail is dominated for
0 < t ≤1 by the absolutely summable series at t = 1, so t ↓0 recovers the next coefficient.

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Each Lx is multilinear and symmetric. Symmetry follows by permuting the momentum variables
and the labels together, using physical symmetry of x and invariance of the product photon measure.
The diagonal identity for real lists therefore polarizes to arbitrary real lists. For example, expand
the diagonal polynomial at h+tu; its derivative replaces one slot by u, and symmetry identifies the
sum with n times that term. Induction on n recovers all mixed terms. Compact tests supported
spacelike to the trace support form a complex vector space, and every such list is a complex
linear combination of real lists by its real and imaginary parts, with the same support restrictions.
Complex multilinearity then gives the polarized identity for all these lists, with complex conjugation
of the lists on the conjugated side.

Transversality removes the polarization matrices in these integrals. For a fixed slot expand ; move the real symmetric matrix Π onto the wave component in the finite sum over spatial labels. That component is unchanged because is transverse. Repeat in every slot. More explicitly, exchanging the selected component label and the summed matrix index is a bijection of the finite label-index pairs; all other slot factors remain fixed. No regularity of wave representatives is used, since there are only finitely many almost-everywhere transverse identities at each particle number.

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Finally take in slot i a list having just the component si nonzero, with scalar test gi. After
the projectors are removed, all label sectors except s vanish in the sum. The surviving polarized
identity is
Z
Z
Y
Y
|ki|bgi,−((ki)+) dλn =
|ki|bgi,+((ki)+) dλn,
xn,s(k)
yn,s(k)
i
i

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where x = σ(f) and y = σ(  ̄f). The Fourier conjugation identity b ̄g+(p) = bg−(p) fixes the signs.
Taking the real shrinking trace bump as described above yields (TR32). Together with (TR26)–
(TR36) and the high-energy contradiction, this completes all the Maxwell exclusions asserted here.

16.6.4 Passing a running trace label through the scaling limit

There are two separate ingredients in the trace limit: convergence of the running label in its finite-dimensional field space, and convergence of the moments evaluated on that label. The latter passage does not require a uniform convergence assertion on an infinite-dimensional space. Let be the fixed free field space of degree, choose a basis, and let be the scaling identifications. The source proves, for every pair of basis vectors,

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⟨Uiea(xi)Uieb(yi)⟩χ −→⟨ea(x)eb(y)⟩free
(TR38)

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along the same chosen refinement and in every selected phase under consideration. The rounded
scaled configuration tends to the target configuration; its rounding error is at most 4n/mi, where
the mesh magnification mi →∞. Generic separated configurations stay in their open admissible
region. The comparison error is bounded by oi(1 + M(xi, yi)), where oi →0 and the finite profile
norm M is continuous at the separated target. This proves (TR38).
If vi, wi ∈Ed tend to v, w, expand them in the chosen finite basis. Every coordinate converges,
and bilinearity expresses the moment as the finite sum
X

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a,b
(vi)a(wi)b⟨Uiea(xi)Uieb(yi)⟩χ.

Taking the limit term by term gives the free moment of. This is the passage used for the running centered trace, rather than an interchange of an uncontrolled varying field with a distributional limit.

The centering convention is also harmless at this step for a specific reason. If is the constructed stress tensor, replace it by1. This preserves symmetry, reality, conservation and its charges: derivatives of the unit vanish, and a constant symmetric unit tensor has zero charge contribution in the stated Ward construction. Its trace differs from the phase-centered trace by a constant multiple of the unit. Under the inverse scaling map the latter contributes

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Bir4
i c 1,

which tends to zero by the running-band estimate. The scaling maps identify the unit exactly. Thus the tensor normalization can first be proved with the convenient centered tensor and then transferred to the exact selected-vacuum trace. In the source’s coupling convention

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b0 = 11 Cadj
48π2 ,
βslope = 2b0,
d2 = −b0
2 .
(TR39)

The last identity follows from the stress coefficient = forceR after substituting the displayed slope; its derivation from the stress comparison is treated with that comparison below. Once the running centered trace converges to 4, the preceding finite-basis argument gives its two-point limit. The free Wick calculation yields 3 dim g). The running charge contributes after multiplication by log). Consequently the constant is

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(2b0)−2(4d2)2 3 dim g
π4
= 12d2
2
b2
0
dim g
π4
= 3 dim g
π4
,

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as claimed in (TR2). The powers and logarithms cancel before taking the limit: the two running
labels each carry Bir4
i , and the prefactor is (log(1/ri)/Bi)2.
The running-clock cancellation used here follows directly from the calibrated clock, including
the extra magnification in the scaling sequence. Write

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bβi = β∗+ g ti,
ri = e−tiai,
g = 2b0 > 0.

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The chosen magnifications satisfy ai ≥1 eventually and 0 ≤log ai ≤Gate( bβi), where Gate(x)/x →
0.
The calibrated coupling sequence and the band charge satisfy βi/ bβi →1 and βi/Bi →1.
Therefore
log(1/ri)
 
ti −log ai
! bβi
βi
Bi
−→1
Bi
=
bβi
βi
g.
(TR40)

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Here ti/ bβi = (1−β∗/ bβi)/g, and the magnification term tends to zero by its bound. Also log(1/ri) →
∞, because ri ↓0 along the scaling sequence. The elementary exponential limit le−4l→0, with
l= log(1/ri), gives

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Bir4
i = log(1/ri)r4
i
log(1/ri)/Bi
−→0.

This proves both the logarithmic factor and the disappearing unit contribution used in the trace argument.

16.6.5 The three finite candidates and their common extraction

All three finite tensor candidates, their individual stencil derivatives and their divergences are retained in one extraction. Thus a divergence is the sum of derivatives of those same finite functions.

The correction does not combine unrelated limiting candidates whose coefficient coordinates happen to agree.

Here are the actual finite functions behind the three columns used below. Write curv(Hol)), with the curvature coordinate and normalization of (SQ1)–(SQ12), and put . At a fixed issued ticket, the three tensors are

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J∅
ij(x) = Meld(J)ij(x),
(
0
i = j,
P
a Pija(x)
i ̸= j,
Jsh
ij (x) =
(TF11)
X
Jsc
ij (x) = δij
a,b
Paab(x).
4

The first is exactly the restored, localized principal tensor (TF0d), including its six antisymmetric differences and its scalar forward difference. The other two use elementary plaquette holonomies; there is no new stationary current in their definition. Their free coefficient vectors in the three-coordinate tensor basis are respectively the principal vector, (0 1), and (0 0). In the source’s three-coordinate convention the principal free vector isweyl. Its physical sign is fixed together with the Ward charge in (TR41)–(TR44); it cannot be changed independently of that normalization.

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All sixteen entries of each tensor, all sixteen individual stencil derivatives, and all four diver-
gences are extracted on the same filter. Explicitly these are the origin functions Jb
ij(0), elmi Jb
ij(0),
and P
i elmi Jb
ij(0), for b = ∅, sh, sc. The tensor entries have degree-four coefficient records and
the derivatives have degree-five records. These records include both the finite-cutofflabel and its
retained limit; extraction of the joint correlation book gives the corresponding fields Li, Ωi, ∆i.
Thus extraction of a divergence is extraction of the sum of its actual finite derivatives, not an
independently selected vector limit.

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The local bounds needed here have a direct source.
A plaquette has four links.
If
each matrix link has derivatives through order k bounded by the chart price d, its holon-
omy has price travelCost(G, k, 4, d).
Applying the curvature map gives price z(d)
=
The bilinear product has price 2k∥⟨·, ·⟩∥z(d)2.
curvatureCost(G) travelCost(G, k, 4, d).
Sum
the finitely many axes and multiply the scalar branch by 1/4. This gives a bound independent of
the torus and the ticket, supported in radius four. One stencil derivative enlarges the radius to
eight. The scheduled small-field extension makes these bounds into the coefficient and correlation
bounds used in extraction. The issued principal tensor has these same bounds by (TF0d) and its
localization estimates. The condition 20(high(x) + rim(x) + 10) < n makes all integer offsets used
in the coefficient tests nonwrapping.

Taylor projection commutes with the stencil divergence on these records: expand the finite translates and use the stencil moments from (TF0b). The degree-four stress coordinate therefore maps to its degree-five divergence coordinate. For the principal branch the antisymmetric difference corrections have zero relevant tensor coordinate; the scalar difference has zero trace coordinate. Consequently its free divergence scalar coordinate vanishes. This is the specific low-degree statement used later; it does not say that its entire finite divergence vanishes. The finite action, localized current, surgery polynomial and coefficient equations are issued by one ticket, so the six terms of

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(LCE6) concern this same first branch. The two elementary branches supply the correction columns
in (TR41).
i U−1
With YiT = r4
T and Pi = BiYi, their normalized limits and correction are precisely
i
(TR41)–(TR44): solve the two divergence coordinates for ci,0, ci,1, then set eTi = BiLi+ci,0Ωi+ci,1∆i.
This specifies the finite origin, normalization, and limiting meaning of all three candidates and both

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correction objects.

16.6.6 Correcting the stress tensor and fixing its amplitude

The two correction coordinates are solved by an eventually invertible finite-dimensional detector matrix. Its kernel agrees with the true divergence kernel on the stated stress space. This converts small detector errors into an exactly conserved tensor, while the weighted scalar sensor fixes the trace amplitude. The local estimates following the inverse argument supply the quantitative bounds needed for this stronger conclusion.

The coefficient in (TR39) is earned by an exact finite-dimensional correction, not by inserting an assumed trace anomaly. The three lifted stress candidates are denoted Ω ∆. Their free limits are respectively the Maxwell stress tensor, a traceless correction, and a scalar tensor with trace 4. All lie in the centered symmetric, real, rotation-covariant stress space. Let be the two-component detector map for their divergences, and define

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Si(c0, c1) = c0Ωi + c1∆i,
Xi = BiKiLi.

The maps converge to an invertible free map. In finite dimensions this gives eventual invertibility and a uniform inverse bound: identify matrices in fixed bases, write the nearby matrix as ), and use the convergent geometric inverse when2. Solve

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(KiSi)ci = −Xi.
(TR41)

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The available detector estimate bounds ∥Xi∥≤CRi, hence ∥ci∥≤C′Ri, with Ri/Bi →0. The
detector map has exactly the same kernel as the true divergence on this stress space. Consequently
eTi = BiLi + Sici is exactly conserved for every sufficiently large i.
The distinguished scalar detector li satisfies, for a nonnegative control ti with Ri ≤ti,

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li(Xi) →−g,
tili(KiΩi) →0,
ti
li(Ki∆i) + 4

 →0.
(TR42)

The coefficient bound therefore kills the two error products. Applying to (TR41) gives

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0 = li(Xi) + ci,0li(KiΩi) + ci,1li(Ki∆i).

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Equivalently,
4ci,1 = li(Xi) + ci,0li(KiΩi) + ci,1
li(Ki∆i) + 4

.

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Thus ci,1 →−g/4 = forceR. The factor four is the exact scalar detector value, and the sign follows
from the displayed equation. The primitive comparisons supplying (TR42) are unfolded with the
detector construction.
Write YiT = r4
i (U−1
i
T), component by component, and Pi = BiYi.
The normalized lifted
candidates satisfy PiLi →TMaxwell, PiΩi →TΩ, and Pi∆i →T∆. Therefore

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Yi eTi = PiLi + ci,0
Bi
PiΩi + ci,1
Bi
Pi∆i −→TMaxwell.
(TR43)

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For the scaled trace, the stronger traceless estimates give Bi tr(PiLi) →0 and ci,0 tr(PiΩi) →0.
The remaining trace tends to
Bi tr(Yi eTi) −→4 forceR Oscale.
(TR44)

Only the first correction coefficient requires a growth bound here; the scalar coefficient has the actual finite limit just proved.

The conserved centered stress space is one-dimensional. To see why this rank statement follows from the same construction, being onto makes onto. Its kernel equals the divergence kernel. The real dimension of the centered stress space agrees with the corresponding free dimension by the scaling and unit identifications. The detector target has the free divergence-target dimension. Rank–nullity and the free calculation, whose divergence kernel has dimension one, therefore show that the physical divergence kernel has dimension one as well. Let be the actual centered stress tensor with its translation Ward charges. It is nonzero.

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Indeed the free derivative map is nonzero: its divergence target contains a nonzero coordinate and
is attained; if every observer vanished on every first derivative, the separating observers would make
that target zero, a contradiction. Choose an observer with a nonzero derivative value y. Its free
translation-flux Ward identity has value y. The corresponding physical shell-flux test of R, with
positions, cutoffand fields scaled together and with the single factor ri prescribed by the derivative,
tends to y. If R = 0, every such test would be zero, contradicting y ̸= 0.
Since the kernel is one-dimensional, eTi = viR for real vi eventually. The shell-flux test of R
tends to y, while its test on eTi also tends to y, by (TR43) and the tested derivative comparison.

Linearity makes the latter times the former. Divide by the nonzero limiting value to obtain 1. Then= 0 eventually, and multiplying (TR43) and (TR44) by transfers both limits to. Centering back to the selected-vacuum trace uses the vanishing unit term proved in (TR40). This proves the running trace-label limit required for (TR2).

There is also an exact scalar-coordinate formulation. The invariant scalar degree-four space has the real basis 1, with injective coordinate map (. A linear left inverse exists and is continuous in this finite-dimensional space. Applying it to the running trace limit gives coefficients with

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U−1
i
(r4
i Θ) = ei 1 + 4di Oscale,
Bidi →d2,
Biei →0.

The exact scalar membership, followed by this left inverse, justifies these simultaneous coefficients without selecting unrelated subsequences. The two correction columns have a concrete free-coordinate description. The free divergence detector is a surjective map with

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ker V = R(1, −1
4, 1).

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The two columns are Ω= (0, 0, 1) and ∆= (0, 1
4, 0). Their images are independent: if xΩ+ y∆=
c(1, −1
4, 1), coordinate zero gives c = 0, and the other coordinates give x = y = 0. They span the
quotient: for w ∈R3,

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(w2 −w0)Ω+ (4w1 + w0)∆= w −w0(1, −1
4, 1).
(TR45)

Thus their detector images form a basis of. This is the free invertibility used in (TR41). Here are the precise error transfers behind (TR42). Let be the moving map from the finite bare degree-five coordinate space to the free divergence target, within operator norm. For a lifted candidate, its detector value and its bare divergence coordinates satisfy

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HN
i (pi −siui) →0
for every fixed N ≥0.
(TR46)

The increasing clock dominates the positive control For the two correction candidates, , with Ωor ∆, respectively. In particular (TR46) at = 0 gives the two column limits.

The scalar functionals are uniformly bounded and approximate a fixed bare functional with the stronger rate sup (TR47)

For each correction candidate the bare comparison also gives0. Decompose the full scalar error into the three terms

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ti(li(pi) −L(u∗)) = tili(pi −siui)
+ ti(li(siui) −L(ui))
+ ti(L(ui) −L(u∗)).

The first tends to zero by boundedness of, and (TR46) at = 1. The second tends to zero by (TR47) and boundedness of. The third is the stated bare comparison. The exact free scalar readings are) = 0 and4, giving the last two limits in (TR42). For the principal candidate, a renormalized bare divergence satisfies

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Xi −sivi →0,
∥vi∥≤CRi,
Ri ≤ti,
L(vi) →−g.
(TR48)

Although need not stay bounded, (TR47) still pays its growth:

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|li(sivi) −L(vi)| ≤εi
ti
∥vi∥≤Cεi →0.

Uniform boundedness of handles the first difference in (TR48), proving. Operator-norm boundedness of, the bound on, and1 eventually give. Thus the same estimates supply both the scalar limit and the norm bound needed to solve the correction equation.

The elementary absorption principle used throughout this step is: if, and 0 in a normed space, then0. It follows immediately by taking norms. In (TR48), the first difference is obtained by applying this principle to the two preexisting comparisons: the physical detector differs from the candidate’s bare coordinate reading by an error smaller than every inverse power of, and the candidate’s renormalized divergence differs from by an error of the same strength. Since, both errors remain negligible after multiplication by This explains why growing running factors can be inserted in (TR42)–(TR44).

The scalar membership used after (TR44) follows from the same free degree-four classification. A real scalar, fixed by all signed orthogonal symmetries, gives a diagonal tensor belongs to the classified free symmetric stress space. Writing it in that space’s real coordinates its trace has the form tr

Thus, as required. The scalar-coordinate map is injective: expectation kills and sends the unit to one, forcing the unit coefficient to vanish; the scalar field is nonzero by its positive free two-point norm, forcing the remaining coefficient to vanish. The running inverse scaling preserves conjugation and signed symmetries, so it preserves this exact scalar membership. The physical map has the exact formula

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(KiT)a = Bir5
i U−1
i,5
(div T)a

.
(TR49)

The real symmetry restrictions make this a map between the stated real tensor spaces. Its prefactor is positive and is invertible, so = 0 if and only if div = 0. This proves the kernel identification used above. The bare-to-physical degree-five map is likewise the componentwise