The centered trace and its comparison exclusions · Part 3

mass to be zero. The surviving finite components have n ≤P
i ai ≤3. For M such components,
the explicit bound | PM
r=1 wr|2 ≤M P
r |wr|2 recovers TR25. Sector zero retains its one-point wave
and constant mass. The local distribution proof of TR29 remains the required dependency for this
finite algebraic conclusion.
For a bounded self-adjoint comparison map P fixing UH, the exact scalar equality

Polynomial slab growth and TR27 give, for some integer d,
Z
|q(k)|2 Y
i
(1 + |ki|)−2ddλ⊗n < ∞.
(TR31)

A sector ball lies in energy slab (n + 1)R; apply TR11 and a sufficiently large product weight.
Cauchy–Schwarz and
R (1 + |k|)−4dλ < ∞show that |q| Q
i(1 + |ki|)−d−2 is integrable, controlling
all ensuing integrals by fixed Schwartz seminorms. Slot dilation preserves the bound by 1 + 2|k| ≤
2(1 + |k|). With bg−(p) = bg+(−p), sector extraction from TR22 gives
Z
q
Y
j
|kj|bgj,−((kj)+)dλn =
Z
 ̄q
Y
j
|kj|bgj,+((kj)+)dλn.
(TR32)

Its order is independent of the other tests, though its multiplier may depend on them. For compact
f away from zero, use spacetime f(x)∂b
t al(t) in TR32. Eventually it is remote; its time Fourier

Weighted locality-data recovery.
For a fixed nonzero sector retain its amplitude and photon
coordinates. Its horizontal wave readout is the same q, so every weighted norm in TR31 is un-
changed. Writing k = (k1, . . . , kn), one has Q
i(1 + |ki|) ≥1 + maxi |ki|; choosing 2d large enough
therefore bounds that product weight by a constant times the inverse radial weight furnished by
polynomial ball growth. In particular Cauchy–Schwarz gives the explicit integrability budget
Z
|q|
Y
i
(1 + |ki|)−d−2dλn

Order-j jets of C2f scale by 23+j, yielding roots 2j for b = 0 and 2j−1 for b = 1. With K = L0+L1+2,
TR29’s polynomial kills all these jets. Its amplitude r therefore satisfies T0[r] = T1[r] = 0 for every
spatial test and every other remote test. For compact smooth momentum w away from zero, use
inverse negative Fourier tests of |k|w for b = 0 and of w for b = 1. The negative parts agree and
positive parts have opposite signs; adding isolates
Z
r|ki|2w(ki)
Y
j̸=i
|kj|bgj,−((kj)+)dλn = 0.
(TR36)

Dilation polynomial and zero-amplitude recovery.
The slot readout retains q, its photon
measure and both data functionals.
Changing variables ui = 2ki gives a factor 2−2 from the
measure and 2−(b+1) from the electric/energy power; d
C2f±(ui) = bf±(ui/2) supplies the remaining
test transformation.
This recovers exactly 2−3−b in TR35.
For the real polynomial PK(z) =
QK−1
a=0 (z −2a−1), iteration gives

Cauchy–Schwarz gives integrability. Orthogonal sector decomposition gives the convergent expan-
sion
⟨x, E(h)⟩= e−∥zh∥2/2 X
in
√
n!
Lx(h, . . . , h).
(TR37)

where x = σ(f), y = σ(  ̄f). The identity b ̄g+(p) = bg−(p) fixes Fourier signs. The real shrinking
trace bump then gives TR32 by its previously paid dominated convergence. With TR26–TR36 and
the high-energy contradiction, this completes the specified Maxwell exclusions. The lifted coherent
source retains the entire convergent sector series, its physical symmetry and transverse constraints.
The scalar parameter reader extracts each coefficient with controlled tail; finite polarization and
projector transfer recover the individual sector reading.
This proves compatibility of the local
matrix reader with TR32 without selecting pointwise representatives or adding separation between
different probe tests.

Each real sum has the same remote support condition; the empty sum is zero for n > 0. Hence the
diagonal equality recovers every mixed real list. Real/imaginary decomposition then transports the
complex identity with conjugated tests on its conjugate side, as specified in the source.
For fixed momentum and all other labels, projector removal is the finite sum identity
X
xa
X
b
Πab(k)vb =
X
(Π(k)x)bvb =
X
xbvb.

It uses real symmetry of Π and the almost-everywhere transverse constraint.
Repetition over
the finite slots and finite labels preserves all other factors. A single chosen component per test
therefore selects the indicated label sector exactly.
Together with b ̄g+ = bg−, this recovers the
source’s isolated two-frequency formula. The already paid shrinking real trace bump yields TR32
with the same domination. Thus the coherent-extraction dependency in the locality-to-dilation
argument is recovered without imposing regular representatives of the waves.

Rounding error is at most 4n/mi →0, generic separated plots remain admissible, and the com-
parison error oi(1 + M(xi, yi)) →0 since M is continuous at the separated target. For convergent
labels vi, wi, bilinearity gives the finite sum P
a,b(vi)a(wi)b⟨Uiea(xi)Uieb(yi)⟩. Coordinate conver-
gence and TR38 give the free moment of their limits. This is the passage used for the running
centered trace, rather than an interchange of an uncontrolled varying field with a distributional
limit. No infinite-dimensional uniformity is used. Centering Tμν by subtracting its expectation
times the unit preserves symmetry, reality, conservation and charges: unit derivatives vanish and
a constant symmetric unit tensor has zero Ward charge. Its trace differs from the phase-centered
trace by a constant unit. Inverse scaling gives Bir4
i c1, vanishing by the running band below, with
exact unit identifications. In the source conventions,

Each running label carries Bir4
i and the remaining prefactor is (log(1/ri)/Bi)2, so the powers and
logarithms are accounted for before the limit. Explicitly, let bβi = β∗+ gti, ri = e−tiai, g = 2b0 > 0.
The magnifications obey eventually ai ≥1 and 0 ≤log ai ≤Gate( bβi), with Gate(x)/x →0.
Calibration and band comparison give βi/ bβi →1, βi/Bi →1. Hence

Here ti/ bβi = (1−β∗/ bβi)/g and the magnification quotient vanishes. Since ri ↓0, l= log(1/ri) →∞
and le−4l→0, giving

Running-label and calibrated-clock recovery.
Keep the free label coordinates fixed and
transport the interacting scaling map as UZ
i
= JiUi. The running-clock cancellation used here
follows directly from the calibrated clock, including the extra magnification in the scaling sequence.
For the selected character readout, the finite matrix of moments is identical:

All rounding addresses, magnifications and the continuous profile norm are retained. Their com-
parison error therefore still tends to zero on the separated moving plots. For vi →v and wi →w
in this finite space, the recovered moment error has the explicit bound


Thus the same finite coordinate reader recovers TR38 for the running trace label.
Its actual
coefficient limit remains a dependency of the stress correction below; this passage does not replace
that coefficient calculation.
The scalar clock source retains ti, ai, bβi, βi, Bi and the gate inequality. In particular