Gaussian short-distance comparison · Part 3

The same holds for the complement when 11. Choose the first with2, then 1. The full tail is the finite dyadic annuli through 2 and that complement. At physical distance at least 0, their bounds sum to 2, giving

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| jointh,D(x) −pieceh,s,D(x)| ≤3CD,R/s.

This is PT27’s tail estimate without requiring an absolute uncut infinite-volume derivative sum.

Finite Fourier bound recovery. The source retains the normalized symbol, centered representatives, zero summand, mesh and difference order. Horizontal readout leaves these scalar entries unchanged. In particular the low-frequency estimate uses the actual inverse-square sum and the exact scaling

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(Nh)−4s|D|−2T −4 = (2π)−4s|D|+2,
T =
2π
Nhs.

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The retained induction constant 432 and symbol constant (π/2)2 therefore recover PT28 with its
original prefactor. No oscillatory cancellation or positive separation is required for this reader, so
its bound still applies at coincidence.
For a smooth normalized annular symbol F, the finite-difference estimate can be written exactly
as
∆v
−TeνF(q) = (−T)v
Z
[0,1]v ∂v
νF
q −T(t1 + · · · + tv)eν

 dt1 · · · dtv.

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The prescribed support and derivative bounds thus give CT v per representative, and counting the
enlarged support gives CT v−4. The recorded boundary and high-frequency alternatives supply the
same bound for the periodic symbol when a representative changes.
This transports the finite
difference of the actual symbol rather than replacing it by a continuum derivative.
Let Fx(f) = P
p χp(x)f(p) be the finite Fourier reader. Index translation is a bijection of the
same torus, so
Fx(∆v
−eνf) = (χeν(x) −1)vFx(f)

is an exact readout identity. It remains so on the horizontal source with the same character and representatives. For nonzero physical distance lift(, the retained chord bound and normalization yield

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(Nh)−4s|D|−2T v−4(Nh/ρ)v
2π
v
,
v = |D| + 3.
= (2π)v−4s|D|+2−vρ−v =
1
(2π)4s
ρ

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This recovers PT30 including its powers and physical normalization. The final complement retains
the same estimate at its allowed scale. The finite dyadic decomposition is an identity of readers
before estimating, and Pq−1
j=0(2js)−1 ≤2/s together with the final complement bound gives exactly
the stated 3CD,R/s tail for ρ ≥R. Thus both the low-frequency anchor and oscillatory tail used
by PT27 are preserved by the same scalar source lift.
For fixed s, set cm = Nmhm, integer on the reporting schedule.
Physical Fourier spacing
is c−1
m.
The normalized annular integrand is jointly continuous in t = hms, displacement and

Fourier variable, including = 0, uniformly bounded and supported in radius 2). Choose integer containing it. Eventually 2, equivalent to 2 1, so all supporting integer addresses are their own centered representatives. The torus sum is exactly unrestricted box quadrature. If) is its representative,. The step function integrates to that quadrature with cell volume , converges pointwise by joint continuity and is

bounded by a constant times the radius- + 5 ball indicator. Dominated convergence proves its moving-displacement limit. Its phase is exact:

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e2πi⟨hm lift(xm),lift(p)/cm⟩= χp(xm).

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The normalized radial variable is also exact since cm = Nmhm.
For continuum heat-uniform oscillation let HD,a(q) = e−a∥q∥2MD(q), a ≥0, where MD is the
normalized smooth annular symbol supported in 1/2 ≤∥q∥≤3. Every fixed derivative order has
a common bound on that annulus, uniform in a. Indeed with y = √a q, order i derivatives of the
heat factor are ai/2 times Gaussian derivatives. For ∥y∥≤1, √a ≤2 gives compact bounds; for
∥y∥≥1, √a ≤2∥y∥and rapid decay give the same bound. Leibniz with MD finishes it. Choose
a coordinate of x at least half its norm and integrate by parts j times; compact support removes
boundary terms, and the derivative L1 bound gives

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∥x∥j
Z
e2πik·xHD,a(k)dk
 ≤CD,j
(a ≥0).

Rescaling contributes; choosing + 3 leaves. The rescaled heat parameter is nonnegative, so this proves the annular estimate uniformly during heat removal and its summable dyadic tail in PT27. The lifted oscillatory source retains periodic representatives, exact momentum differences, normalized phases, cell volumes and heat parameter. PT29 transports the reader by finite index translation; PT30 bounds its actual tail. The quadrature reader recovers the continuum annulus by dominated convergence with moving displacement, and uniform heat bounds preserve recovery when its regulator is removed.

Moving annular integral and heat-uniform recovery. The annular source records the mesh/period pair and the physical Fourier address = lift( The phase recovery is an identity before any limit, since

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⟨hm lift(xm), kp⟩= ⟨lift(xm), lift(p)⟩
Nm
.

Thus its exponential is the retained torus character. The radial argument and cell volume are also retained exactly; on the supporting box the annular reader is

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c−4
m
X
u∈Z4
Fm(u/cm) =
Z
R4 Fm(rm(k)) dk.

Only finitely many summands are nonzero. Eventual centered-address injectivity follows from 1. All scalar ingredients have the same horizontal readout, so the equality also holds for the lifted annular reader. Joint continuity with the retained moving displacement and0 gives pointwise convergence. The common bound is integrable and remains identical under readout. Consequently dominated convergence recovers the continuum annulus with its actual phase and normalization.

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For explicit heat-uniform control, for a multi-index α of length i and a > 0 the derivative of the
heat factor is
q e−a∥q∥2 = ai/2(∂αe−∥·∥2)(√a q).
∂α

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On the recorded annulus ∥q∥≥1/2, putting y = √a q bounds √a by 2 when ∥y∥≤1, and by 2∥y∥
otherwise. Thus a finite constant controls this derivative uniformly in a > 0. The case a = 0 is the

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constant heat factor, with all positive derivatives zero. Leibniz’s formula therefore gives uniform
L1 bounds for every prescribed finite derivative of HD,a on its fixed compact support.
For x ̸= 0 choose |xν| ≥∥x∥/2. The exact continuum reader identity is
Z
e2πik·xHD,a(k) dk = (−2πixν)−j
Z
e2πik·x∂j
νHD,a(k) dk.

It retains no boundary term and transports unchanged. Its derivative bound recovers the uniform oscillatory estimate. After rescaling, at + 3; the new heat parameter still ranges over nonnegative values. Summing the dyadic annular bounds gives 2 uniformly during heat removal. This retains the precise continuum tail used in PT27 and completes recovery of the moving annular step of the Gaussian comparison.