FINAL SOURCE BINDING AND INDEPENDENT CODE/EVIDENCE REVIEW
18 October 2026. Internal AI review, research only.

FINAL SOURCE
revised_stopping_theorem.txt
SHA256 f5406cc1ccedf46b7bbcf0d7d0544f7ea84891632f7b6dbb48c91a6bc1675089

PRESERVED REVIEWED SOURCE
reviews/source_before_notation_cleanup.txt
SHA256 22d4268df41449978b3eb08cae62460f1d9111048513745a70ef9074a57516ef

DISPOSITION
PASS for final-source equivalence and the stated diagnostic-code/evidence
scope. No substantive correction is required in the inspected material.
This report does not independently re-prove every analytic estimate, certify
worldwide novelty, supply finite-sample calibration, or authorize publication.

1. EXACT DIFF AND REVIEW BINDING
Read both complete sources and examined their complete unified diff.
There are precisely two changed blocks:
  (a) The second-line status changes from proof candidate for review to an
      analytic theorem/proof research edition. This is an editorial status
      change, with no altered mathematical assumption or conclusion.
  (b) Equation (17) now states directly, for every j>=2,
        sqrt(n)||E_(j,n)||_2 <= E_n+K_n epsilon^(1/2-d/4)2^(-j),
      using common E_n=o_p(1), K_n=O_p(1). The preserved version already
      states this exact equivalent interpretation explicitly immediately
      after its ambiguous displayed supremum notation. The final source
      removes that ambiguity and does not strengthen the bound.
No other text, theorem, calculation, hypothesis, endpoint, or proof changes.

I read the following assessments, checked their old-source hash against the
preserved source, and inspected their relevant scope and limitations:
  reviews/independent_proof_review.txt
    SHA256 aa61c844983016c7ac52b1d827b1e514b6c9347771781eaec3847bfb344ba1b2
  FINAL_ORIGINALITY_AND_VALUE.txt
    SHA256 94f3a03b735476a191596089bad0a84be38537a5eb0defc31dbd4a3258f62f22
Their mathematical and literature/value assessments therefore also apply to
the final f5406c... source, with their original qualifications intact. This
is authenticated reuse after a complete source comparison, not a claim that
the previous reviewers independently reviewed a new mathematical edition.
The originality report remains conditional on proof verification; it does
not become a worldwide priority guarantee through this binding.

2. SCRIPTS AND ISOLATED REPLAY PROVENANCE
Read code/verify_revision.py, code/endpoint_constants.py, and
code/d2_population_probe.py completely. Checked their Python syntax without
executing the producers or modifying their outputs.

verify_revision.py SHA256:
  0143f163abef2bcbe240f0e0f0977f9e42b6921a05f6c065638d0e18584bfc16
endpoint_constants.py SHA256:
  d0f90311fd90b27b1d11ebbcc0b4d38db997413f49ccf06c3123b9595561c81a
d2_population_probe.py SHA256:
  6d3810f1d6bfc134fb609682481010a132026700e60bc08d6450341a1703239a
results/independent_verification.json SHA256:
  46ff74fe2a913838d20f6eb1c747ce38c175e59ec90161a0a2306aeba0a5ede3

Fresh read-only checks confirmed:
  - The verifier's recorded source hash matches the verifier actually read.
  - Both recorded producer hashes match the source scripts actually read.
  - Each isolated_replay/code copy is byte-identical to its source.
  - Each preserved original and isolated replay JSON is byte-identical,
    as well as equal after JSON parsing.
  - endpoint_constants.json's embedded source hash matches its producer.

Original and replay output SHA256 values are respectively identical:
  endpoint_constants.json
    1a8dc3ccd496bf8fb06d6de57e5666a267bca062207a1255a4571c08a0dfa5be
  d2_population_probe.json
    47bce4aac3e7d07ace9c42861dcb8a97042816f273420c7de48bc82ebf2832c2

The verifier creates a separate replay tree and refuses to run if that tree
already exists. Producer path calculations correctly redirect output into
that isolated tree when copies are executed there. The verifier does write
its own new verification record; it does not overwrite producer records or
earlier-edition evidence. I did not rerun it. Its saved receipt and inspected
copies support the reported replay, but my contribution here is a separate
source/provenance audit rather than a second execution of that replay.

Exact parsed JSON reproducibility tests source/output consistency under the
used runtime, not numerical enclosure or mathematical correctness. The
record does not embed Python/NumPy version metadata or prove that arbitrary
platforms produce identical floating values. This is a minor reproducibility
limit, accurately subordinate to the analytic theorem.

3. ENDPOINT CONSTANTS: INDEPENDENT ANALYTIC DERIVATION
Write Lambda(xi)=exp(-2 pi^2 |xi|^2). Then
  int_Rd Lambda^2 = (4pi)^(-d/2),
so the initial residual constant is
  c_d = (1/2)(4pi)^(-d/4).
For each coordinate, the Gaussian second-moment identity gives
  4 int_Rd (2pi xi_1)^2 Lambda^2 = 2(4pi)^(-d/2) = W_d.
Mixed-coordinate integrals vanish by symmetry.

For d=2, angular integration and t=2pi^2 r^2 give
  J_2 = (1/(2pi)) int_0^infinity t/(1+exp(t))^2 dt.
If L(t)=1/(1+exp(t)), then L^2=L+L'. Integration by parts yields
  int t L^2 dt = int t L dt - int L dt = pi^2/12-log(2).
Consequently the stated exact formula is correct:
  J_2 = (pi^2/12-log(2))/(2pi).

The saved numerical values agree with these formulas:
  d=1: c=0.26556298300679926;
       W=0.5641895835477563, 4J=0.25571941851229246.
  d=2: c=0.14104739588693907;
       W=0.15915494309189535, 4J=0.0823275752929957;
       J=0.020581893823248924.
In every nonzero coordinate direction, Lambda^2 is strictly greater than
[Lambda/(1+Lambda)]^2 on a set of positive measure. Thus W_d I_d-V_d is
positive definite analytically, without reliance on floating inequalities.
The radial integrand, spherical factor, coordinate factor 1/d, lattice
rescaling epsilon^(d/2+1), and reported lattice cutoffs in the verifier all
match the theorem's normalization. This checks constants, not the separate
triangular-array and stopping arguments needed for the endpoint theorem.

4. WHAT THE DIAGNOSTICS ACTUALLY CHECK
endpoint_constants.py uses finite one-dimensional Fourier sums of the
continuum j=0 residual proxy. It excludes the constant mode correctly and
shows approach to the predicted c_1. It does not simulate actual sampled
iterations, and its Fourier tails are not enclosed.

The verifier compares a finite radial trapezoid integration to a separate
Cartesian lattice sum in d=1,2, and to the analytic closed form in d=2.
The tests are floating-point checks without validated tail or rounding
bounds. They are appropriate independent sanity checks of the constants;
they are not certified numerical proof of the asymptotic theorem.

d2_population_probe.py is a 128-by-128 periodic FFT population quadrature
for one fixed smooth density. The density weights sum to one on this grid
and are strictly positive. Its heat multiplier and half-self updates match
the theorem. The coefficient
  h=-Delta(f^(3/2))/(12 f^(3/2))
agrees analytically with the first-order log-error coefficient:
  (1/4)[Delta(sqrt(f))/sqrt(f)-Delta(f)/f].
The weighted standard deviation of h is the correct first residual
coefficient because the leading local residual map is the identity as
epsilon decreases on fixed smooth functions.

The finite grid includes spectral truncation and nonlinear aliasing and
solves only that discretized population equation to a floating tolerance.
It neither samples iid point clouds nor verifies random empirical residual
concentration, same-sample normalization, the first-hit rule, confidence
coverage, the full density class, or the continuum d=2 mixed-rate estimate.
In the saved range, e2/epsilon^2 changes visibly; the finite probe alone
would not establish an O(epsilon^2) continuum theorem. The record correctly
labels itself a finite population diagnostic rather than an asymptotic
certificate, and the analytic proof does not use these numbers as premises.

5. LIMITATIONS AND PRESERVATION
The final source's fixed positive C6 periodic-density, d=1,2, fixed-query,
strict fixed-gamma and sufficient-bandwidth restrictions are retained.
Unit-coefficient upper-endpoint failure is class-level through uniform f,
not failure for every nonconstant density or arbitrary endpoint prefactor.
The application statements do not claim selected-query inference, global
process control, mode location, pathwise Langevin accuracy, finite-n
calibration or runtime gains. Those limitations are appropriate.

No inspected original, prior edition, script, producer output, replay,
review or manuscript was changed. Only this new review file was written.
The broader prior-edition preservation audit is outside this narrow report.
Internal AI reviews and hashes do not constitute external human refereeing.
