FINAL INDEPENDENT MATHEMATICAL REVIEW
18 October 2026, America/Chicago

Reviewer: stopping_paper_adversary, separate AI reviewer.
Package: statistical_self_transport_stopping_publication_20261018
Paper: Residual Stopping and Pointwise Inference for Nonuniform Entropic Self-Transport

DISPOSITION: PASS for the exact mathematical scope of the six manuscript sources
identified below. No unresolved mathematical objection found. The research proof
has been expanded into a coherent, self-contained analytic manuscript. This is an
internal independent AI review, not external human refereeing, a formal proof
certificate, originality clearance, or permission to publish.

EXACT SOURCE BINDING
All six current file hashes were recomputed and matched
evidence/manuscript_source_review1.json at the time of this final review.

geometry.tex
044c471f72eace8a3a115af6db4cb66ecee1fb5579e909e91d3c79c2eb419823
inference.tex
1c8c7bd26da5424767dc1def4137ae780d9c0127eba1d3e0cfd34d94bc11475e
main.tex
87b2bf549bbd99c3898c8ad322195f3a79921592c32f847027663137a32a7a26
references.tex
b9a56069f940bb459fdf8994db45ca45402f183f2a0b855b2734f35bda25a194
stopping.tex
3219540581a156a2176e8c7ed5b705ef455d29110e096d80d687dd7184074acf
trajectory.tex
8f37894e57f6b1bc1a2759456b190733936ccdaa1bc867eea026b7d856cd35fb

METHOD AND INDEPENDENCE
I independently derived the baseline research proof from
D:/LUMINARY/research/self_transport_nonuniform_revision2_20261018/revised_stopping_theorem.txt
SHA256 7d2ed0072718a9fbb84581ed4c67fe75d4781468592bafa3198e163bbcd12edd
before reading its earlier proof reviews. I subsequently read every manuscript
source in full, checked the new explanatory lemmas and inference discussion, and
checked the exact final changes and hashes. I supplied objections and suggestions
but did not author or edit the manuscript. No old proof, evidence, or edition was
changed. No numerical simulation was treated as proof of an asymptotic assertion.

RESOLVED FINDINGS
1. The research note compressed the population existence and Gram steps. The
   manuscript now proves the C2 contraction, uniform heat-plus inverse,
   Gaussian-RKHS operator comparison and lattice sum. These details close the
   identified publication-presentation obligation without changing conclusions.
2. The first manuscript reused K_n for both empirical convolution and a scalar
   probabilistic envelope. The final source uses M_n for the latter.
3. The first uniform early-hit paragraph stated only r1>t_n. This alone would not
   imply r0>t_n from r1<=C r0. The final source explicitly derives r1/t_n->infinity
   from r2/t_n->infinity and then r0/t_n->infinity. The missing logical link is
   now explicit. No stronger hypothesis or altered theorem was needed.
4. The high-band fourth-moment envelope in the preserved research source is
   retained correctly, and the final low-band parameter is explicitly positive.

DETAILED ANALYTIC CHECKS

Model, scaling and hypotheses. The normalized periodic Gaussian and its bandwidth
convention agree throughout. Finite scaling exists uniquely: positive diagonal
terms give a coercive separable lower bound for Phi and a positive-definite
Hessian contribution. The off-sample reciprocal extension is the branch used in
all scores. Under H_d, eta,zeta->0, sqrt(n)epsilon^2->0, A_n->infinity and
sqrt(n)eta*zeta->0 for d=1,2. The strict power windows are correct. The threshold
goes to zero but sqrt(n)t_n diverges for every fixed 0<gamma<d/4.

Population geometry. Writing q=a sqrt(f) gives the stated C2 nonlinear equation.
Its derivative is M_sqrt(f)(I+H_epsilon) plus an O(sqrt(epsilon)) C2 operator
error. The identity-minus-Schwartz multiplier argument gives a uniform inverse;
the small quadratic contraction is justified with the stated C6 assumption.
The PSD energy uniqueness identity is correct. In the same RKHS, C differs from
C_H by O(epsilon) in operator norm because a^2 f=q^2=1+O(epsilon). Resolvent
identities and the lower-order k_x section contribution reduce the Gram limit
to the displayed Gaussian lattice sum. It has exactly J_d/f(x), with the stated
score factor 4. Min-max on C and C_H proves fixed interior-band counts without
assuming Fourier eigenvectors for the nonuniform operator.

Empirical branch. Feature second moments give zeta and the Hilbert--Schmidt
covariance error eta without finite-dimensional concentration. The covariance
and stationary orthogonality identities hold exactly. The reciprocal remainder
bound uses one RKHS sup evaluation and one covariance-controlled L2 evaluation;
its Lipschitz version proves the local positive empirical branch and the eta*zeta
remainder. Positive finite-scaling uniqueness identifies the branch with the
actual empirical solution. Value and weighted-gradient errors and random
operator/section bounds follow with the displayed powers.

Initialization and same-sample effects. The refined normalized-row calculation
uses d1(y)-d1(x)=O(epsilon dist(x,y)), so weight replacement costs
epsilon^(3/2)eta. The Gaussian-image distance bound is valid on the torus.
The fixed-scale numerator has exactly zero own diagonal at X_i; its conditional
bias and variance are of the stated orders. Averaging conditional second moments
therefore proves empirical L2 control without treating queries as independent.
Taylor expansion at deterministic d1 and subsequent control from e2 avoid a
nonnegligible root-n epsilon*eta term in dimension two. All displayed mixed
rates vanish under H_d.

Common trajectory, score and residual. The Markov log map is sup-nonexpansive;
PSD gives half-filter norm at most 1/2. Hence e2 controls every later nonlinear
remainder on a common event. The initializer commutator has zero own diagonal,
size sqrt(epsilon)zeta and the necessary geometric 2^(-j) propagation. Its fixed-j
root-n error is not negligible in d2 and is correctly retained. Derivative-row
L2 bounds give the explicit nonlinear score error and preserve the C factor in
the score proxy. Summing pointwise proxy second moments permits the same-sample
random index. The residual's empirical mean uses the exact stationary identity,
not an unsupported mean-zero nonlinear iteration assumption. Its norm comparison
correctly loses a square root of covariance error and keeps D_n epsilon^(-d/8)
2^(-j) until comparison with a threshold or a spectral lower bound.

Stopping count and overshoot. The squared Hilbert sample-mean variance identity
includes the nonconstant diagonal fourth-moment term. Fixed-band counts and the
chosen beta_* make the Chebyshev sum over every j<=j_- vanish. A single-j lower
bound would not be sufficient, but the manuscript provides the necessary union
event. The j_+ trace upper bound and Markov inequality give a hit before budget.
Local projective residual comparability and the explicit divergent early ratios
exclude both j=1 and j=0. The high-band projected envelope is sharper than the
global 4^(-J) envelope, so its narrow-band relative variance vanishes. Positive
spectral-sum log-convexity then yields the later ratio limit 1/2, and the retained
norm error is o_P(t_n) at crossing. This proves every c<1/2, without claiming an
exact half event or a unique crossing-phase limit. The leading count difference
for two fixed exponents follows by subtraction on the same sample.

Exact and stopped CLT. The score influence has the correct plus-resolvent and
sign. The stationary covariance subtraction vanishes by <theta,v>=0. The Gram
limit and uniform influence bound establish the triangular-array multivariate
CLT. The bandwidth bias condition kills the O(epsilon) population bias. The
uniform later-iteration point-score tail transfers the limit to the actual first
hit. The KDE plug-in is positive and consistent; same-sample Slutsky establishes
the feasible normal and chi-square pivots. The pointwise confidence ball and
predetermined directional consequences are valid asymptotically for fixed f.

Endpoint and constants. For uniform f, the initial centered normalized residual
has the stated H delta_mu/2 proxy, constant c_d=(1/2)(4pi)^(-d/4), and smaller
empirical norm/mean and nonlinear errors. Translation invariance makes the
individual diagonal variance vanish. Thus the coefficient-one endpoint rule
stops at zero with high probability. The initial score covariance is
W_d=2(4pi)^(-d/2)I, strictly greater in every nonzero direction than 4J_d I.
The displayed J2=(pi^2/12-log 2)/(2pi) also checks independently by polar
coordinates and the identity F^2=F+F' for F(u)=(1+e^u)^(-1).

Finite-sample scope paragraph. The smooth oscillatory densities have score q e1
at x0, are eventually between 1/2 and 3/2, and their one-sample TV distance from
uniform is |q|/(2pi^2 m). The n-sample TV bound follows by product coupling.
For fixed n and every fixed q this tends to zero. Positive uniform coverage of
all those scores by an almost surely bounded interval would force that interval
under uniform data to cover every q with the same positive probability, which
contradicts dominated convergence as |q| grows. This is a valid standard
discontinuous-functional example. It is not used to assert finite-sample
coverage for the theorem or claimed as a new inferential construction.

LIMITS OF THIS REVIEW
Validity is supported for a fixed positive C6 density on the periodic unit torus,
d=1 or 2, deterministic H_d bandwidth, fixed interior gamma, and fixed spatial
query. Neither uniformity in density norms or varying f_n/gamma_n, endpoint
multipliers, bandwidth optimality, finite-sample confidence calibration,
data-selected queries, spatially simultaneous bands, boundaries, d>=3, runtime
gains, nor global diffusion/Langevin consequences are proved or advertised.
The bibliography and contribution wording were inspected for mathematical
compatibility; fresh priority/literature verification is a separate review.
This review binds source content, not any later altered source or an uninspected
PDF artifact. Final rendering and package-preservation audits remain separate.
