FINAL MANUSCRIPT MATHEMATICAL CROSS-REVIEW
18 October 2026. Review-1 source snapshot.

Verdict: PASS within the stated model and asymptotic scope. No unresolved
mathematical defect was found in the reviewed sections.

Independence and scope
----------------------
This AI reviewer authored manuscript/geometry.tex and its derivation notes.
It therefore does not claim independent or blind review of that section.
It separately read and checked root-authored main.tex, trajectory.tex,
stopping.tex and inference.tex after those sections were written. The
geometry lemmas were used as supporting premises, with their algebraic
content also checked for consistency. This is a section-level independent
cross-check, not external human refereeing, and is not a separate primary-
literature originality assessment. Bibliographic priority and package
release authority require their separate checks.

Reviewed frozen sources (SHA256, lower case)
--------------------------------------------
main.tex
87b2bf549bbd99c3898c8ad322195f3a79921592c32f847027663137a32a7a26
trajectory.tex
8f37894e57f6b1bc1a2759456b190733936ccdaa1bc867eea026b7d856cd35fb
stopping.tex
3219540581a156a2176e8c7ed5b705ef455d29110e096d80d687dd7184074acf
inference.tex
1c8c7bd26da5424767dc1def4137ae780d9c0127eba1d3e0cfd34d94bc11475e

Supporting/context source hashes in the same snapshot:
geometry.tex (authored by this reviewer, not independently reviewed here)
044c471f72eace8a3a115af6db4cb66ecee1fb5579e909e91d3c79c2eb419823
references.tex (context only, no fresh literature audit by this reviewer)
b9a56069f940bb459fdf8994db45ca45402f183f2a0b855b2734f35bda25a194

The six hashes match evidence/manuscript_source_review1.json and direct
fresh SHA256 calculation. The preserved source theorem was read at:
D:/LUMINARY/research/self_transport_nonuniform_revision2_20261018/revised_stopping_theorem.txt
SHA256 7d2ed0072718a9fbb84581ed4c67fe75d4781468592bafa3198e163bbcd12edd.
No preserved source or prior edition was edited by this reviewer.

Checks and findings
--------------------
1. Model and normalization. The integral-one periodic Gaussian and its
Fourier multipliers use the same epsilon convention throughout. The exact
finite scale exists uniquely by the strictly convex coercive objective;
its positive diagonal terms supply strict convexity and coercivity. The
off-sample reciprocal-convolution extension and projective residual agree
with the sampled iteration. The sufficient power windows are correctly
(2/7,2/3) in d=1 and (1/4,1/3) in d=2.

2. Refined nonuniform initialization. The population numerator is
H_eps(af*d1)-d1*H_eps(af). Each heat remainder is O(eps^2) since
||d1||C2=O(eps) and ||af||C2 is bounded. The normalized empirical row
weight replacement ahat->a costs eps^(3/2)*eta: the row mean of
eps*dist(x,y) is eps^(3/2), uniformly, by the Gaussian image bound and
the empirical KDE bound at 2eps. Conditional on a sample site, the
remaining numerator has zero own diagonal, mean O(eps^2/n), and variance
O(eps^(3-d/2)/n). Denominators are bounded below. These estimates close
the two-dimensional root-n transient gap; the coarser eps*eta bound
would not suffice in d=2, and is not used in its place.

3. Nonlinear trajectory and derivatives. Starting the small-log argument
at e2 gives both the common L2 contraction and the nonlinear product
(eps^2+eta)(eps^2+zeta). The listed root-n remainders vanish under the
joint window. The first-update commutator's diagonal-zero conditional
calculation supplies sqrt(eps)*zeta; its 2^(-j) decay is retained rather
than discarded. Exact empirical intertwining and the two norm-1/2
filters justify power replacement. The derivative row is Sn*hat(theta),
and its section norm supplies the score linearization and exponential/
logarithmic quadratic remainders without differentiating sample values.
The score-tail conclusion is correctly fixed-query.

4. Residual norm comparison. The projectively normalized square-root
field is centered correctly. Its exact feature mean uses Sn1=vhat and
Bv=0; sqrt(n)*zeta^2=eta tends to zero. The square-root covariance-norm
comparison retains sqrt(eta)*zeta*2^(-j), which is generally too large
to discard at early iterations. The argument correctly uses its decay
and compares it with the relevant proxy or threshold, not a uniformly
root-n negligible assertion.

5. Hilbert quadratic concentration. The variance formula includes the
variable diagonal fourth-moment term. The covariance term, individual-
vector envelope, deterministic spectral projection, and high-band
sharpened envelope give the stated global and band relative variances.
The Chebyshev union bound through j_- has total order bounded by a
constant times (eps^(d/2)+1/n)*eps^(-2*u_-*(sigma-1)), which tends to
zero by the specified beta choice. Thus the lower bracket excludes all
early j>=2, rather than only testing the final lower bracket point.
The upper Markov bracket and logarithmic budget are compatible with
the same joint window.

6. Unrestricted first hit. For small centered log oscillation, the
linear residual is (I+Pn)e/2 on the mean-zero subspace. PSD and Markov
self-adjointness bound its norm between ||e||/2 and ||e||. The nonlinear
remainder is relatively small; Psi kills constants, so centering does
not require nonlinear mean preservation. This proves r(Psi(e))<=C*r(e).
The revised text explicitly derives r1/t->infinity from r2/t->infinity
and then r0/t->infinity in the uniform case. Fixed nonconstant densities
instead have a positive deterministic limit for r0. No compulsory
burn-in is silently imposed, and global residual monotonicity is not
assumed.

7. Attained residual. High-band concentration uses its narrower spectral
envelope, including the diagonal term. The low/high expectation ratio
vanishes at J. Positive spectral-sum log-convexity then controls every
later proxy ratio, tending uniformly to 1/2. The actual residual's common
threshold-relative error is negligible for j>=J. Applying it at the
first hit and its predecessor proves every lower constant c<1/2. No
single limiting residual ratio or finite-sample lower guarantee is
claimed.

8. Exact/stopped score CLT and inference. The reciprocal branch gives the
exact influence 2<theta,Rg>; its scaled remainder is sqrt(n)*eta*zeta,
which tends to zero. Rv=v/2 and <theta,v>=0 remove the rank-one covariance
term. The directional Lindeberg summand bound is C/(sqrt(n)*eps^(d/4)),
which vanishes. Population bias vanishes by n*eps^(d/2+3)->0. The
same-sample random stopping index is treated using a simultaneous
score tail, without an independence or optional-stopping assumption.
The feasible KDE plug-in, confidence balls and directional statements
follow by ordinary Slutsky and continuity, for predetermined queries.

9. Constants, endpoint, and scope. Direct integration confirms
J2=(pi^2/12-log(2))/(2*pi), and Wd=2*(4*pi)^(-d/2)*I. At f=1,
the empirical mean identity mu_n h=||h||H^2 and the translation-invariant
zero diagonal variance give r0/zeta -> (1/2)*(4*pi)^(-d/4)<1. Thus the
coefficient-one rule stops at zero for gamma>=d/4; the initial KDE
score covariance is strictly larger than the exact-score covariance.
The claim is correctly restricted to a class counterexample with this
threshold convention. The sinusoidal finite-sample impossibility example
has score q*e1 at x0 and product total-variation bound n*|q|/(2*pi^2*m).
It does not contradict the fixed-density theorem. The paper properly
limits its results to d=1,2, positive fixed C6 densities, periodic geometry,
predetermined queries, asymptotic coverage, and iteration counts.

Corrections requested during this review
-----------------------------------------
The scalar common-bound symbol K_n conflicted with convolution K_n; it
was renamed M_n before the frozen snapshot. This was a notation change.
The overshoot proof now explicitly takes 0<beta_L<1/2. The uniform
early-hit implication now states the diverging residual ratios needed
to exclude r0 and r1; no new assumption or mathematical claim resulted.
All requested points are resolved in the hashes above.

Limitations of this review
---------------------------
This review does not certify worldwide priority, external human peer
review, practical finite-n coverage, floating-point reliability of a
solver, a formal proof-assistant verification, or publication approval.
The mathematical proof is analytic; diagnostic numerical outputs are
not premises. Any mathematical source change after this snapshot needs
source-difference assessment and updated review binding.
