INDEPENDENT PREPUBLICATION PROOF AUDIT
18 October 2026, America/Chicago

Reviewer: stopping_paper_adversary, a separate AI mathematical reviewer.
Task: challenge the mathematical proof, not author the paper or certify novelty.

Source actually read in full:
D:/LUMINARY/research/self_transport_nonuniform_revision2_20261018/revised_stopping_theorem.txt
SHA256 7d2ed0072718a9fbb84581ed4c67fe75d4781468592bafa3198e163bbcd12edd

Disposition: no theorem-level counterexample or unresolved rate obstruction found
in this research source. This is a preliminary analytic review. Final manuscript
identity and sufficiently detailed publication presentation remain to be checked.

Independence: I read and independently derived the argument from the source before
reading its previous proof reviews. The older reviews were then inspected to check
the documented concentration repair and scope, not counted as votes for validity.
No manuscript, source proof, numerical evidence, or prior edition was changed.

1. ESSENTIAL PUBLICATION EXPANSIONS

The research note compresses its population construction and Gram limit. They
appear correct but should become actual proved lemmas in a self-contained paper.

For population construction write F=sqrt(f), q=a F and
G(q)=q H(Fq)-F. At q=1 its derivative is
  D G(1)=M_(HF)+H M_F=M_F(I+H)+O_(C2->C2)(sqrt(epsilon)).
The last estimate follows from HF-F=O_C2(epsilon) and [H,M_F]=O_C2(sqrt(epsilon)).
The inverse of I+H is uniformly bounded on C2: its multiplier is identity minus
a periodized rescaled Schwartz multiplier. A small quadratic contraction gives
q=1+O_C2(epsilon). C6 is sufficient. Uniqueness follows from PSD and the identity
obtained by multiplying the difference of v=K(1/v) equations by
(v1-v2)/(v1 v2). The paper should not simply cite the research note as proof.

The population Gram limit has a short direct RKHS argument. Define
  C_H=integral k_y tensor k_y dy,
which is diagonal on Gaussian RKHS Fourier features with eigenvalues lambda_k.
Since a^2 f=q^2=1+O_infinity(epsilon), the quadratic-form bound gives
  ||C-C_H||op <= ||q^2-1||infinity ||C_H||op=O(epsilon).
Resolvent identities imply
  ||R C R-R_H C_H R_H||op=O(epsilon).
Also theta_x=a(x) grad k_x+(grad a(x))k_x, with ||k_x||/||grad k_x||=O(sqrt(epsilon)).
After multiplication by epsilon^(d/2+1), the Gram therefore reduces to a(x)^2
times the Gaussian lattice sum, with vanishing O(sqrt(epsilon))+O(epsilon) error.
This proves the stated J_d/f(x) limit without an unexplained derivative-row
conjugation. The same C_H comparison and min-max prove interior-band counts.

Score expansion should explicitly state the derivative-row L2 bound and the
nonlinear error C epsilon^(-d/4-1/2)||e||infinity ||e||2. At e2 this is negligible
after multiplying by A_n because sqrt(n)(epsilon^2+eta)(epsilon^2+zeta)->0.
This makes the bridge from log-trajectory approximation to fixed-query inference
fully checkable. These are presentation obligations, not changes in conclusions.

2. ACTUAL FIRST HIT, SAME-SAMPLE EFFECTS AND d=2

The d=2 row improvement is valid. d1(y)-d1(x)=O(epsilon dist(x,y)) and normalized
weight replacement costs eta times the empirical mean absolute difference, hence
epsilon^(3/2) eta. The Gaussian image inequality dist*k_epsilon <= C sqrt(epsilon)
k_(2epsilon) supplies the uniform row-distance bound. In the fixed-scale centered
numerator at Xi the diagonal is zero, conditional bias is O(epsilon^2/n), and
conditional variance is O(epsilon^(3-d/2)/n). No independent-query assumption is
needed when averaging those conditional second moments in empirical L2.

Taylor expansion at deterministic d1, followed by nonlinear control from e2,
avoids the invalid root-n epsilon*eta term in d=2. The retained commutator
sqrt(epsilon)*zeta has root-n size epsilon^(1/2-d/4), so its 2^(-j) decay must
remain explicit; the final source does retain it. Likewise empirical/population
residual norms lose a square root of covariance error, producing
n^(-1/4)epsilon^(-d/2)2^(-j). That term is compared with the threshold or with a
spectral lower bound rather than incorrectly asserted root-n negligible uniformly.

Local projectively centered residual comparability is sufficient to remove the
two-update delay: r2<=C r1, and r2/t_n diverges. Nonconstant fixed f gives a positive
limit for r0; the uniform case is in the same small-log neighborhood already at
j=0 and gives r1<=C r0. No global actual-residual monotonicity is required.

3. SPECTRAL COUNTS, VARIANCE, OVERSHOOT

The Hilbert sample-mean squared-norm variance retains the nonconstant individual
diagonal fourth-moment term. Its stated envelope yields equation (21). The union
Chebyshev bound through j_- is necessary and the exponents in (22) suffice. A
single-j lower bound would not prove the actual first-hit statement, but the
source supplies the simultaneous event. The j_+ trace upper bound plus Markov
and common error bounds establishes the complementary upper bracket.

For the one-half overshoot claim, the final narrow high-band vector has envelope
epsilon^(-d/2) beta_H2^(2J), not the global epsilon^(-d/2)4^(-J) envelope. Its
relative variance is bounded by
[epsilon^(d/2)+1/n](beta_H2/beta_H)^(2J), and the positive band width can make
this vanish. The lower/high trace ratio then vanishes, while log-convexity makes
successive proxy norm ratios nondecreasing. Transfer by the threshold-relative
error yields every fixed c<1/2; it does not prove an exact 1/2 event or a single
crossing phase limit. The final source has this distinction right.

4. TRUE-SCORE LAW AND ENDPOINT

The exact potential influence is Rg, and the score influence is 2<theta,Rg>.
The stationary covariance subtraction vanishes because <theta,v>=0. The direct
Gram calculation gives 4J_d/f(x), and the normalized summand bound
C/(sqrt(n)epsilon^(d/4))->0 proves triangular-array Lindeberg. The O(epsilon)
population bias is negligible precisely under n epsilon^(d/2+3)->0.

For uniform f, the actual initial residual has linear proxy H delta_mu/2 with
constant c_d=(1/2)(4pi)^(-d/4)<1. Empirical norm and nonlinear errors are smaller
by sqrt(eta) and eta. Hence the unit-coefficient endpoint rule stops at j=0
with high probability. Its covariance involving Lambda^2 is strictly larger than
the exact covariance involving [Lambda/(1+Lambda)]^2 in every nonzero direction.
This proves the stated model-class counterexample, not a necessity result for
every nonconstant f or arbitrary endpoint multiplier.

5. REVIEW LIMITS

Supported scope: fixed positive C6 density on T^d, d=1 or 2, deterministic
bandwidth in H_d, fixed interior gamma, fixed spatial query, asymptotic laws.
There is no proof of finite-sample tolerance calibration, honest uniform coverage
over unrestricted density classes, data-selected queries, higher dimensions,
boundaries, runtime improvement, invariant-measure error, or worldwide priority.
Final manuscript and compiled package review remain pending. Internal AI review
is not external human peer review or publication permission.
