INDEPENDENT CROSS-REVIEW: SECOND-EDITION NONUNIFORM STOPPING THEOREM
18 October 2026, America/Chicago. Research proof review only.

FINAL SOURCE REVIEWED
D:/LUMINARY/research/self_transport_nonuniform_revision2_20261018/revised_stopping_theorem.txt
SHA256 7d2ed0072718a9fbb84581ed4c67fe75d4781468592bafa3198e163bbcd12edd

INITIAL SOURCE REVIEWED
SHA256 f5406cc1ccedf46b7bbcf0d7d0544f7ea84891632f7b6dbb48c91a6bc1675089
Preserved by the author at reviews/source_before_projected_band_clarification.txt.

DISPOSITION
PASS for the exact hypotheses and asymptotic conclusions stated in the final
source. One concentration-argument clarification was identified independently,
reported to the author, repaired in the final source, and rechecked below. No
unresolved substantive mathematical objection was found after that repair.
This is an internal independent AI analytic review. It is not a formal proof
assistant certificate, external human refereeing, originality clearance or
publication permission. No predecessor, root proof or frozen package was edited
by this reviewer.

METHOD AND INDEPENDENCE
I read the full root source and formed the objections/calculations below before
reading reviews/independent_proof_review.txt. I then inspected that earlier review
and the relevant population/empirical geometry in the preserved first nonuniform
proof. The final-source change was checked directly. This review concerns proof
validity and mathematical scope; the independent literature/value review is a
different task. No numerical calculation was treated as proof of an asymptotic
claim.

1. FOUND ISSUE AND VERIFIED DISPOSITION

In the initial source, Section 7 invoked the global relative-variance inequality
(21) for a narrow high-beta spectral band. Narrow width controls the off-diagonal
covariance term. If (21)'s global 4^(-J) fourth-moment envelope is used literally,
width alone does not control its diagonal term for an arbitrary beta_H: a further
condition placing beta_H sufficiently near 1/2 would be needed. This was a missing
explanation in the concentration argument, rather than a counterexample to the
theorem.

The final source instead explicitly projects the individual Hilbert vector Z
onto beta in [beta_H,beta_H2]. On that subspace,
  ||Z_H||^2 <= C epsilon^(-d/2) beta_H2^(2J).
The band trace is at least c epsilon^(-d/2) beta_H^(2J), with both endpoint
lambda values fixed and positive. The covariance and diagonal relative-variance
terms are consequently bounded together by
  C[epsilon^(d/2)+1/n](beta_H2/beta_H)^(2J).
Its stipulated width condition
  2 u_- log(beta_H2/beta_H)/log 2 < d/2
makes the first term tend to zero. The second also tends to zero since
n epsilon^(3d/2)->infinity. This repairs the inference and verifies the claimed
high-band concentration. The repair changes no theorem claim or hypothesis.

2. HYPOTHESES, NORMALIZATION AND WINDOWS

The result is for d=1 or d=2, exact integral-one periodic Gaussian convolution,
one fixed positive C6 density bounded below on the unit torus, deterministic
epsilon->0, and
  n epsilon^(3d/2)->infinity,
  n epsilon^(d/2+3)->0.
For epsilon=n^(-kappa), the intervals (2/7,2/3) and (1/4,1/3) follow directly.
No endpoint equality is included. The assumptions imply eta,zeta->0,
A_n->infinity, n epsilon^(d/2)->infinity, sqrt(n)epsilon^2->0 for d<=2,
and sqrt(n)eta zeta->0. A fixed gamma satisfies 0<gamma<d/4. Then
t_n->0 but sqrt(n)t_n=epsilon^(-gamma)->infinity. These facts are used in
the early-hit exclusion and attained-residual conclusions. Constants may depend
on this fixed f and gamma; no uniformity for f_n or gamma_n is asserted.

The square-root-mass residual is normalized by its own empirical mean. This is
projectively invariant and distinct from an unnormalized marginal residual.
The actual empirical scale, residual, iterate and query all use the same sample;
the proof never replaces them by independent samples. The fallback stopping
definition gives a finite procedure on every sample.

3. POPULATION AND EMPIRICAL GEOMETRY

Writing q=a sqrt(f), F=sqrt(f), the population equation is q H(qF)=F.
At q=1 the linearization is M_F(I+H)+[H,M_F]. The heat-plus inverse is bounded
on C2 by its identity-minus-periodized-Schwartz multiplier representation.
Commutators on C2 are O(sqrt(epsilon)); the forcing H F-F is O_C2(epsilon).
A local quadratic contraction therefore gives q=1+O_C2(epsilon). C6 is more
than the regularity needed for these stated C2 bounds. The PSD uniqueness
identity identifies the positive scale. In particular d1=O_C2(epsilon) and the
population score bias is O(epsilon).

The definitions S,S*, C=SS*, P=S*S are compatible with the weighted L2 space.
P is a self-adjoint PSD Markov contraction, ||S||=1, and the stationary feature v
has norm one, Cv=v and S*v=1. The nonzero spectra of C and P agree. Under the
sqrt(f) unitary map, P becomes M_q H M_q, differing from H by O(epsilon) in
operator norm. Min-max and fixed interior heat bands consequently give the
needed order epsilon^(-d/2) eigenvalue counts. They are fixed-band counts;
the proof does not assume a shrinking-shell Weyl asymptotic.

For the Gram limit, S*(I+C)^(-1)=(I+P)^(-1)S*. After the unitary change,
the derivative row is a(x) times the derivative heat row up to relative
O(sqrt(epsilon))+O(epsilon) L2 errors. Resolvent comparison costs O(epsilon).
The Fourier lattice sum yields exactly J_d/f(x), with no missing density,
score factor, or bandwidth power.

The empirical forcing has exact covariance (C-v tensor v)/n and is orthogonal
to v. Hilbert second moments and Hilbert--Schmidt covariance moments give zeta
and eta without finite-dimensional concentration. The local reciprocal equation
has derivative I+C0n with inverse norm at most one. Its remainder is bounded by
C epsilon^(-d/4)||h||_H^2 using one sup evaluation and one covariance-controlled
L2 evaluation. Since epsilon^(-d/4)zeta=eta=o(1), its positive local branch
is the unique empirical scale. This gives the stated eta*zeta remainder and
the value/weighted-gradient bounds. Multiplication by positive scalar weights
then gives C_hat-C=O_p(eta) and the theta-section comparison.

4. THE NEW D2 ROW ESTIMATE AND LATER NONLINEAR TERMS

The essential cancellation in B_n d1 is legitimate. The deterministic difference
d1(y)-d1(x) is O(epsilon dist(x,y)). Comparing normalized a_hat and a rows costs
eta times their empirical mean absolute difference, which is O(epsilon^(3/2)).
The Gaussian-image inequality dist*k_epsilon <= C sqrt(epsilon) k_2epsilon,
the same empirical embedding bound at 2epsilon, and positive row denominators
control this mean uniformly. This produces epsilon^(3/2)eta, not the invalid
coarser epsilon*eta error at root-n scale in d2.

For the remaining centered fixed-scale numerator evaluated at Xi, the diagonal
is exactly zero. Conditional on Xi=x, the bias is minus its population mean/n,
of size O(epsilon^2/n). Conditional variance is bounded by
  C epsilon^2/n * integral k(x,y)^2 dist(x,y)^2 dmu(y)
  <= C epsilon^(3-d/2)/n.
Averaging these conditional second moments proves the required empirical-L2
bound without an independent-query assumption. Denominator fluctuations multiply
the O(epsilon^2) population numerator. This proves (13).

Taylor expansion of the log map at deterministic d1, rather than at zero before
separating the population term, gives the tilted-row operator error O(epsilon)
and noise remainder eta*zeta in (14). Its infinity estimate is permitted to use
the coarser row estimate because epsilon*eta is absorbed by eta there.

Starting the full nonlinear comparison from e2 gives the product
(epsilon^2+eta)(epsilon^2+zeta). The four root-n-scaled pieces vanish:
  sqrt(n)epsilon^4,
  epsilon^(2-d/4),
  epsilon^(2-d/2),
  (n epsilon^(3d/2))^(-1/2).
The other burn-in errors also vanish, including
sqrt(n)epsilon^(3/2)eta=epsilon^(3/2-d/2). Sup-norm nonexpansion keeps the
same small log ball invariant; PSD makes the L2 linear half-step at most 1/2,
so a deterministic 3/4 contraction applies on one common high-probability event.

5. ITERATION-DEPENDENT PROXIES AND SAME-SAMPLE MEANS

The initializer commutator kernel [1-v(x)a(y)]k(x,y) vanishes at y=x.
Its conditional variance is O(epsilon^(1-d/2)/n); its conditional bias is
O(epsilon/n). Thus the sample-L2 error is O_p(sqrt(epsilon)zeta). Its
propagated root-n error is epsilon^(1/2-d/4)2^(-j), which must be retained in
d2. The final source does retain it with common random bounds for all j>=2.
Intertwining the empirical sample and feature operators and telescoping the
half-filters adds only uniformly bounded multiples of eta*zeta, since
j2^(-j) is bounded.

The score proxy 2<theta,C B^j Rg> has the correct extra C smoothing factor.
Derivative-row bounds send a sample-L2 error into a point-score error with
factor epsilon^(-d/4-1/2), exactly canceled by A_n except for sqrt(n).
Logarithmic nonlinear terms use e2's smaller radius. The fixed-query second
moment is bounded by C4^(-j); summing the tail validates evaluation at the
same-sample random stopping index. Spatially uniform remainder bounds do not
amount to a spatially uniform stochastic limit, and the source distinguishes
them.

For the residual, (I+P_n)e/2 is the correct linear square-root-mass field.
Intertwining cancels R against I+C. Its empirical mean is controlled by the
exact identity <1,S_n*B^j g>_n=<v_hat-v,B^j g>, because Bv=0. This is
O_p(zeta^2 2^(-j)); its root-n scale eta*2^(-j) tends to zero. No mean-zero
property of the nonlinear log iteration is assumed.

The empirical/population norm comparison genuinely loses a square root of the
covariance error. Its root-n size is
n^(-1/4)epsilon^(-d/2)2^(-j)=D_n epsilon^(-d/8)2^(-j).
It is not uniformly root-n negligible at small fixed j. The source retains
this exact term and compares it with spectral lower bounds or the tolerance;
these are the needed comparisons.

6. QUADRATIC CONCENTRATION AND THE LOGARITHMIC COUNT

For j>=1 the stationary rank-one covariance disappears and the Q_j mean is
T_j/(4n), T_j=tr(C^2B^(2j)). Expanding a centered Hilbert sample-mean norm
gives the stated variance formula, including its nonconstant diagonal term.
The individual-vector fourth-moment envelope gives the second term of (21).
Cross terms with a singly occurring independent centered variable vanish by
conditioning; no symmetry or zero third-moment assumption is required.

For the lower bracket, choosing beta_* close to 1/2 gives
u_- sigma<u_* and u_-(sigma-1)<d/8. Summing the Chebyshev bounds over every
2<=j<=j_- is valid: the off-diagonal exponent is
d/2-2u_-(sigma-1)>d/4 and the diagonal term is controlled by H_d. A single
fixed-j bound would not suffice, but this proof supplies the full union bound.
The smallest retained proxy exceeds epsilon^(-gamma), and the measurement-error
ratio is bounded by D_n epsilon^(d/8-u_-(sigma-1))->0. The remaining common
root-n errors are O_p(1) or o_p(1), smaller than that diverging lower scale.

At j_+, the trace upper bound and Markov inequality make q_j/t_n vanish.
The measurement term divided by t_n is D_n epsilon^(d/8+delta)->0.
This brackets every actual first hit among j>=2 and yields the leading count.
H_d also makes these bracket indices smaller than the prescribed 4 log n
budget. Subtracting the two fixed-gamma convergence statements is valid on the
same sample, giving the iteration-count difference without requiring independence.

7. REMOVING DELAYED STOPPING AND THE ONE-HALF OVERSHOOT

For a small log oscillation, projective centering leaves the residual unchanged.
On centered e, its linear field (I+P_n)e/2 has norm between ||e||2/2 and
||e||2. Taylor remainders are relative O(||e||infinity). Thus r(Psi(e))<=C r(e)
with a common finite constant; this does not require nonlinear mean preservation
or global monotonicity of the actual residual. The established e1=o_p(1) in
infinity norm and r2/t_n->infinity exclude j=1. For fixed nonconstant f, r0
tends to the strictly positive coefficient of variation of sqrt(f) under f dx.
For f=1, e0 is already small and the same local comparison excludes j=0.
This closes the actual unrestricted first-hit assertion, including uniform f.

For a realized g, Q_j is a positive spectral sum. Cauchy--Schwarz gives its
log-convexity, so q_(j+1)/q_j is nondecreasing and bounded by 1/2. The high-band
concentration repaired in item 1 makes every fixed low-beta portion negligible
at J=j_-. Hence the ratios approach 1/2 uniformly for later j. The norm-error
bound is uniformly o_p(t_n) there because its worst measurement exponent is
d/8-delta>0. Applying it to both the hit and its predecessor gives every lower
constant c<1/2, and no assertion of an exact 1/2 lower event or a unique crossing
phase limit is needed. The Q ratio would approach 1/4; the proof correctly uses
the q ratio for its 1/2 claim.

8. TRUE-SCORE CLT, STUDENTIZATION AND ENDPOINT FALSIFICATION

The exact empirical potential expansion gives score influence 2<theta,Rg>.
The stationary subtraction vanishes because <theta,v>=grad(a v)=0. The Gram
limit gives covariance 4J_d/f(x) times the identity. Each normalized summand
is bounded by C/[sqrt(n)epsilon^(d/4)]->0, establishing the triangular-array
Lindeberg condition. The O(epsilon) population bias is killed precisely by
n epsilon^(d/2+3)->0. The later-iteration score tail transfers this law to
the stopped estimator.

The same-sample heat KDE is positive and pointwise consistent under H_d, so
Slutsky justifies the feasible normal pivot and chi-square statistic. These
are pointwise confidence balls, predetermined directional intervals and a
fixed-query score-zero test. They do not locate a stationary point, distinguish
a mode from a saddle/minimum, cover selected queries, or supply simultaneous
bands. The proof states those limits.

For f=1, the actual initial residual has continuum linear field H delta_mu/2.
Its squared mean is (4n)^(-1)sum_(k!=0)lambda_k^2, with relative variance
O(epsilon^(d/2)). Empirical-norm and nonlinear corrections are smaller by
sqrt(eta) and eta; its empirical mean correction is also negligible. The
Gaussian Riemann sum gives the stated c_d=(1/2)(4pi)^(-d/4)<1. Thus the unit
threshold at every gamma>=d/4 hits at j=0 with probability tending to one.

The actual initial score has covariance W_d involving Lambda^2, whereas the
exact score has V_d involving [Lambda/(1+Lambda)]^2. W_d-V_d is strictly
positive definite: every nonzero direction has positive integral of its squared
linear projection times a strictly positive radial weight. The true kernel
triangular-array expansions have negligible nonlinear remainders under H_d.
Different limit laws therefore genuinely preclude numerical equivalence at this
endpoint. This is a model-class counterexample for coefficient one, not failure
for every density or a characterization of all endpoint multipliers.

9. PRECISE REVIEW LIMITS

This review supports analytic validity in the stated periodic d=1,2 fixed-density,
fixed-query asymptotic setting after the documented repair. It does not certify
worldwide priority, bandwidth optimality, nonasymptotic coverage, finite-n
calibration, runtime savings, d>=3, boundary models, varying densities, adaptive
bandwidths, data-selected queries, global stochastic-process laws, Langevin
invariant measures or diffusion-map spectral consequences. Those require their
own hypotheses and proofs. Research source preservation and hashes establish
version identity; they do not themselves prove mathematics.
