SECOND EDITION: FINAL ORIGINALITY, VALUE AND APPLICATIONS REVIEW
18 October2026. Independent AI literature/value assessment; research only.

EXACT SOURCE AND REVIEW SCOPE
Read revised_stopping_theorem.txt completely, SHA256
22D4268DF41449978B3EB08CAE62460F1D9111048513745A70EF9074A57516EF.
Also read IMPROVEMENT_CANDIDATE.txt and the prior nonuniform proof during
the preceding reviews. Prior sources/packages were not changed. This is
an originality/value/application assessment, not a replacement for separate
proof verification. I have not independently certified every estimate or
claimed external human peer review. Exact literature access is recorded in
LITERATURE_AND_APPLICATIONS.txt and the preserved first-edition review.

VERDICT
Conditional on independent proof verification, this is a meaningful focused
improvement over the first nonuniform edition. The result is no longer
only a special delayed-check counterexample: it gives the genuine standard-
start first-hit rule in d1 and d2, every fixed strict exponent0<gamma<d/4,
a class counterexample at/above the upper exponent, and a leading stopping-
iteration law, while preserving true-score pointwise inference. No checked
primary theorem directly subsumes this combined actual-trajectory statement.
The contribution is limited and specialist in scope; it is not a major new
score-estimation method, a new Sinkhorn algorithm, an optimal-bandwidth
result, or a general discovery of statistical early stopping. No worldwide
priority guarantee follows from focused literature coverage.

The core value lies in the nonlinear empirical trajectory and stopping
analysis. The confidence regions and inversion of a geometric residual
rate are useful consequences, not additional conceptual breakthroughs.
The new edition should be treated as a strengthened successor in the same
result family, not as multiple independent discoveries corresponding to
each changed exponent, dimension or overshoot constant.

WHAT THE SOURCE ACTUALLY ESTABLISHES, SUBJECT TO VALIDITY REVIEW
Fixed positive C6 density on T^d, d1 or d2; exact normalized periodic Gaussian;
same iid sample for scaling, iterations, residual and score. Conditions:
epsilon->0,n epsilon^(3d/2)->infinity,n epsilon^(d/2+3)->0, giving power
windows d1:2/7<kappa<2/3; d2:1/4<kappa<1/3 for epsilon=n^-kappa.
For t_gamma=n^-1/2 epsilon^-gamma,0<gamma<d/4, the first hit from b0=1:
  - occurs before its logarithmic budget with high probability;
  - has tau_gamma/log(1/epsilon)->(d/4-gamma)/log2;
  - attains residual between c*t_gamma and t_gamma with high probability
    for every fixed0<c<1/2, hence sqrt(n)r_tau diverges;
  - has negligible numerical error at each fixed spatial query under
    A=sqrt(n)epsilon^(d/4+1/2), retaining the exact empirical true-score CLT
    with covariance[4J_d/f(x)]I_d.
Uniform f=1 already stops at zero with a different score limit if gamma>=d/4
for the specified coefficient-one threshold. This is class-level failure,
not failure for every f or every possible endpoint prefactor. Gamma0,
gamma depending on n, nonpower frontiers and endpoint prefactor classification
are not covered. The attained residual ratio need not converge to1/2;
discrete crossing phase remains. The count difference for two exponents is
an asymptotic comparison of iteration numbers, not a runtime result.

CLOSEST PRIMARY PRIOR ART: EXACT OVERLAP AND REMAINING DISTINCTION
1. Kokot, Variational Problems in Feature Selection and Data Compression,
UW PhD dissertation, publicly issued16 September2026:
https://hdl.handle.net/1773/57893
Primary PDF:
https://digital.lib.washington.edu/server/api/core/bitstreams/7873395e-7e5e-4796-a3b3-413c310e39b6/content
The primary repository API verifies issue/accession date, Open Access and
CC BY; PDF, extraction and metadata were archived in the first nonuniform
research folder. Read Chapter3 Theorem3.6/proof, Corollary3.3/proof and
Lemma3.29/proof: fixed-epsilon Gaussian-RKHS scaling differentiation,
self-diagonal(I+T)^-1 derivative and Hilbert CLT are direct antecedents of
our exact-scaling block. Read Chapter4 Section4.4.1,Lemma4.4/Corollary4.3
with proofs: under q=b^-2 the same population half-self update recovers the
first-order converged scaling after two updates by heat-Taylor cancellation.
These methods and burn-in mechanism cannot be claimed new. The checked
chapters do not provide the present joint sampled-trajectory first-hit
law, full exponent interval/endpoint obstruction or its iteration count.
Attribution in Section10 of the revised source is appropriately explicit.

2. Mordant, The statistics of entropic optimal transport with decreasing
regularisation, arXiv2412.12007v2,29 April2026:
https://arxiv.org/html/2412.12007v2
Assumptions1 AND2, Theorem12 and Algorithm1 were read/refreshed. Exact
decreasing-regularizer score inference and the half-self algorithm already
exist. Its Euclidean compact-convex quadratic-cost theorem is not directly
the present periodic Gaussian theorem. The d2 candidate window is stricter
than Mordant's printed exact-score power window; it does not improve that
sampling range. The new target is actual first residual stopping while
preserving the estimator's law. Its report of two-to-three iterations
typically is not a leading count theorem in this joint statistical regime.

3. Wormell--Reich, SIAM J Numer Anal59(3),1687--1734(2021),
DOI10.1137/20M1344093; arXiv2006.02037:
https://wormell.perso.math.cnrs.fr/preprints/WR20-preprint.pdf
Prior audits read nonuniform iid torus scaling/operator estimates, sup/Hardy
plus-resolvent bounds and selected proofs; Theorem4.1 was refreshed here.
Normalization, derivative controls, spectral convergence and ASSA stopping
are existing. ASSA has a different local filter -P(I-P)/2. Its fixed-operator
geometric rate/stopping theorem is not the current joint iid half-self score
first-hit conclusion. The proposed RKHS route uses established ingredients,
not a newly invented normalization or convergence principle.

4. Cheng--Landa, arXiv2206.11386v3; Information and Inference13,iaae026(2024),
DOI10.1093/imaiai/iaae026:
https://arxiv.org/html/2206.11386v3
Theorem4.2 and approximate constrained scaling discussion refreshed. Early-
terminated approximate normalization retaining nonuniform graph-Laplacian
statistical consistency is a direct precedent for statistical stopping of
normalized kernels. The current paper cannot claim that broad observation.
Its output metric/statistic differs, and it gives neither the present
fixed-query true density-score Gaussian law at the actual first hit nor the
attained residual, exponent boundary and leading half-self iteration law.

5. Maeda--Yao--Nitanda, arXiv2510.22560v1,Theorems1--2:
https://arxiv.org/html/2510.22560v1
Actual empirical Sinkhorn bridge iterations already have combined sample
and optimization error analysis in path total variation. Wang's
arXiv2607.00665v1 already provides deterministic low-temperature paths on
fixed finite supports. Neither checked target is this nonlinear half-self
spatial-score stopping problem. Belomestny arXiv2608.29152v1 treats exact
empirical potential stability/rates and explicitly excludes finite-iteration
optimization error. Greco et al. and Greco--Tamanini already prove gradient
convergence/stability. Broad claims of finite-iterate statistics, potential
stability or gradient convergence therefore belong to existing work.

6. Celisse--Wahl,JMLR22(76),1--59(2021):
https://www.jmlr.org/papers/volume22/20-358/20-358.pdf
Effective-dimension definitions, residual/bias/variance comparisons,
Proposition13 and Theorem15 refreshed, with earlier Theorems12,19--20 proof
reads retained. Blanchard--Hoffmann--Reiss,JUQ2018,
DOI10.1137/17M1154096 is another foundational discrepancy-stopping source.
Data-dependent spectral stopping, random-design norm comparison and noise
effective dimension are not new. These regression/inverse-problem results
do not directly supply the candidate's nonlinear sampled scaling trajectory.
Its value must be claimed at that precise application level, not as a new
general statistical stopping principle or oracle-adaptation result.

7. Generic count classification is longstanding. Knight2008,
DOI10.1137/060659624, gives classical Sinkhorn spectral convergence
background. Its official page was opened, but full theorem text was not
newly available here after report-PDF failure. As a concrete accessible
recent primary example, SinkSLOT arXiv2608.28262v1,AppendixA.4:
https://arxiv.org/html/2608.28262v1
explicitly derives fixed-matrix quotient spectral convergence and a
log-tolerance iteration count using classical Jacobian/fixed-point theory.
This does not cover the current joint statistical coefficient. Once the
two-sided actual residual has log-scale
-.5log n-(d/4)log epsilon-j log2+o(log(1/epsilon)), solving for its crossing
immediately gives(d/4-gamma)/log2. The inversion and two-threshold difference
are routine; proving that residual law for the actual varying-epsilon iid
trajectory is the additional mathematical work.

STANDARD INGREDIENTS VS THE CANDIDATE CONTRIBUTION
Standard: half-step filter, PSD/Markov contraction, implicit equations,
RKHS covariance and Hilbert CLT, population heat expansions, spectral min-max,
Hilbert quadratic variance, concentration, positive-sum log-convexity,
geometric crossing arithmetic, triangular-array CLT and Slutsky.
Candidate-specific work: sharper empirical normalized-row burn-in needed
in d2; retaining its nonnegligible fixed-j commutator with its2^-j decay;
uniform nonlinear empirical trajectory control; relative covariance/norm
errors at threshold; simultaneous lower residual concentration through the
logarithmic crossing range; excluding early0/1 hits; controlling attained
overshoot; and validating inference at that same random index. These form
one coherent theorem, with no reliance on an engineered initialization or
an assumed generic residual direction.

PROPORTIONATE MATHEMATICAL VALUE AND PUBLICATION ASSESSMENT
This improves the earlier theorem materially: d2 adds a genuine mixed
deterministic/stochastic estimate; the ordinary first-hit rule removes an
artificial procedural restriction; a tolerance interval plus its upper
counterexample explains how far the relaxation can go; the leading count
connects that interval to computation. Together they answer the project's
necessity question more informatively than a single uniform countermodel.
The result is still specialized: fixed smooth positive periodic density,
low dimension, sufficient sampling windows and asymptotics only. It does
not establish minimax improvement, dimension-general usefulness, measured
speedup or a practically calibrated tolerance. It also does not contradict
the earlier arbitrary-candidate worst-case certificate, which never said
the standard solver must realize its adverse directions.

If the entire exact source passes independent proof review, the theorem is
coherent material for a focused specialist methods/theory note that credits
all these precursors and leads with actual stopped-score behavior. That is
a proportionate publication-merit assessment, not a venue recommendation,
acceptance prediction or instruction/authorization to create a manuscript.
A broad claim of a new statistical Sinkhorn framework or major advance
would overstate the evidence. If proof verification fails, this assessment
does not rescue the theorem merely because its application is attractive.

CONCRETE APPLICATIONS THAT THE POINTWISE LAW SUPPORTS
With same-sample KDE ftilde(x)=n^-1 sum k_epsilon(x,X_i),
ftilde(x)->f(x)>0: heat bias vanishes and variance is O(1/(n epsilon^(d/2))),
which tends to zero under the stated windows. Slutsky requires no
independence between ftilde and the stopped score. Hence
  Z_n=A sqrt(ftilde(x))/(2sqrt(J_d)) [s_tau(x)-grad log f(x)]
      =>N_d(0,I_d).
This yields the feasible confidence ball
  {u: A² ftilde(x)||s_tau(x)-u||²/(4J_d)<=chi²_(d,1-alpha)},
with fixed-query asymptotic coverage1-alpha. Fixed directional intervals
follow; d1 half-width is z_(1-alpha/2)*2sqrt(J_1)/(A sqrt(ftilde(x))).
These are standard, useful inference consequences, not original core claims.

The same statistic with u=0 tests grad log f(x)=0 at a predeclared query,
with asymptotic level alpha under each fixed null density and consistency
against each fixed alternative with nonzero score because A->infinity.
For circular/phase or pairs-of-phase data, this can quantify local log-density
slope or screen stationarity at a scientifically specified location, provided
the positive smooth iid periodic model is appropriate. This does not prove
a nearby mode, distinguish maximum/minimum/saddle, estimate a mode location,
handle a query chosen after inspecting the data, or justify simultaneous
testing. Local-alternative power needs changing-density theory not supplied.

Goldfeld--Kato--Rioux--Sadhu,EJS18(1),980--1041(2024),
DOI10.1214/24-EJS2217,Example2 already constructs confidence bands for EOT
maps in its fixed-regularizer functional setting:
https://arxiv.org/pdf/2207.08683
It was refreshed in this revision's application audit. The general inference
principle is prior art, and its bands do not transfer to our shrinking-epsilon
fixed-query theorem without additional process control.

NOT YET PROVED APPLICATIONS
Langevin drift values at a fixed point can be rescaled from the score law;
diffusion path accuracy, invariant distributions, mixing and generative
quality require uniform or path-weighted drift bounds plus stability and
discretization analysis. Diffusion-map operator/eigenvalue/eigenvector
consistency requires separate global operator results. The present theorem
proves none of those by itself. No simultaneous spatial confidence bands,
bootstrap validity, Hessian inference, selected-query guarantees, d>=3,
boundaries, density-uniform minimax theorem, finite-n calibration or wall-
clock improvement follows. The source appropriately retains these limits.

EXTERNAL COVERAGE AND LIMITS
Actual Google was used at revision start, during application review, during
the count question, and at the final exact-source check. Final query:
  "Sinkhorn" "score" "stopping" "asymptotic"
Visible results inspected; snippets/AI overviews were leads, not evidence.
Earlier actual Google queries and supplemental web queries are recorded in
LITERATURE_AND_APPLICATIONS.txt. Primary statements and selected proofs
were read/refreshed to the explicitly stated extent. Kokot's primary PDF
and repository metadata bindings are reused from verified local downloads;
other primary version/access limits remain in the literature ledger.
This is focused related-work coverage, not an exhaustive forward-citation
census or a guarantee that an equivalent theorem is absent elsewhere.

FINAL DISPOSITION
Meaningful and more complete specialist-level theorem-family extension,
conditional on exact-source proof verification; bounded originality in the
actual nonlinear stochastic stopping result, not its standard ingredients.
Concrete pointwise inference is justified if that theorem is valid. Wider
path/global/practical claims remain open. No source or prior edition was edited.
