FINAL ORIGINALITY AND VALUE REVIEW: NONUNIFORM STOPPING CANDIDATE
18 October 2026. Research checkpoint only; no publication recommendation.

SOURCE BINDING
Read the complete joint_nonuniform_theorem.txt, SHA256
788216E8D3F8780CD5CAD1712BFE1D74464355013E09F57106AC1007CAC223C2.
Also read the complete independent PROOF_CANDIDATE.txt route and earlier
assigned sources. This review judges originality and value; independent
proof-review completion has not been verified in this file. Neither a
source hash nor a related-work judgment certifies proof correctness.

SHORT VERDICT
Conditional on proof verification, the exact candidate has a meaningful,
limited mathematical increment. It passes the research-value test that the
previous uniform-only theorem did not: it handles a fixed nonuniform density
and a genuine true-score inference window. The actual first residual hit
after two mandatory updates has attained sqrt(n) residual diverging, yet
retains the exact estimator's true-score limit. This is stronger than
contraction plus a worst-case certificate and Slutsky, and is not subsumed
by any checked primary statement. It is a focused one-dimensional stochastic
solver theorem, not a major general advance in Sinkhorn statistics.
No worldwide priority claim follows from this focused search.

The important new precursor found in this review materially reduces credit
available for the two-update population argument, but does not displace the
full stochastic stopping statement. Credit it explicitly as detailed below.

EXACT CANDIDATE INCREMENT
Fixed positive C6 density on the circle; same iid sample for all computations;
exact periodic Gaussian; b0=1 half-self iteration; two mandatory updates;
then first observed residual crossing t=n^-1/2 epsilon^-1/16 with log budget.
Under epsilon->0, n epsilon^1.5->infinity and n epsilon^3.5->0, equivalently
strict power window 2/7<kappa<2/3 for epsilon=n^-kappa:
  * the crossing occurs after growing J and before the budget;
  * its attained residual lies in [t/8,t] with high probability;
  * pointwise numerical score error is negligible at sqrt(n)epsilon^.75;
  * exact and stopped scores share the proved periodic CLT centered at f'/f,
    with variance 4J1/f(x).
This is not the first hit starting at j=0; the delayed check is part of the
procedure. The source correctly proves a periodic CLT rather than importing
Mordant's Euclidean theorem. The residual comparison (18) is threshold-relative:
its decaying 2^-j measure term is retained, not silently promoted to a uniform
root-n negligible error. This distinction is essential to the full window.

CLOSEST NEW PRECURSOR: KOKOT, PUBLIC BEFORE THIS WORK
Alex Kokot, Variational Problems in Feature Selection and Data Compression,
PhD dissertation, University of Washington, 2026. Stable primary handle:
https://hdl.handle.net/1773/57893
Primary PDF:
https://digital.lib.washington.edu/server/api/core/bitstreams/7873395e-7e5e-4796-a3b3-413c310e39b6/content
Primary repository item API (accessed 18 October2026):
https://digital.lib.washington.edu/server/api/core/items/dd92c3f1-9d82-4abc-b13a-2d87a02a45d7
Repository metadata verifies issued date 2026-09-16, accession timestamp
2026-09-16T18:35:50Z, Open Access, and CC BY. This is a verified prior public
disclosure, not merely a copyright-year inference. Metadata is saved at
results/Kokot_repository_metadata.json; downloaded PDF SHA256
CA5927E14428F9F81D6C14559C56B7FE4D8BC5286FA6F5C7D4E791239027795E.

Read Chapter4 introduction/related work, score expansion Corollary4.2,
Section4.4.1 equation4.33, Lemma4.4/proof and Corollary4.3/proof (printed
pp171--173; PDF pp187--189). Its population update starts q0=1 and uses
q_next=sqrt(q)P_epsilon(h²/sqrt(q)). With q=b^-2 and density h², this is
the same half-self update, allowing for Gaussian variance conventions.
Lemma4.4 already proves first-order converged scaling is recovered after
two symmetric updates, by explicit cancellation of the first-order heat
terms. Corollary4.3 gives o(epsilon²) potential error on compact sets.
Therefore two-step population matching and its heat-Taylor cancellation
are prior results. Our quantified C6 periodic burn-in estimate is a technical
refinement serving the stochastic theorem, not an independent major novelty.

The checked chapter holds a population fixed while epsilon shrinks, compares
scores/densities/diffusions, and transfers a two-step kernel to coreset
approximation. It does not give the candidate's iid empirical trajectory
relative to its own exact empirical fixed point, random residual first-hit
law, attained root-n-large residual or true-score sampling CLT at that hit.
No full stochastic subsumption is established by this precursor.

SOURCE-SPECIFIC OVERLAP WITH OTHER PRIMARY WORK
Mordant, arXiv:2412.12007v2 (29 April2026), Theorem12/Algorithm1:
https://arxiv.org/html/2412.12007v2
Already provides this half-self iteration and exact-estimator true-score
asymptotics with decreasing regularization. Both Assumptions1 AND2 apply;
Euclidean compact convex support and a uniform rescaled-potential regularity
premise differ from our periodic kernel. For d1 its printed power window is
also (2/7,2/3). Claim no new score-estimation construction or general CLT;
the new target is finite actual stopping in a matched periodic proof.

Wormell--Reich, SIAM J Numer Anal59(3),1687--1734 (2021),
DOI10.1137/20M1344093:
https://wormell.perso.math.cnrs.fr/preprints/WR20-preprint.pdf
Read Sections3--4,6,8--9 as recorded in LITERATURE.txt. Nonuniform iid torus
normalization, sup/Hardy inverse tools, empirical operator concentration and
derivative control are established. ASSA has a different two-transform
iteration with local filter -P(I-P)/2 and an error stopping certificate.
The candidate's RKHS perturbation and spectral-shell calculations do not
create a new general convergence mechanism. The checked results do not
give its actual half-self density-score first-hit separation.

Cheng--Landa, arXiv:2206.11386v3, Theorem4.2 and approximate scaling setup;
Information and Inference13,iaae026 (2024), DOI10.1093/imaiai/iaae026:
https://arxiv.org/html/2206.11386v3
Already justifies early-terminated approximate scaling at statistical
accuracy for nonuniform graph-Laplacian consistency. This is the strongest
precedent against a broad claim that statistical noise permits looser
normalization accuracy. It does not supply a pointwise true density-score
CLT, same-solver first crossing, or attained root-n-large residual. The
candidate is a more specific result about a different inferential functional,
not the invention of statistical stopping for normalized kernels.

Maeda--Yao--Nitanda, arXiv:2510.22560v1, Theorems1--2:
https://arxiv.org/html/2510.22560v1
Actual empirical bridge iterates already have algorithmic/statistical error
analysis in path total variation. Wang arXiv:2607.00665v1 already gives
low-temperature deterministic finite-support paths. Neither checked target
is the proposed half-self iid score/residual first-hit theorem.

Belomestny arXiv:2608.29152v1 explicitly treats exact empirical potentials
and excludes finite-iteration optimization error; residual inverse stability
with polynomial temperature constants is prior work. The newer fixed-point
ERM paper (PMLR337:521--548,2026) uses a different estimator; full linked PDF
was inaccessible to the web tool, so no detailed covering claim is based on
its abstract. Greco et al. COLT2023 and Greco--Tamanini Bernoulli2026 already
give gradient convergence/stability, not this iid stopping limit. Primary
links, precise access and proof-read limits are in LITERATURE.txt.

Celisse--Wahl JMLR22(76),2021 and Blanchard--Hoffmann--Reiss JUQ2018 already
develop spectral discrepancy stopping, random-design and oracle results.
The candidate is not an oracle inequality, adaptivity theorem or discovery
of spectral statistical stopping. Its current proof uses standard kernel
CLT, Hilbert covariance/U-statistic variance, PSD contraction, spectral
min-max, log-convexity and Slutsky. These are tools, not separate discoveries.

WHY THERE IS STILL SOME MATHEMATICAL VALUE
The unrestricted arbitrary-candidate certificate has adverse directions.
The candidate does not artificially exclude them or impose a good initial
scaling: it analyzes the actual fixed start and same noisy empirical sample.
It shows an observed, attained residual can remain above the worst-case
root-n tolerance while the particular score error is negligible. Establishing
this requires uniform nonlinear trajectory approximation, same-sample
commutator control, a nonuniform quadratic variance with its diagonal term,
and residual lower/no-overshoot bounds. Neither generic contraction nor
Slutsky provides those lower bounds. Moving to a nonconstant f and a bias-
removal window eliminates the principal scientific weakness of the earlier
uniform zero-bias demonstration.

LIMITS TO KEEP PROMINENT
One-dimensional, fixed C6 density with min f>0; periodic Gaussian, no
Euclidean boundary claim; pointwise stochastic score conclusion; mandatory
two-step burn-in; nonoptimal threshold/window constants. No d>=2 theorem,
varying-density uniformity, optimal tolerance frontier, calibrated finite-
sample rule, minimax gain, practical speedup or iteration-complexity
improvement is demonstrated. The theorem refutes necessity only for this
actual delayed-check procedure/model. It does not retract the sharp
arbitrary-candidate bound or show that every larger tolerance is safe.
Its meaning is most direct as a structural answer to the project's own
necessity question; significance should not be inflated by attributing an
unjustified universal-necessity assertion to the previous paper.

SEARCH AND CONFIDENCE LIMITS
Used actual Chrome Google at this checkpoint for
  "self-transport" "stopping" score
which discovered the consequential Kokot thesis. Earlier start/during
Google queries are recorded in LITERATURE.txt. Supplemental web search
(engine unspecified, not labeled Google) covered inexact score inference,
computational error CLTs, symmetric statistical stopping, and Kokot's
two-update result. Read primary statements/proofs to the explicitly listed
extent, normalized q=b^-2 before comparing, and verified public repository
dates through its API. No exhaustive citation census or human peer review
is claimed. Thesis and metadata are archived locally; external arXiv/PDF
versions otherwise follow the access limitations in LITERATURE.txt.

Research decision: the candidate is not displaced by checked literature and
has enough focused mathematical content to reach an independent proof/value
checkpoint. This is not a claim of a major advance or a publication decision.

FINAL-SOURCE RECONCILIATION AND FIXED-EPSILON RKHS PRECEDENT
18 October 2026. Limited follow-up requested by root; no new research branch.

Read Kokot's Chapter3 Theorem3.6/proof, Corollary3.3/proof (diagonal derivative),
Lemma3.29/proof, and their surrounding setup in the same archived primary
university PDF. The thesis introduction explicitly says Chapter3 perturbs
marginals at fixed epsilon>0, whereas Chapter4 fixes the population and sends
epsilon to zero. Chapter3 fixes a Gaussian RKHS and scaled tangent spaces.
Theorem3.6 is Hadamard differentiability of the two scalings into H²;
Corollary3.3 explicitly identifies the self-diagonal derivative as
(I+T_P)^-1 J_q gamma. Lemma3.29 applies the Hilbert CLT to the centered
empirical feature average and invokes that derivative to obtain root-n
empirical scaling tightness in H. This is a direct stochastic RKHS precursor,
not merely a population antecedent. Theorem3.6 is on printed pp127--129,
Corollary3.3 pp129--131, Lemma3.29 p143 (PDF pp143--145,145--147,159).

Normalization: Kokot's Chapter3 q is the reciprocal scaling v in our notation,
so T_P=K_mu(a² .) is exactly our feature covariance C (up to the Gaussian
parameter/normalizing convention); J_q empirical measure error is our g_n.
The exact-scaling first variation Rg_n, its diagonal plus resolvent, and a
fixed-epsilon Hilbert CLT therefore already appear in essentially the same
RKHS coordinates. They must be credited as prior ingredients. Our shrinking-
epsilon RKHS branch remainder with explicit eta,zeta and derivative
normalization is a quantitative extension/verification of that block,
not invention of a new stochastic sensitivity method.

No simultaneous shrinking-epsilon empirical trajectory or random residual
first-hit theorem is asserted in those checked Chapter3 results. Combining
them formally with Chapter4's population two-step expansion does not by
itself provide uniform nonlinear iid remainders, the threshold-relative
2^-j comparison, attained residual lower/no-overshoot control, or the
same-model true-score inference window. Those remain the exact candidate
increment. This closer stochastic antecedent narrows attribution further
but does not change the value/subsumption verdict above.

Final current joint_nonuniform_theorem.txt SHA256:
4CD5289F95224CD85F112D6070A7FB8B0853AA1AA12E289515E62BC78E94E02E.
Read its updated attribution paragraph explicitly citing Kokot Chapter3
Lemma3.29 and Section4.4.1; checked the parenthesized A_n exponent and fixed
spectral-band interior-margin clarification. These changes do not alter the
C6 one-dimensional theorem, bandwidth window, delayed first-hit rule or
claimed stochastic conclusion. The earlier hash above remains the accurately
identified first reviewed snapshot. The final-source originality/value
verdict remains: meaningful, narrowly scoped increment conditional on proof
verification; no checked direct subsumption; no broad method or priority claim.
