ACTUAL SELF-SINKHORN STOPPING: ORIGINALITY AND MATHEMATICAL VALUE
Research checkpoint audit, 18 October 2026. No publication recommendation.

Reviewed source: joint_stopping_theorem.txt, SHA256
6B310FDF366EAA21B83B3DD9FFA16CFC321BBABB6E6A7E12B5872AABFB6E2336.
Read the complete statement and proof. This is an originality/value review;
two separate proof reviews are pending according to the parent. Earlier
LITERATURE.txt and LITERATURE_UPDATE.txt remain unchanged.

SHORT VERDICT
Conditional on proof verification, this is a real but modest mathematical
increment over the fixed-epsilon candidate. It establishes a precise
counterexample to necessity of the unrestricted root-n residual tolerance
along actual iid, standard-start trajectories, and makes that separation
observable through an actual residual hitting rule. It is more than a CLT
plus Slutsky or a local Jacobian calculation. Its ingredients and basic
mechanism are standard; its value lies in carrying them through the nonlinear
empirical iteration and proving the stopping residual itself stays large.
This does not yet establish broadly useful relaxed stopping for statistical
score estimation. Uniform population and the very slow bandwidth window
are material limits. I would call it a focused structural separation theorem,
not a substantial general advance in Sinkhorn statistics or numerical
inference. No directly covering primary result was identified in the
coverage below; that does not establish global priority.

EXACT INCREMENT
For iid uniform torus data, the periodic Gaussian, actual b0=1 half-self
updates, and n epsilon^(6m)->infinity with integer m>d/2+1, the source proves:
  * uniform-in-all-j nonlinear score and squared-residual expansions;
  * pointwise numerical score negligibility after a deterministic growing J;
  * the first observed residual hit at n^-1/2 epsilon^-d/16 occurs after J;
  * the attained residual is between one eighth and one times that threshold,
    hence sqrt(n) r_tau diverges, rather than merely an upper bound diverging;
  * the stopped score retains the exact empirical pointwise Gaussian law.
The last assertion uses a proved kernel CLT within the matched periodic model.
The first-hit lower bound and overshoot control are necessary to make the
nonnecessity claim concrete. A loose upper tolerance alone would not do so.

KNOWN INGREDIENTS TO CREDIT
The damped self-update, its B=(I-P)/2 derivative, local geometric contraction,
Sobolev algebra and smoothing estimates, implicit scaling expansion, empirical
Fourier covariance, quadratic U-statistic variance, positive spectral-sum
log-convexity, triangular-array kernel CLT, and Slutsky are established tools.
The broad facts that numerical error need only be small in the inferential
functional and that typical noise need not attain worst-case operator norms
are also established principles. These should not be independently advertised
as discoveries. The threshold exponent d/16 and n epsilon^(6m) window are
convenient proof choices, not sharp phenomena.

CLOSE PRIMARY COMPARISONS
Mordant, The statistics of entropic optimal transport with decreasing
regularisation, arXiv:2412.12007v2, 29 April 2026:
https://arxiv.org/html/2412.12007v2
Refreshed Theorem 12 and proof opening plus Section 6.1/Algorithm 1. The
actual zero-log initialization and half update already appear there, with
rapid few-step convergence observed numerically. Exact score asymptotics
are existing theory. The paper does not prove this same-data first residual
hit, attained-residual lower bound, or stochastic nonnecessity statement.
Theorem 12 requires Assumptions 1 AND 2 and the window
n^-2/(d+2) << epsilon << n^-2/(d+6). The new theorem does not extend that
true-score inference result to relaxed stopping in its bandwidth regime.

Maeda--Yao--Nitanda, Statistical Analysis of the Sinkhorn Iterations for
Two-Sample Schrodinger Bridge Estimation, arXiv:2510.22560v1:
https://arxiv.org/html/2510.22560v1
Refreshed Theorems 1--2 and their algorithmic-error interpretation. Statistical
and finite-iteration errors for actual empirical bridge iterates are already
analyzed together. The metric is path total variation and the algorithm is
two-marginal. Our distinct claim is a spatial self-score law at a measured
residual crossing with an attained residual larger than the unrestricted
inference tolerance. Merely combining sample and computational errors would
not be a contribution relative to this paper.

Wang, Effective dynamics of the Sinkhorn algorithm in the regime of low
entropy regularization, arXiv:2607.00665v1, 1 July 2026:
https://arxiv.org/html/2607.00665v1
Theorems 4.1--4.2 are actual low-temperature path theorems for deterministic
alternating two-marginal iterations on fixed finite supports. Their constants
depend on marginals/cost, and phase count can grow exponentially in support
sizes. They do not supply a uniform growing-iid-support score hitting-time
theorem. Thus low-temperature trajectory analysis generally is not new,
whereas the present stochastic self-score comparison is different.

Belomestny, Uniform Statistical Convergence of Empirical Sinkhorn Potentials
with Exponential and Polynomial Dependence on the Regularization Parameter,
arXiv:2608.29152v1:
https://arxiv.org/html/2608.29152v1
Theorem 4.4 and Proposition 4.6 already connect polynomial epsilon-dependent
statistical rates to quantitative residual stability/local inverses. They
concern exact empirical potentials in quotient sup norm, not an actual
half-self iid trajectory retaining the point-score law at a loose crossing.
Polynomial dependence or a local-inverse proof alone is therefore not new.

Celisse--Wahl, Analyzing the discrepancy principle for kernelized spectral
filter learning algorithms, JMLR 22(76), 1--59 (2021):
https://www.jmlr.org/papers/volume22/20-358/20-358.pdf
Previously read Theorems 12 and 19--20 and nearby proofs. Random-design
spectral filtering, residual stopping, smoothed discrepancies and stopping
time oracle inequalities are established theory. Blanchard--Hoffmann--Reiss,
Optimal adaptation for early stopping in statistical inverse problems,
arXiv:1606.07702 / DOI 10.1137/17M1154096, supplies further precedent.
These regression/inverse-problem results are not a direct nonlinear
Sinkhorn theorem. The current result is not an oracle inequality or an
adaptivity result, and should not claim that general conceptual territory.

Goldfeld--Kato--Rioux--Sadhu, Gonzalez-Sanz--Loubes--Niles-Weed, and
Gonzalez-Sanz--Hundrieser exact-potential limit/linearization results are
discussed with primary links and sections in LITERATURE_UPDATE.txt. They
substantially overlap fixed-bandwidth E1--E2 methods. They do not erase the
quantitative all-iteration and attained hitting-residual obligations here.

New discovery checked: arXiv:2609.07954v1 HTML is titled Transport Topology
for Fixed-Support Sinkhorn Layers, Dylan B. Forde, 7 September 2026:
https://arxiv.org/html/2609.07954v1
Read scope, first-order half-step proposition and finite-cycle extension.
It analyzes quotient derivative/cotangent kernels for fixed support,
including actual finite cycles. Its use of score means transport logits,
not a density's spatial log-gradient. It supplies no iid inference or
shrinking-bandwidth residual nonnecessity theorem. Search metadata returned
a different title; the primary HTML title is recorded here deliberately.

Additional lead opened: Faster Wasserstein Distance Estimation with the
Sinkhorn Divergence, primary PDF hosted by an author/university:
https://www.normalesup.org/~mfathi/docs/enseignement/articles%20TO%20M2/sinkhorn.pdf
Its error decomposition includes finite iterations and statistical errors
for distance estimation. Targeted PDF finding then failed; no detailed
covering-theorem claim is made from that incomplete access. The broad
statistical/computational error-budget idea is already established regardless.

WHY THE PRESENT VALUE IS NARROW
Uniformity is doing substantial work: the population scale is exactly one,
the continuum start is already exact, the true score is zero, Fourier modes
diagonalize every population filter, and all initialization error comes from
sample fluctuations. The exact empirical estimator still has nondegenerate
noise, so preserving its law while observing a large residual is nonvacuous.
But this says nothing yet about deterministic burn-in, bias or useful stopping
for a fixed nonuniform density. Returning a good estimate of a nonconstant
score is not demonstrated by this model.

For d=1, m=2 permits only epsilon >> n^-1/12. Mordant's true-score
undersmoothing requires epsilon << n^-2/7, so the windows do not overlap.
More generally 6m>d+6, hence 1/(6m)<2/(d+6); the conservative window is
incompatible with that upper undersmoothing requirement in every dimension.
Uniform zero bias removes this conflict for the theorem itself; it does not
solve it for nonuniform inference. Constants selected for a short proof do
not provide practical calibration, a demonstrated runtime saving, or an
optimal stopping order. A particular diverging residual is shown sufficient;
no sharp typical-trajectory tolerance frontier is established.

The logical counterexample is useful specifically against elevating a
worst-case necessary order over arbitrary candidate scalings into a necessary
order for the standard stochastic solver. If nobody had made that inference,
the theorem is principally a refinement of interpretation. It remains a
valid answer to this project's research question, but its significance should
not be inflated by contrasting it with an unjustified universal necessity
claim that the existing certificate did not actually make.

RESEARCH CHECKPOINT ASSESSMENT
Validity: full source read; separate proof-verification disposition pending.
Originality: candidate-specific nonlinear hitting-time separation not found
in checked primary statements; known ingredients and adjacent broad claims
must be credited. Focused search cannot certify priority.
Value: modest, precise structural information about an actual solver in one
matched model. Stronger than fixed-epsilon delta method, weaker than evidence
for broadly relaxed numerical accuracy in score inference. The natural next
scientific test would be a fixed nonuniform density and an inference-useful
bandwidth regime; this file does not authorize or initiate that extension.

SEARCH COVERAGE AND LIMITS
On 18 October 2026 direct Chrome Google query:
  "half-Sinkhorn" "stopping"
Google showed no exact quoted match and fallback leads including Knight;
this absence is not evidence of priority. Earlier direct Google queries and
primary proof reads are recorded in LITERATURE_UPDATE.txt. Supplemental web
search (engine unspecified, not called Google) refreshed self-Sinkhorn score
stopping, half-Sinkhorn statistical residual stopping and residual
nonnecessity. Primary statements were opened/refreshed as listed above.
Wang, Belomestny and the general discrepancy literature were also read in the
immediately preceding audit. This is a focused related-work comparison, not
an exhaustive forward-citation review. No manuscript or publication package
was created and no original was edited.
