PATH 3: SUPPLEMENTAL LITERATURE / SCOPE AUDIT
18 October 2026. Research only. Preserves LITERATURE.txt and all originals.

Input reviewed
proof_candidate.txt, SHA256
84582D98F5D300B36EE25B56FECD3F42ED0B0A8379650BDFD0E44983C33AE884.
This snapshot ends at sections E1--E2 (fixed-bandwidth actual-iterate
expansion/CLT), F (shrinking-bandwidth linear proxies), and G (remaining
obligations). The later uniform-in-j quantitative proposal was communicated
by the parent during this audit; its full proof has NOT been independently
checked here. Originality assessment below is conditional on that proof.

DECISION
E1--E2 are a useful explicit calculation for the actual half-self iteration,
but their mathematical method and distributional conclusion are routine
smooth-functional asymptotics. They should not be presented as a new general
Sinkhorn CLT. The explicit B^j filter and joint residual/score law identify
the mechanism that a stronger theorem would need. A quantitative theorem
for actual iid iterates, uniformly over iteration number with an explicit
shrinking-bandwidth window and an actual residual hitting-time conclusion,
would add a real, narrowly scoped mathematical statement beyond E1--E2.
None of the primary results checked here supplies that exact statement.
This is an overlap assessment, not a worldwide priority claim.

1. WHY FIXED EPSILON E1--E2 ARE STANDARD
The candidate works in H^-m, m>d/2+1, with fixed smoothing kernel. Empirical
measure fluctuations have an H^-m Hilbert CLT; the exact self-scaling is a
smooth implicit functional because its derivative I+H is invertible;
finitely many smooth update compositions preserve this differentiability.
Taylor expansion and continuous mapping then give the score Gaussian limit
and residual Gaussian-norm limit. The latter is NOT a scalar Gaussian CLT.
Its squared version is a quadratic Gaussian limit. Shared self samples use
one empirical bridge, not independent copies of two bridges.

Goldfeld, Kato, Rioux and Sadhu, Limit theorems for entropic optimal transport
maps and Sinkhorn divergence, EJS 18(1), 980--1041 (2024),
DOI 10.1214/24-EJS2217; arXiv:2207.08683:
https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-18/issue-1/Limit-theorems-for-entropic-optimal-transport-maps-and-Sinkhorn-divergence/10.1214/24-EJS2217.pdf
https://arxiv.org/pdf/2207.08683
Read Theorem 1, discussion comparing prior limits, Theorem 3/proof starting
Section 6.1, and proof of Theorem 1. They obtain Gaussian limits for exact
potentials in smooth function spaces through Hadamard differentiability and
the functional delta method, with gradient-map consequences. Their
second-order analysis of Sinkhorn divergence already illustrates a
degenerate first derivative followed by a quadratic limit. Our specific
finite-step residual is a different statistic; extending the differentiable
functional by finitely many smooth iterates is routine specialization.

Gonzalez-Sanz, Loubes and Niles-Weed, Weak limits of entropy regularized
Optimal Transport: potentials, plans and divergences, arXiv:2207.07427:
https://arxiv.org/html/2207.07427
Read Theorem 2.2 and nearby operator linearization, plus smooth-composition
appendix. Smooth exact empirical potentials and resolvent expressions are
already central. Self-diagonal perturbation simplifies the two-marginal
linear system; recognizing (I+H)^-1 is not a new inverse-function method.

Gonzalez-Sanz and Hundrieser, Weak Limits for Empirical Entropic Optimal
Transport: Beyond Smooth Costs, arXiv:2305.09745:
https://arxiv.org/pdf/2305.09745
Read Theorem 3.1 context, Corollary 3.2, Theorems 3.3/3.5, and Section 5 proof
structure. Exact empirical-potential linearization in empirical L2, followed
by CLTs for cost and plan functionals, is existing theory even for bounded
nonsmooth costs. This is especially close at the linearization/resolvent
level, but it is not our spatial derivative or finite damped trajectory.

Hundrieser, Klatt and Munk, Limit Distributions and Sensitivity Analysis for
Empirical Entropic Optimal Transport on Countable Spaces,
arXiv:2105.00049:
https://arxiv.org/pdf/2105.00049
Read statistical-limit section and Section 5 Hadamard sensitivity, including
Theorem 5.5 and plan sensitivity. This is a separate paper, not the
Gonzalez-Sanz--Hundrieser result. Its geometry is countable spaces; it
reinforces that sensitivity plus empirical-process delta methods is standard,
not a direct theorem about the torus spatial score.

Numerical M/Z-estimation interpretation: the generic result that a solver
error negligible at the statistical scale preserves a limit law is standard.
Nor does a fixed finite update automatically eliminate the first-order error:
here its derivative is B=(I-P)/2, generally nonzero, unlike a suitably
preconditioned Newton one-step correction. E1's nonzero finite-j numerical
score covariance is an explicit realization of this elementary distinction.
No complete primary numerical-M/Z-estimation theorem was newly verified in
this refresh: Newey--McFadden chapter PDF access failed; secondary textbook
and one-step leads were used only for discovery, not as evidence of an exact
covering theorem. The primary functional-delta-method proofs above suffice
for the stated fixed-bandwidth overlap judgment.

2. MORDANT AND ACTUAL ITERATIONS
Mordant, The statistics of entropic optimal transport with decreasing
regularisation, arXiv:2412.12007v2, 29 April 2026:
https://arxiv.org/html/2412.12007v2
Refreshed Theorem 11/proof opening, Assumption 2/Remark 3, Proposition 25,
and Section 6.1 Algorithm 1. Algorithm 1 is exactly the damped self update
from zero log-potential (b0=1), up to kernel/domain conventions. The text
reports typically two or three iterations empirically. It does not turn that
observation into the proposed residual-hitting-time score theorem. Assumption
2 is separate uniform C4 regularity of rescaled population self-potentials;
do not silently attribute it to Assumption 1. Neither the algorithm nor its
half-update derivative should be claimed as new. Its decreasing-regularizer
exact-estimator statistics are essential background, not the new theorem.

3. NEW WANG PAPER REALLY DOES STUDY ACTUAL LOW-TEMPERATURE PATHS
Guillaume Wang, Effective dynamics of the Sinkhorn algorithm in the regime
of low entropy regularization, arXiv:2607.00665v1, submitted 1 July 2026:
https://arxiv.org/abs/2607.00665v1
https://arxiv.org/html/2607.00665v1
Read introduction, Theorems 4.1--4.2 and surrounding qualification, Theorem
5.2, and fixed-support convergence propositions. Ordinary alternating
two-marginal iterates, rescaled in time by temperature, approach a piecewise
linear cold dynamics. Theorem 4.1 gives O(tau log(1/tau)^(2L+1)) uniform path
error through L phases; Theorem 4.2 extends the time interval. Constants
depend on the finite marginals and cost, with L potentially exponential in
support sizes. The paper does not track uniform growing-sample constants.
Thus deterministic low-temperature trajectory analysis already exists in a
strong form. It does not directly cover half-self iid torus data, changing
support size, spatial-score inference, or residual-adaptive stopping.

4. OTHER CLOSE SOURCES / IMPORTANT EXCLUSIONS
Maeda, Yao and Nitanda, Statistical Analysis of the Sinkhorn Iterations for
Two-Sample Schrodinger Bridge Estimation, arXiv:2510.22560v1:
https://arxiv.org/html/2510.22560v1
Refreshed Theorem 2: expected squared path-TV algorithmic error has explicit
geometric k decay and epsilon-dependent factors. Joint statistical and
iteration error analysis is already its purpose. The proposed result must
be distinguished by its actual half-self spatial score and weaker residual
threshold from stochastic modal structure, not by merely combining errors.

Belomestny, Uniform Statistical Convergence of Empirical Sinkhorn Potentials
with Exponential and Polynomial Dependence on the Regularization Parameter,
arXiv:2608.29152v1 (29 August 2026):
https://arxiv.org/html/2608.29152v1
Read Theorem 4.4, Proposition 4.6, Corollary 4.7, and envelope discussion.
Polynomial epsilon-dependent statistical rates under quantitative residual
stability/local inverse assumptions are existing. Its target is exact
empirical potentials in quotient sup norm. It supplies no actual iterated
score/residual covariance or hitting-time gain. A conservative polynomial
window alone therefore would not establish distinctive value.

Celisse and Wahl, Analyzing the discrepancy principle for kernelized spectral
filter learning algorithms, JMLR 22(76), 1--59 (2021):
https://www.jmlr.org/papers/v22/20-358.html
https://www.jmlr.org/papers/volume22/20-358/20-358.pdf
Read Theorems 12, 19--20 and nearby proof descriptions. Residual stopping,
spectral filters, random-design issues, and data-dependent stopping-time
oracle bounds are established subjects. Their regression observation model
and target differ. No claim that residual-adaptive statistical stopping or
typical spectral improvement is new in general is justified.

Blanchard, Hoffmann and Reiss, Optimal adaptation for early stopping in
statistical inverse problems, arXiv:1606.07702; SIAM/ASA JUQ (2018),
DOI 10.1137/17M1154096:
https://arxiv.org/abs/1606.07702
https://epubs.siam.org/doi/10.1137/17M1154096
Primary abstract verified: residual-based stopping and oracle adaptation for
general spectral regularization precede this project. Full proof not read in
this refresh. It is background against broad novelty, not a direct covering
result for a nonlinear empirical Sinkhorn solver.

Knight, Feydy, Wormell--Reich, Cheng--Landa, semigroup and initialization
sources remain covered in LITERATURE.txt; no new full reread of each was
performed here. They prevent crediting standard local spectral convergence,
positive-kernel stability, empirical graph convergence, or initialization
dependence as new effects. Their combination does not by itself supply the
specific nonlinear iid first-hitting-time comparison below.

5. STRONGEST TARGET NOW WORTH VERIFYING
Parent's emerging proposal is uniform population on a torus, heat kernel,
integer m>d/2+1, actual start b0=1, all-iteration nonlinear expansion with
score remainder Op(n^-1 epsilon^-2m), squared-residual remainder
Op(n^-3/2 epsilon^-3m), and explicit conservative window
n epsilon^(6m) -> infinity. With
J=floor[d log(1/epsilon)/(16 log 2)] and
tolerance=n^-1/2 epsilon^-d/16,
the intended conclusion is: the first actual residual crossing occurs after
J with high probability and its numerical pointwise score error is
negligible at A_n=sqrt(n) epsilon^(d/4+1/2), even though sqrt(n) times the
tolerance diverges. This would show the unrestricted worst-case residual
condition is unnecessarily strict for this particular stochastic trajectory.

Substance if valid: quantitative propagation through the actual nonlinear
solver, a uniform event permitting a data-dependent index, and the mismatch
between residual energy and a differentiated smoothing functional for iid
modal noise. Generic contraction plus Slutsky does not supply the joint
residual lower bound needed to certify that this looser threshold is reached
only after the necessary burn-in. The filtered linear Fourier variances are
elementary; the valuable step is closing uniform nonlinear and empirical
remainder estimates and actual stopping consequences.

Exclusions: no new algorithm, no uniform gain for arbitrary initializations
or residual directions, no density-general assertion, no new exact-estimator
CLT, no optimal bandwidth window, no minimax result, no iteration-complexity
improvement. A diagonal existence argument choosing epsilon_n sufficiently
slow from fixed-epsilon limits is also routine and does not replace an
explicit window. Distinguish negligibility relative to the exact empirical
score from accuracy relative to the true population score. Do not claim
uniformity over evaluation x unless independently proved; the proposed
pointwise result and spatial supremum are different targets.

6. ACTUAL SEARCH / ACCESS LIMITS
Google through Chrome used at the start of this update for
"Sinkhorn potentials functional delta method Hundrieser finite iterations",
and during the changed quantitative target for
"Sinkhorn statistical residual stopping score shrinking regularization".
Read visible results; AI summaries were not evidence. Supplemental web search
(engine unspecified, NOT called Google) covered statistical stopping,
finite-iteration CLTs, small regularization, Mordant iterations and numerical
M/Z-estimation. Primary arXiv HTML/PDF, Project Euclid and JMLR statements and
selected proofs were read as specified above. Some arXiv HTML endpoints
failed and were replaced by their PDFs. Unpinned PDF endpoints for older
papers were read as served on 18 October 2026; no independent hash/version
archive of those external PDFs was made. This is focused coverage, not an
exhaustive forward-citation census. No absence-of-search-result priority
inference, external peer review, or publication authorization is asserted.
