REVISION 2: LITERATURE, SCOPE AND POINTWISE INFERENCE APPLICATIONS
18 October 2026. Research only. Baselines preserved; exact revised proof
and independent validity review pending at this initial writing.

CURRENT TARGET, NOT YET ASSUMED PROVED
Improve the fixed nonuniform periodic half-self theorem to d=1,2; for d2
use epsilon->0,n epsilon^3->infinity,n epsilon^4->0 (strict power interval
1/4<kappa<1/3). Remove compulsory two-step delay, checking the actual first
residual hit from b0=1. Allow t=n^-1/2 epsilon^-gamma for each fixed
0<gamma<d/4; show failure of a class-uniform guarantee at gamma>=d/4 via
the uniform-density countermodel; improve overshoot lower constant toward
any fixed c<1/2. These are proposed improvements until exact source review.

1. CLOSEST PREDECESSORS AND ATTRIBUTION
Alex Kokot, Variational Problems in Feature Selection and Data Compression,
University of Washington PhD dissertation, 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 DSpace metadata was verified in the preceding branch: issued
2026-09-16; accessioned2026-09-16T18:35:50Z; Open Access; CC BY. Archived
PDF/text and metadata remain read-only in that branch's results directory.
Chapter3 Theorem3.6, Corollary3.3 and Lemma3.29 explicitly give scaling
Hadamard differentiability, self-diagonal plus-resolvent derivative and a
fixed-epsilon Hilbert CLT in the Gaussian RKHS. Chapter4 Section4.4.1,
Lemma4.4/Corollary4.3 explicitly give first-order population matching after
two symmetric updates. Normalize q=b^-2 before comparison. These are
direct method/population precedents, not merely vaguely related work.
The fully checked sections do not establish a shrinking-epsilon empirical
trajectory first-hit score theorem or its attained residual asymptotics.
Removing an explicit delay does not make the population two-step mechanism
new. Keep attribution even if early hits are shown asymptotically impossible.

Wormell--Reich, Spectral Convergence of Diffusion Maps: Improved Error Bounds
and an Alternative Normalization, 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 relevant Sections3--4,6,8--9 and proofs. Nonuniform iid
periodic Gaussian normalization, uniform plus-resolvent sup/Hardy bounds,
empirical operator/weight convergence and derivative losses are existing.
Its ASSA algorithm has filter -P(I-P)/2, unlike half-self (I-P)/2. Its
convergence/stopping theorem does not directly yield our density-score
first-hit law. Spectral contraction and a posteriori stopping bounds are
not independent discoveries of this revision.

Cheng--Landa, Bi-stochastically normalized graph Laplacian: convergence to
manifold Laplacian and robustness to outlier noise, arXiv2206.11386v3;
Information and Inference13,iaae026 (2024), DOI10.1093/imaiai/iaae026:
https://arxiv.org/html/2206.11386v3
Refreshed Theorem4.2 and its nearby rate/algorithm discussion. It already
justifies approximate constrained scaling at statistical accuracy and
early termination for nonuniform graph-Laplacian consistency. This is the
closest precedent against claiming that statistical noise permits looser
normalization accuracy generally. Its empirical-L2 graph output is not a
fixed-query density-score Gaussian limit. The actual unrestricted first hit,
attained residual and admissible gamma interval remain different targets.

Mordant, The statistics of entropic optimal transport with decreasing
regularisation, arXiv2412.12007v2,29 April2026:
https://arxiv.org/html/2412.12007v2
Refreshed Theorem12; previous audit read Assumptions1 AND2, Algorithm1,
score-proof linearization. Exact estimator true-score CLTs and the half-self
update already exist. Its Euclidean compact-convex quadratic-cost setting
is not our periodic heat kernel. d2 printed score window is
n^-1/2 << epsilon << n^-1/4; proposed d2 (1/4<kappa<1/3) is a stricter
nonempty subset, not an improvement in exact-estimator sampling range.
The new contribution, if valid, concerns finite actual stopping.

Maeda--Yao--Nitanda arXiv2510.22560v1 already analyzes actual empirical
bridge iterations with statistical and algorithmic errors in path TV:
https://arxiv.org/html/2510.22560v1
Wang arXiv2607.00665v1 already proves deterministic low-temperature
finite-support path theorems. Belomestny arXiv2608.29152v1 already treats
exact empirical potential rates via polynomial residual inverse stability
and explicitly excludes finite-iteration optimization error. Greco et al.
COLT2023 and Greco--Tamanini Bernoulli2026 already control gradients of
Sinkhorn iterates. These sources were read as recorded in prior branch
reviews; no claim that all their full proofs were newly reread here.

Celisse--Wahl, Analyzing the discrepancy principle for kernelized spectral
filter learning algorithms, JMLR22(76),1--59 (2021):
https://www.jmlr.org/papers/volume22/20-358/20-358.pdf
Refreshed Section4 smoothed discrepancy, Proposition13 and Theorem15,
building on previously read Theorems12,19--20. Spectral effective dimension,
residual smoothing, random-design change of norm and data-dependent stopping
are established. Blanchard--Hoffmann--Reiss JUQ2018,
DOI10.1137/17M1154096 is further prior discrepancy-stopping theory. A noise
dimension determining a permissible stopping order is not conceptually new;
the possible new item is its precise realization for actual nonlinear
half-self density-score inference and a proved threshold exponent boundary.

2. HOW TO ASSESS THE PROPOSED SHARP GAMMA CLAIM
For uniform density, initial residual natural size is
n^-1/2 epsilon^-d/4, the square root of a kernel effective-dimension trace.
The coefficient-one threshold with gamma>=d/4 can therefore stop at j0.
For Gaussian multipliers Lambda(xi)=exp(-2pi²|xi|²), the anticipated
constant is
  n epsilon^(d/2) r0² -> (1/4) integral Lambda²
                         =(1/4)(4pi)^(-d/2)<1.
This must be proved for the actual nonlinear projective residual, not only
the proxy. Its initial numerical score error relative to exact scaling has
nonzero statistical-scale covariance. If these statements hold, they refute
a density-class uniform guarantee at gamma>=d/4 for the specified threshold.
They do not prove failure for every density, every tolerance prefactor,
every candidate or every initialization. In particular the boundary case
can depend on its fixed prefactor. Call gamma=d/4 an exponent boundary of
this theorem family, not an optimal universal stopping tolerance.
Likewise any c<1/2 overshoot assertion must be supported by late-trajectory
spectral concentration; generic 1/2 upper contraction alone is insufficient
for a matching lower overshoot factor.

3. CONCRETE POINTWISE INFERENCE: STANDARD COROLLARIES
All conclusions in this section are conditional on a verified revised
fixed-query CLT
  A_n(s_tau(x)-s(x)) => N_d(0, [4J_d/f(x)] I_d),
where A_n=sqrt(n)epsilon^(d/4+1/2), s=grad log f, fixed d and fixed x.
No new core novelty is claimed for studentization, Slutsky, Wald regions,
or testing. These are useful concrete outputs of a valid stopped-score law.

Consistent density plug-in using the same data:
  ftilde_n(x)=n^-1 sum_i k_epsilon(x,X_i)>0.
Heat bias tends to zero at fixed smooth f, and
Var(ftilde_n(x))=O(1/(n epsilon^(d/2))). Thus epsilon->0 and
n epsilon^(d/2)->infinity imply ftilde_n(x)->f(x) in probability. These
conditions follow from the proposed windows. No independence between the
plug-in and stopped score is needed. A differently tuned consistent KDE is
also allowed, provided its own bandwidth conditions are explicitly stated.

Studentized vector pivot:
  [A_n sqrt(ftilde_n(x))/(2sqrt(J_d))](s_tau(x)-s(x))
      =>N_d(0,I_d).
For alpha in (0,1), the confidence region
  {u in R^d: A_n² ftilde_n(x)||s_tau(x)-u||²/(4J_d)
                                      <=chi²_(d,1-alpha)}
has asymptotic coverage1-alpha at that fixed x. In d1 this is the interval
  s_tau(x) +/- z_(1-alpha/2) 2sqrt(J_1)/(A_n sqrt(ftilde_n(x))).
For any fixed nonzero direction ell, replace the interval half-width by
2sqrt(J_d)||ell||/(A_n sqrt(ftilde_n(x))) for ell dot s(x).

Fixed-query stationarity test H0:s(x)=0:
reject when A_n² ftilde_n(x)||s_tau(x)||²/(4J_d)>chi²_(d,1-alpha).
It has asymptotic level alpha at each fixed null density in the theorem's
class. Under each fixed alternative with s(x)!=0, consistency and A_n->infinity
give rejection probability tending to one. A gradient-zero test does not
distinguish maxima, minima and saddles, prove a mode exists nearby, or locate
a mode. Local-alternative power requires triangular-density stability not
supplied merely by a theorem for each fixed f. A data-selected x, sample-site
query, simultaneous band, multiple-query calibration or bootstrap requires
additional results and must not be silently substituted for fixed x.

Goldfeld--Kato--Rioux--Sadhu, EJS18(1),980--1041 (2024),
DOI10.1214/24-EJS2217, arXiv2207.08683:
https://arxiv.org/pdf/2207.08683
Refreshed Example2 and bootstrap discussion: confidence bands for EOT maps
are already a primary application of entropic transport limit theory at
fixed regularization. This supports treating inference as a standard
corollary rather than claiming novelty for confidence regions generally.
Their functional theorem is stronger spatially in its own fixed-epsilon
setting; our shrinking-epsilon pointwise theorem does not inherit its bands.

4. APPLICATIONS THAT REMAIN CONDITIONAL OR UNPROVED
Langevin drift at a predetermined point is a score value multiplied by a
known convention-dependent constant (often 1/2). Its pointwise uncertainty
then follows by linear scaling. A simulated diffusion visits random,
path-dependent points; convergence of its law, invariant measure, mixing,
sampling accuracy or generative-model quality needs uniform/Lp path-weighted
drift control, stability and often time-discretization analysis. None follows
from the fixed-query CLT alone. Do not label those applications proved.
Similarly, diffusion-map graph operator/eigenvector consistency concerns
an integrated or global operator and its spectrum. A point-score CLT is
not such a result; Cheng--Landa/Wormell--Reich have separate hypotheses and
proofs. No automatic new diffusion-map algorithm guarantee is obtained.
Hessian inference, uncertainty for critical-point location, global tests of
uniformity, and confidence bands likewise require additional mathematics.

5. SEARCH COVERAGE AND CURRENT STATUS
Actual Chrome Google at revision start:
  Sinkhorn statistical residual stopping score estimation confidence interval
Visible results inspected, including Maeda et al. Search snippets/AI
summaries were not used as theorem evidence. Supplemental web search
(engine unspecified, not called Google) used stopping threshold/asymptotic
normality, score-estimation confidence, self-transport residual theorem,
inexact/computational-error and symmetric statistical-stopping variants.
Primary statements were refreshed as identified above; prior primary proof
reads and Kokot metadata bindings are explicitly reused, not falsely described
as new rereads. Full new proof review is pending exact source availability.
No identified source directly subsumes the proposed revised full statement;
this focused coverage does not establish worldwide priority. No manuscript,
publication, external-chat message or baseline edit was made.

SUPPLEMENT: PROPOSED LEADING ITERATION COUNT
18 October2026. Exact revised proof still pending at this entry.

Proposed actual stopping law for each fixed 0<gamma<d/4:
  tau_gamma/log(1/epsilon) -> (d/4-gamma)/log2 in probability.
Two fixed exponents gamma1<gamma2 on the same data/initialization would imply
  (tau_gamma1-tau_gamma2)/log(1/epsilon)
                         ->(gamma2-gamma1)/log2.
No independence is needed for subtracting two probability limits.

Mathematical interpretation: after obtaining, in the logarithmic iteration
range, a two-sided actual residual scale
  log r_j = -.5log n -(d/4)log epsilon -j log2
                                          +o(log(1/epsilon)),
the count follows by solving for the crossing of n^-1/2 epsilon^-gamma.
This logarithmic inversion is generic geometric-convergence arithmetic.
The initial factor epsilon^-d/4 is a kernel effective-dimension noise scale;
the factor1/2 is the limiting half-step spectral contraction. Neither
principle is new. The genuinely additional task is proving matching upper
and lower bounds for the ACTUAL nonlinear iid trajectory with epsilon
shrinking, at j proportional to log(1/epsilon), rather than inserting the
worst-case contraction upper bound alone. Uniformly small nonlinear error
must be relative to the crossing scale, and two-sided residual concentration
must hold in that range. Fixed bands with beta arbitrarily close to1/2 can
suffice at leading logarithmic order; a thin-shell limit is not necessary.

Close primary checks refreshed:
* Wormell--Reich Theorem4.1 (author-hosted preprint), local geometric ASSA
  rate and stopping certificate. Different filter -P(I-P)/2; not the joint
  iid half-self count. https://wormell.perso.math.cnrs.fr/preprints/WR20-preprint.pdf
* Mordant arXiv2412.12007v2 Section6.1: reports two-to-three iterations
  typically, but gives no asserted formula for a statistical shrinking-
  bandwidth first-hit count. https://arxiv.org/html/2412.12007v2
* Cheng--Landa arXiv2206.11386v3 approximate normalization at statistical
  tolerance remains the close numerical/statistical precedent. Its graph
  consistency theorem is not an exact leading count for this score rule.
* Celisse--Wahl JMLR22(76),2021 effective-dimension and discrepancy stopping
  sections: stopping times controlled through trace/bias/variance comparisons
  are standard. Our source-specific novelty must not be sold as discovery
  of such a methodology.
* SinkSLOT: Sinkhorn via Sparse Lifted Optimal Transport,
  arXiv2608.28262v1 (28 August2026), AppendixA.4:
  https://arxiv.org/html/2608.28262v1
  Its fixed-support standard-cycle Jacobian analysis explicitly invokes
  Knight2008 and standard differentiable-fixed-point convergence to obtain
  spectral linear convergence and a log-tolerance iteration count. Read the
  stated argument; it supplies an example of generic iteration arithmetic,
  not a theorem covering our growing iid kernel self-score regime.
  Knight DOI10.1137/060659624 official page opened; the report PDF failed and
  full theorem text was not newly accessible, so no newly verified detailed
  Knight covering claim is made.

Actual Chrome Google at this subcheckpoint:
  "Sinkhorn" "statistical" "iteration count" residual
Visible results inspected; AI overview not evidence. Supplemental web search
(engine unspecified) used half-Sinkhorn rate, symmetric statistical iterations,
asymptotic stopping time/residual, and leading iteration count. This focused
coverage found no verified exact covering half-self joint statistical law;
absence does not establish priority. At most the iteration law strengthens
and interprets the specific stopping theorem. It is an asymptotic count
comparison, not measured runtime savings or a new faster algorithm.
