NONUNIFORM SELF-TRANSPORT: PROOF CHECKPOINT
18 October 2026. Research only; no manuscript or publication package prepared.

OUTCOME
The extension reaches a meaningful, focused theorem in ONE dimension. It
handles a fixed positive nonconstant smooth density and a genuine bandwidth
range for inference about that density's true score. It is no longer limited
to the uniform zero-bias benchmark or the earlier unusably slow bandwidths.

Authoritative proof: joint_nonuniform_theorem.txt
Final SHA256: 4cd5289f95224cd85f112d6070a7fb8b0853aa1aa12e289515e62bc78e94e02e
An independently written derivation is PROOF_CANDIDATE.txt. Its C-infinity
assumption is stronger than necessary; both final reviews check the root's
stated positive C6 density assumption.

EXACT ESTABLISHED STATEMENT
Let f be a fixed positive C6 probability density on the unit circle, with
min f>0. Sample iid from f and use the exact integral-one periodic Gaussian.
Use actual damped self-Sinkhorn updates from b0=1. Let epsilon_n->0 with
  n epsilon_n^(3/2)->infinity, n epsilon_n^(7/2)->0.
For epsilon_n=n^-kappa the window is
  2/7<kappa<2/3.

Perform TWO mandatory updates, then stop at the first observed projective
square-root-mass residual
  r_j<=t_n=n^-1/2 epsilon_n^(-1/16),
with the stated logarithmic iteration budget as a fallback on rare samples.
With probability tending to one the threshold is hit before the budget and
  t_n/8<=r_tau<=t_n.
Thus sqrt(n)r_tau diverges, while at every fixed query x,
  sqrt(n)epsilon_n^(3/4)|s_tau(x)-s_exact(x)| ->0 in probability.
The stopped and exact empirical scores have the SAME true-score law,
  sqrt(n)epsilon_n^(3/4)[s(x)-f'(x)/f(x)]
      => Normal(0,4J1/f(x)),
  J1=int_R (2 pi xi)^2/[1+exp(2 pi^2 xi^2)]^2 dxi.

The two mandatory updates are part of the algorithm. The statement is not
the first residual crossing from iteration zero. The pointwise limit is
proved for this same periodic model; no theorem for a Euclidean support
with boundary is silently transplanted.

MATHEMATICAL INCREMENT
The result proves that a residual above the arbitrary-candidate root-n
tolerance can be attained by an actual solver while retaining inference
about a nonconstant true score. It closes the two limitations of the earlier
uniform result in this one-dimensional model: population burn-in and an
inference-useful bandwidth range.

The essential new argument is a quantitative same-sample RKHS trajectory
expansion, with separate sup and L2 relative-log control. The residual
comparison retains a 2^-j empirical/population measurement term. That term
is not uniformly negligible at root-n scale in the full bandwidth window;
it becomes negligible relative to the observed stopping threshold. General
Hilbert quadratic variance, including its nonconstant diagonal term, and
spectral log-convexity then justify the first hit and its attained residual.

INDEPENDENT VERIFICATION
Two separately tasked AI mathematical reviewers read the full proof, checked
its rates, normalization, regularity, same-sample dependence, all-iteration
bounds, spectral concentration, random stopping and true-score CLT. Both
reported no substantive gap. The separate derivation agrees. Two precision
clarifications were applied: an explicitly parenthesized A_n exponent and
positive-width spectral intervals counted through interior margins. Prior
work attribution was added without changing the mathematics. Final-source
reconciliation is recorded in the reviews. These are internal AI reviews,
not external human refereeing or proof-assistant formal verification.

See reviews/independent_proof_review.txt and
reviews/independent_cross_review.txt. Code and numerical independence are
recorded in reviews/independent_code_review.txt and the finite-check files.

ORIGINALITY AND VALUE
FINAL_ORIGINALITY_REVIEW.txt judges the result a meaningful but limited
mathematical increment. No checked primary statement subsumes the complete
stochastic stopping theorem. This is not a worldwide priority guarantee.

Known ingredients include scaling, RKHS covariance and implicit-function
methods, the half-step filter, contraction, kernel CLTs, spectral min-max,
quadratic variance, log-convexity and Slutsky. Existing work already justifies
statistical early stopping for other Sinkhorn-derived objects.

A close precursor was found and credited: Alex Kokot's UW dissertation,
publicly deposited 16 September 2026, Section4.4.1/Lemma4.4, already gives
the two-update population mechanism. Chapter3/Theorem3.6 and Lemma3.29 also
precede the fixed-regularization RKHS scaling derivative and Hilbert CLT.
Our credit concerns the shrinking-bandwidth actual empirical trajectory and
its observed, attained residual crossing with the preserved true-score law.
Primary repository record: https://hdl.handle.net/1773/57893

Mordant, Wormell--Reich, Cheng--Landa, Maeda--Yao--Nitanda and spectral
discrepancy-stopping theory are compared in the literature files. The result
does not introduce the score estimator, two-update population matching,
or statistical stopping as a general concept.

CALCULATIONS: WHAT THEY DO AND DO NOT SHOW
18 actual iid trajectories used f=1+.4 cos(2 pi x), n=512,2048,8192,
kappa=.35,.5,.6 and two seeds. Independent replay reproduced the stored
values apart from runtime. Separate dense spatial-kernel/reference and
derivative/modulus checks agreed. The population initialization probe checks
the O(epsilon) first-step and O(epsilon^2) second-step algebra numerically.
All these are floating-point, finite-Fourier/grid/quadrature diagnostics;
the analytic proof does not use them as premises.

IMPORTANT FINITE-SAMPLE LIMIT: the original run's threshold summaries allow
j=0/1 and therefore are not the theorem's delayed rule. A separate immutable
postprocessing of its raw rows applies j>=2 and gamma=1/16. All 18 delayed
hits occur at j=2. Their r_hit/t values range .0311--.3933; 14/18 are below
1/8. Thus the finite calculations do NOT demonstrate the theorem's asymptotic
lower bound or calibrate its onset. This is not hidden as a successful
finite-sample theorem test. The lower bound is a probability-tending-to-one
statement with conservative constants and no finite-n assertion.

One original diagnostic code comment reverses the direction of a grid
minimum's relation to the continuous minimum. The audit corrects its
interpretation: the grid minimum is an upper bound on the global minimum.
The code and outputs are preserved; no continuum positivity certificate is
claimed. Sample/query numerical checks do not replace exact kernel positivity.

REMAINING LIMITATIONS AND STOPPING DECISION
Meaningful mathematical value remains, so the user-requested no-value stop
condition was not triggered. The new result is a focused one-dimensional
stochastic solver theorem; it is not a general Sinkhorn breakthrough.
Dimensions >=2, boundaries, sharper/optimal bandwidths and tolerances,
uniformity over changing density classes, practical runtime gains, finite-n
calibration, and external human mathematical review remain unresolved.

Stop at the authorized Proof checkpoint. Do not prepare a paper or change
the specialist publication package in this research turn. The appropriate
next decision is whether to incorporate this verified extension into a
future revision, after expert review, rather than multiplying publication
packages for overlapping results.

PRESERVATION
PRESERVATION.json checks the original and corrected publication manifests
and the earlier uniform research manifest without modifying them. The new
MANIFEST.json inventories this branch. Hashes establish identity, not proof
correctness or originality. All superseded candidate arguments, rate notes,
raw experiments, replays, primary-source copies and critiques remain present.
