PATH 3 PROOF CHECKPOINT -- 18 OCTOBER 2026
Actual standard-start self-Sinkhorn trajectories and statistical score accuracy

Active objective and boundary
Scott authorized Riemann B to assign Path 3 to Riemann C: determine whether
actual damped self-Sinkhorn trajectories from b0=1 encounter the adverse
worst-case residual modes, or permit weaker observable stopping while
preserving score inference. The assignment covers development and verification
through the Proof checkpoint, not preparation or publication of a new paper.
The older publication handoff was already completed; revisiting it after
compaction was an objective-tracking error. Research has resumed and reaches
this checkpoint. No existing manuscript or original evidence was changed.

MAIN ESTABLISHED RESULT
File: joint_stopping_theorem.txt
Final SHA256: 92d7805cea6b158192a4dbfe8215d1af611fdc006d52f065219610b039a68756

For iid UNIFORM data on T^d, the exact normalized periodic Gaussian kernel,
and actual damped self-Sinkhorn updates starting at b0=1, fix an integer
m>d/2+1 and epsilon_n->0 with n epsilon_n^(6m)->infinity. Define
  A_n=sqrt(n) epsilon_n^(d/4+1/2),
  t_n=n^-1/2 epsilon_n^(-d/16).
Stop at the FIRST observed projective square-root-mass residual r_j<=t_n,
with the stated ceil(4 log n) budget to define the algorithm on every sample.
With probability tending to one the threshold is hit strictly before the
budget and the attained residual satisfies
  t_n/8 <= r_tau <= t_n.
Thus sqrt(n) r_tau diverges. Nevertheless, at each fixed spatial query x,
  A_n |s_tau(x)-s_exact(x)| ->0 in probability.
The stopped and exact empirical scores have the same nondegenerate Gaussian
pointwise limit, proved directly for this matched uniform periodic model.

This establishes a genuine separation: r=o_p(n^-1/2), which suffices for
arbitrary positive candidates under the existing worst-case theorem, is not
necessary for these actual stochastic trajectories. The attained residual
lower bound is essential; this is not merely a loose upper tolerance whose
actual output might silently satisfy the old tolerance.

What makes the new proof work
1. Negative-Sobolev empirical fluctuations plus polynomial heat smoothing
   give a quantitative local scaling expansion uniformly in epsilon.
2. The exact relative-log iteration has a single high-probability invariant
   contraction basin. Its nonlinear score and SQUARED residual expansions
   hold uniformly over ALL iterations, permitting a data-dependent first hit.
3. Exact Fourier/U-statistic moments show a large residual remains at a
   growing deterministic iteration, while the normalized point-score tail
   vanishes. Fixed annuli avoid an unproved growing-frequency Riemann sum.
4. Positive spectral-sum log-convexity controls overshoot at the first hit.
5. A directly checked triangular-array kernel CLT supplies the reference
   pointwise law. No CLT for a different geometry is imported.

OTHER ESTABLISHED RESULTS
proof_candidate.txt records independently checked preliminary results:
* Arbitrary nearby positive initializations can reach every local direction
  after any fixed finite number of updates (finite-dimensional inverse map).
* Starting from 1, varying a smooth density near uniform at fixed epsilon
  and fixed iteration can approach the full existing worst-case modulus.
  It is a supremum over engineered density paths, not necessarily attained
  by one smooth density tangent. No fixed-density joint limit follows.
* Fixed-bandwidth finite-iteration empirical expansions and limit laws follow
  by a standard smooth implicit-function/delta-method argument.
These are useful supporting calculations; they are not a substantial general
advance or a replacement for the quantitative joint theorem above.

VERIFICATION COMPLETED
Mathematical: two separate AI proof reviews read the full joint proof and
found no substantive gap. A second complete derivation was written separately.
The one minor strict-before-budget wording issue was resolved by the same
estimate at L-1. Final source reconciliation is recorded in the reviews.
These are internal AI checks, not external human refereeing or formal proof.

Computation: 12 actual iid trajectories used uniform and cosine densities,
n=256,512,1024, two seeds each, and epsilon=.64/sqrt(n). All reached r<=1e-10
in 21--24 updates. A separate replay matched the authoritative run_02 exactly
apart from runtime. An independent spatial Gaussian implementation, dense
Newton reference and dense local-modulus checks agreed. The maximum independent
modulus discrepancy was 5.934e-13; derivative checks and score identities passed.
These are floating-point diagnostics, not interval-certified premises.

Finite trajectory directions were not uniformly orthogonal to adverse rows:
late pointwise error/(r*local modulus) ranged approximately .097--.911.
This does not contradict the new theorem: the sample sizes, bandwidth rule,
normalization and statement are different. In particular these numerical
bandwidths lie OUTSIDE the theorem's conservative window. The analytic theorem
does not rely on the experiments as proof premises.

Independent review files:
  reviews/independent_proof_review.txt
  reviews/joint_stopping_proof_review.txt
  reviews/joint_stopping_cross_review.txt
  reviews/joint_stopping_independent_derivation.txt
  reviews/independent_code_review.txt
  reviews/independent_finite_checks.py and .json
Authoritative experiment: results/run_02/
Independent replay: results/independent_replay_01/

ORIGINALITY AND MATHEMATICAL VALUE
See ORIGINALITY_AND_VALUE.txt, LITERATURE.txt and LITERATURE_UPDATE.txt.
The originality review read the complete first joint-proof snapshot. The
final snapshot differs only in review-status wording and the explicit L-1
budget clarification; its mathematical statement and argument are unchanged.

The half update, its spectral derivative, local contraction, implicit scaling,
kernel CLTs, U-statistic moments and Slutsky are standard ingredients. Prior
work already combines statistical and algorithmic Sinkhorn errors and studies
low-temperature trajectories. The candidate-specific increment is the uniform
nonlinear control plus actual first-hit/attained-residual separation.

No directly covering primary result was identified in the searches performed.
That supports plausible narrow novelty, not worldwide priority. The literature
review judges the result a real but MODEST structural contribution, rather
than a general advance in score inference or an established practical saving.

Material limitation: uniformity removes population initialization bias and
diagonalizes all population filters. For d=1, m=2, the sufficient window is
epsilon >> n^-1/12. It does not overlap Mordant's nonuniform true-score
undersmoothing condition epsilon << n^-2/7. More generally the new window
excludes that undersmoothing regime in every dimension. The theorem itself
is valid because its uniform periodic population has zero score bias and
its own reference law is proved. It does NOT establish relaxed stopping
under Mordant's Euclidean/nonuniform assumptions.

REMAINING FRONTIER AND RECOMMENDATION
The primary unresolved question is a fixed NONUNIFORM density in an
inference-useful bandwidth regime: separate deterministic population burn-in
from sample fluctuations, control nonlinear/operator errors sharply enough,
and retain an observable stopping guarantee. This is the highest-value next
extension. Optimal typical tolerances, useful runtime improvements, boundaries,
vanishing densities, and external human proof/originality review remain open.

Do not create another publication package at this checkpoint. Preserve this
verified result as a possible later extension and prioritize the nonuniform,
useful-bandwidth question if further research is authorized. Stop here at
the explicitly assigned Proof checkpoint, as required by the mathematical-
research skill. Existing publication packages remain independent and unchanged.

Preservation and file identity: PRESERVATION.json and MANIFEST.json.
Source hashes establish version identity only, not correctness or priority.
