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First-Hit Stopping and Bias-Corrected Score Inference in Dimension Three

Scott Schubr

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Abstract

We study the actual first residual crossing of the symmetric half-Sinkhorn iteration for observations from a fixed positive C⁸ density on the three-dimensional torus. At two heat scales, computed from the same sample, a linear combination removes the first-order population score bias. For ε = n^(−κ) with 2/17 < κ < 4/9 and every fixed 0 < γ < 3/4, stopping each run at its first projective mass residual below n^(−1/2) h^(−γ) preserves the corrected estimator’s fixed-query Gaussian limit. The stopping index is asymptotic to (3/4 − γ) log(1/h)/log 2, and the attained residual lies between c times the threshold and the threshold with probability tending to one for every fixed 0 < c < 1/2. The covariance retains the dependence between the two scales. An explicit cubic population coefficient gives a fixed analytic density for which the limit is shifted at κ = 2/17 and the normalized bias diverges below it, also under actual stopping. The formal Gaussian range extends to κ < 4/9; consistency and shrinking uncertainty require κ < 2/5. A sharp heat-spectral trace determines logarithmic corrections to the stopping count and phase-dependent residual limits, proving that the constants 1/2 and 1 in the residual bracket are sharp.

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Edition files

Available editions

  1. Edition 5 · 2026-11-01 · public
  2. Edition 4 · 2026-10-31 · public
  3. Edition 3 · 2026-10-27 · public
  4. Edition 2 · 2026-10-20 · public
  5. Edition 1 · 2026-10-20 · public

Research details

Subjects
Mathematics · Probability and Statistics
Keywords
entropic self-transport, symmetric Sinkhorn, score estimation, first-hit stopping, bias correction
Manuscript date
2026-10-20
Reader reuse
No additional permission granted

Scope

Fixed positive C⁸ density on the unit three-dimensional torus, a fixed deterministic query, exact periodic heat kernel, the same observations at both scales, and fixed power exponents. Corrected first-hit Gaussian inference holds for 2/17 < κ < 4/9 and every fixed 0 < γ < 3/4; consistent estimation and shrinking uncertainty require 2/17 < κ < 2/5. Sharp stopping-clock and residual-phase geometry has the separate range 2/19 < κ < 2/3. The strict lower 2/17 obstruction concerns this specified correction and stopping rule.

Disclosures

AI tools assisted mathematical development, literature discovery, drafting and checking. The author is responsible for the content. Existing score central limit theorems, entropic bias correction, operator expansions and statistical stopping methods are credited.

Reproduction

Manuscript version 2.1, October 20, 2026; next edition of the existing Dimension Three paper. The thirteen-member package contains the complete 31-page manuscript PDF, nine LaTeX components, a portable Python build helper, README and citation metadata. The proof is analytic. Rebuild using Python 3.9 or later and Tectonic with its LaTeX resources installed; the README describes a new output directory, optional compiler/cache paths and offline mode. Rebuilt PDF bytes may differ with compiler resources or metadata. Version 1.1 contained the corrected estimator, same-sample covariance, lower obstruction and earlier stopping results; version 2.0 added the wider range, complete second-order trajectory and sharp clock/phase law. Version 2.1 adds notation and revision-history clarification only. Prior public editions and the separate dimensions-one-and-two paper remain preserved.

References and related resources

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