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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 C^8 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)/log2, 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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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

Actual first-hit bias-corrected score inference on T³ with the same-sample two-scale covariance, Gaussian range 2/17 < κ < 4/9, consistency range κ < 2/5 and sharp residual phase constants under the stated smooth positive density assumptions.

Disclosures

AI tools assisted mathematical development, literature discovery, drafting and checking. The author is responsible for the content. This research received no external funding.

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References and related resources

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