First-Hit Stopping and Bias-Corrected Score Inference in Dimension Three
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Abstract
Kernel normalization algorithms are stopped using residuals, whereas score estimation uses derivatives of their off-sample fields. We prove that the first observable residual crossing preserves a bias-corrected score estimator's Gaussian limit for a fixed positive C⁸ density on the three-dimensional torus. Two heat scales use the same observations; their joint covariance is retained after cancellation of the leading population bias. For heat time ε = n⁻κ, consistent inference with shrinking uncertainty holds when 2/17 < κ < 2/5. At each scale h, every residual is checked from constant initialization against n⁻¹ᐟ²h⁻γ, for any fixed 0 < γ < 3/4. The stopping count is asymptotic to (3/4 − γ) log(1/h)/log 2, and the attained residual lies between any fixed fraction below one-half of the threshold and the threshold with probability tending to one. A second-order empirical trajectory expansion controls the same-sample numerical error. The normalized Gaussian law extends to κ < 4/9. An explicit analytic density establishes necessity of the lower endpoint for this two-scale rule. A heat-spectral trace expansion resolves the logarithmic correction and integer phase of the stopping count, and proves sharpness of the residual constants one-half and one. For a fixed positive C∞ density, a finite higher-order correction extends genuine first-hit inference to every fixed 0 < κ < 4/9, with shrinking uncertainty for κ < 2/5. A same-sample density estimate gives feasible Gaussian pivots and pointwise confidence balls.
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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
- none
Scope
Pointwise bias-corrected score inference for fixed positive densities on the three-dimensional torus under genuine residual first-hit stopping. The C⁸ two-scale theorem has Gaussian range 2/17 < κ < 4/9; fixed finite higher-order correction for a C∞ density covers each fixed 0 < κ < 4/9. Shrinking uncertainty holds for κ < 2/5. The results retain same-sample cross-band covariance and include feasible confidence balls, sharp residual phases and the two-scale lower-endpoint example.
Disclosures
This research received no external funding. AI tools assisted mathematical development, literature discovery, drafting and checking. The author is responsible for the content.
Reproduction
The manuscript has 35 pages. The source package contains 15 files: bibitems.tex, endpoints.tex, exact_second_order.tex, foundation.tex, higher_order.tex, higher_order_results.tex, main.tex, related_work.tex, sharp_hit.tex, stopping.tex, trajectory_expansion.tex, main.pdf, README.txt, CITATION.cff, MANIFEST.json. Extract the files into one directory and compile main.tex with a current LaTeX installation, for example: tectonic main.tex. Keep the included TeX files together. The paper is analytic and requires no empirical dataset.