First-Hit Stopping and Bias-Corrected Score Inference in Dimension Three
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Abstract
We study whether residual stopping preserves statistical inference for spatial scores computed by symmetric heat-kernel normalization. For a fixed positive C⁸ density on the three-dimensional torus, two heat scales on the same observations give a bias-corrected score whose Gaussian limit is preserved at the first checked residual crossing from constant initialization. Each run checks its residual against n⁻¹ᐟ²h⁻γ for fixed 0 < γ < 3/4. With heat time ε = n⁻κ, consistent inference with shrinking uncertainty holds for 2/17 < κ < 2/5; the normalized Gaussian law holds up to the strict upper exponent 4/9. A complete second-order empirical trajectory expansion controls numerical error at these sample-dependent indices, retaining cross-band covariance and repeated-observation terms. Finite correction order p requires only C⁴ᵖ⁺⁴ regularity; for a smooth density, a suitable fixed order covers every fixed positive power below 4/9. We also give feasible joint inference at finitely many heat-scale-separated queries, with their full covariance. A heat-spectral trace calculation determines the stopping clock through its logarithmic corrections and integer phase, and proves sharpness of the attained-residual constants one-half and one. An analytic density establishes the necessary lower bandwidth endpoint for the two-scale estimator.
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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
Actual residual first-hit stopping and bias-corrected pointwise score inference for positive densities on the three-dimensional torus. The two-scale C⁸ theorem has Gaussian range 2/17 < κ < 4/9 and shrinking uncertainty for κ < 2/5. Finite correction order p uses C⁴ᵖ⁺⁴ regularity; smooth densities admit a suitable fixed order for every fixed 0 < κ < 4/9. Same-sample covariance, finite moving-query inference and sharp clock phases use their displayed conditions.
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 36 pages. The source package contains 17 files: bibitems.tex, clustered_queries.tex, clustered_query_results.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 all files into one directory and compile main.tex with a current LaTeX installation, for example: tectonic main.tex. Keep included TeX files together. The mathematical results are analytic.