Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores
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Abstract
We study how numerical errors in symmetric kernel scaling affect the spatial score obtained from its smooth kernel extension. A full square-root-mass residual gives a sharp kernel-energy bound for every positive candidate scaling, including singular positive semidefinite Gram matrices. For Gaussian kernels this yields computable, scale-invariant pointwise score certificates. For iid samples from a strictly positive C³ density on the d-dimensional torus, we prove a simultaneous nonlinear score bound and determine the exact local modulus in a shrinking-bandwidth regime, with leading constant C_d/√f(x). Under the stated normalization and sampling assumptions, residual o(1/√n) suffices for negligible normalized score error; positive sample-dependent candidates with residual asymptotic to c/√n attain nonzero normalized error. A reproducible solver experiment illustrates the certificate on actual iterates.
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Research details
- Subjects
- Mathematics · Probability and Statistics
- Keywords
- entropic self-transport, matrix scaling, residual error bounds, score estimation, empirical sharpness
- Manuscript date
- 2026-10-18
- Reader reuse
- No additional permission granted
Scope
The contribution is a specific computable projective residual-to-pointwise-score certificate, uniform empirical nonlinear comparison and positive worst-case candidate witnesses in the stated periodic Gaussian model. For generic nonzero O(ε) true-score bias, the usual bias-negligible undersmoothing regime does not overlap the stated sufficient sampling regime in dimensions d ≥ 6; finite certificates, regularized centering and explicitly conditional results remain valid. Sharpness concerns arbitrary positive candidates. Established kernel-energy, RKHS and spectral-sensitivity methods are credited in the manuscript.
Disclosures
AI tools assisted mathematical development, computation and editing. The author is responsible for the final manuscript. Independent AI reviews are not external human peer review or proof-assistant verification. Company affiliation is retained from the manuscript.
Reproduction
Corrected October 18, 2026 specialist preprint. The sealed complete package includes the manuscript, PDF, solver parameters, raw historical iteration records, fresh solver and independent diagnostics, revision/preservation records and checksum manifest. The source-only ZIP supplies the documented reproduction dependencies. The results have analytic proofs. The publication intake verified exact sealed file hashes; it does not relabel prior checks as a new numerical replay. Original release files and archives remain preserved.
Publication package corrected to contain publication materials only. The manuscript and edition are unchanged.