Research Article · Preprint Mathematics
Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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. The sufficient sampling regime is not claimed optimal. A reproducible solver experiment illustrates the certificate on actual iterates. These numerical-accuracy results do not assert a new statistical central limit theorem or an iteration lower bound.
Keywords entropic self-transport · matrix scaling · residual error bounds · score estimation · empirical sharpness
Scope of the work
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. The sufficient sampling condition nεᵈ/log n → ∞ is not claimed optimal. 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 and need not occur along a particular solver trajectory. No new statistical central limit theorem, minimax bound or iteration-complexity/lower-bound result is claimed. Established kernel-energy, RKHS and spectral-sensitivity methods are credited in the manuscript.
Reproduction notes
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 analytic proofs do not depend on floating-point experiments. 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.
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Scott Schubr. (2026). Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/residual-certificates-empirical-sharpness-entropic-self-transport-scores/v/1
BibTeX
@techreport{luminary_residual_certificates_empirical_sharpness_entropic_self_transport_scores_v1,
title = {Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/residual-certificates-empirical-sharpness-entropic-self-transport-scores/v/1}
}Loading discussion…