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Research Article · Preprint Mathematics

Local and eventual mixing-weight entropy concavity under Gaussian smoothing

Scott Schubr

Luminary Oddities

  • Mathematics
  • Probability and Statistics
  • Classical and Real Analysis
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

For independent copies of a symmetric variance-one input, we study entropy of their variance-preserving weighted sum after Gaussian smoothing with retained signal fraction q. For every Q<1/2, we prove all-weight strict entropy concavity, simultaneously for 0<q≤ Q, in an explicit infinite-dimensional Gaussian-relative L^∞ neighborhood. We also prove a common all-weight threshold for symmetric all-moment families with a uniform moment envelope and a finite nonzero Hermite discrepancy. Consequently, among symmetric variance-one log-concave inputs, any counterexample sequence with q↓0 must converge weakly and in every fixed absolute moment to the Gaussian. Each fixed symmetric non-Gaussian all-moment input has eventual all-weight strict concavity, and the whole log-concave class has strictly negative endpoint curvature for 0<q≤1/20. An exact moment-matching exponential-tail construction delineates the remaining near-Gaussian gap. No common all-weight threshold for the full symmetric log-concave class is established.

Keywords Probability and Statistics · Classical and Real Analysis · mathematical research

Scope of the work

Local Gaussian-relative and fixed-input eventual all-weight entropy concavity, uniform finite-Hermite-discrepancy families, and a common endpoint-curvature theorem for symmetric log-concave inputs.

Reproduction notes

14-page manuscript and 6 package members. Compile main.tex directly or run python build.py --tectonic tectonic using Python 3. The script fixes SOURCE_DATE_EPOCH; Tectonic uses its standard support bundle. The mathematical proof and bibliography are contained in main.tex, with no external numerical dataset.

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Author disclosures

AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Local and eventual mixing-weight entropy concavity under Gaussian smoothing (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/local-and-eventual-entropy-concavity-under-gaussian-smoothing/v/2

BibTeX
@techreport{luminary_local_and_eventual_entropy_concavity_under_gaussian_smoothing_v2,
  title = {Local and eventual mixing-weight entropy concavity under Gaussian smoothing},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/local-and-eventual-entropy-concavity-under-gaussian-smoothing/v/2}
}

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