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Research Article · Preprint Mathematics

Moment certificates for homogeneous stable count laws

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Probability and Statistics
Primary subject
Mathematics
Subject subfields
Probability and Statistics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
4 · Permanent link
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Overview

Abstract

For a real vector z with ∑zᵢ = 0 and ∑zᵢ² = n, this paper proves the sharp elementary-symmetric-polynomial bound |eₖ(z)|² ≤ binomial(n,k), simultaneously in every degree and coefficient order. For a normalized nonnegative homogeneous real-stable count law P with positive retained mean vector t, comparison with M = Mult(n,t/n) gives χ²(P∥M) ≤ ∑ₖ₌₂ⁿ(2W)ᵏ and TV(P,M) ≤ min(1,2W), where W = (m − 1 − ∑ᵢVar(Aᵢ)/tᵢ)/2 after removing mean-zero categories. The universal scalar coefficient one and linear total-variation coefficient two are optimal. Classical root extrema, the Han–Niles-Weed centered-coefficient and orthogonal tensor framework, a unimodal contour estimate, reciprocal recurrences and complete rational polynomial certificates cover all real coefficients. The paper also identifies the equivalent binomial-orthonormal Krawtchouk bound at an integer mean and retains the Gaussian mutual-TV limit along powers of a fixed stable law. Arbitrary category counts, nonfactorizable stable polynomials and singular covariance deficits are included.

Keywords real-stable polynomials · multinomial approximation · total variation · chi-square divergence · elementary symmetric polynomials · covariance certificates

Scope of the work

Sharp scalar coefficient one for centered real vectors at every coefficient order; no complex-vector or matrix extension. The normalized n = 1 scalar domain is empty. Stable count laws have a finite chi-square covariance certificate and optimal universal linear TV coefficient two. No sample-complexity, practical testing, sampler-construction or computational-speedup theorem is asserted.

Reproduction notes

The ten-member package contains the complete manuscript, LaTeX, structured references and three standard-library Python files. With Python 3.10 or later, keep the files together and run verify_scalar_one.py in normal and optimized modes; complete exact JSON outputs must agree. The verifier reconstructs all finite certificates and specified domains. Analytic extrema, contour, probability-transfer and local-limit arguments remain manuscript proofs.

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Author disclosures

Generative AI systems assisted mathematical exploration, literature discovery, proof drafting, exact-code development and independent adversarial checks. These are internal AI-assisted checks, not external human refereeing; the author retains responsibility for the manuscript.

References and related publications

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Cite this edition

Scott Schubr. (2026). Moment certificates for homogeneous stable count laws (Version 4). Luminary Research Archive. https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/4

BibTeX
@techreport{luminary_moment_certificates_homogeneous_stable_count_laws_v4,
  title = {Moment certificates for homogeneous stable count laws},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 4},
  url = {https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/4}
}

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