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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
6 · Permanent link
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Overview

Abstract

This paper compares a homogeneous real-stable count law with its mean-matched multinomial law through the normalized covariance deficit D. Finite chi-square bounds retain Tr(D) and Tr(D²), with an exact covariance-determined quadratic likelihood component and a uniform remainder. Matching mean and covariance gives a sharp total-variation rate of order Tr(D)^{3/2}. The optimal centered real elementary-symmetric coefficient inequality is proved in every order with its equality classification. A sharp centered complex degree-five coefficient bound in six variables yields the optimal factor 3/4 for the fifth likelihood component of every degree-six stable count law, at arbitrary covariance rank. Together with the other orders, this establishes the complete-homogeneous factor-one target throughout degree six. General degrees retain a factor-24 envelope, with stronger order-five bounds and structured spectral regions. Higher-interior factor one in degrees at least seven remains open. A fixed-law Gaussian mutual-total-variation limit is retained. Classical root-extremum, determinantal, centered-coefficient and orthogonal-expansion methods are credited; standalone code reconstructs finite exact certificates.

Keywords real-stable polynomials · covariance spectrum · multinomial approximation · total variation · elementary symmetric polynomials · exact certificates

Scope of the work

The all-order real scalar theorem and sharp complex-six theorem have the stated centered norm normalizations. The degree-six factor-3/4 fifth-order bound holds at every covariance rank and is attained by positive stable product laws. It does not assert an all-degree complex coefficient-one theorem or unrestricted higher-interior factor one in degrees at least seven. Analytic transfer and limit arguments are distinct from finite polynomial certificates.

Reproduction notes

The twelve-member package includes the 45-page manuscript, editable source, bibliography, five standard-library Python files, citation metadata and manifests. With Python 3.10 or later, keep the four legacy verifier files together and run verify_spectral.py --output certificate.json and verify_complex_six.py --output complex-six-certificate.json. Repeat each in optimized mode and compare each normal/optimized exact output pair. Follow README.md for full instructions. Build main.tex with Tectonic or two pdflatex runs. Finite certificates do not replace the analytic reductions, stable-law transfer or limit 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.

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A stable scholarly record

Cite this edition

Scott Schubr. (2026). Moment certificates for homogeneous stable count laws (Version 6). Luminary Research Archive. https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/6

BibTeX
@techreport{luminary_moment_certificates_homogeneous_stable_count_laws_v6,
  title = {Moment certificates for homogeneous stable count laws},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 6},
  url = {https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/6}
}

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