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Covariance-Spectrum Certificates for Homogeneous Stable Count Laws

Scott Schubr

Luminary Oddities

  • Mathematics
  • Probability and Statistics
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
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7 · Permanent link
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Overview

Abstract

We compare a homogeneous real-stable count law with the multinomial law having the same mean. Finite chi-square bounds retain both the trace and the spectral square of the normalized covariance deficit D. The likelihood ratio has a covariance-determined quadratic component of exact squared norm nTr(D^2)/[2(n-1)], and an explicit bound controls the full higher-order remainder. This yields convergence for bounded trace and vanishing spectral square without a lower bound on positive category means. Two such stable laws with the same degree, mean and covariance agree in total variation to order (TrD)^(3/2) as the deficit tends to zero; a cubic example makes this exponent sharp. Nonfactorizable positive stable cubic laws also obstruct a stronger dimension-uniform third-order Frobenius estimate. The proof combines centered real roots with an orthogonal multinomial expansion and sharp coefficient bounds. We prove the optimal real inequality |e_k(z)|^2≤binom(n,k) for centered vectors of squared norm n, classify equality, and obtain the optimal universal total-variation coefficient. A sharp complex fifth-coefficient inequality in six variables yields the degree-six complete-homogeneous bound at every covariance rank. General-degree envelopes and structured subclasses are treated separately. A Gaussian mutual-total-variation limit along powers of a fixed law is a further consequence. We credit classical root-extremum and determinantal results and the coefficient, tensor and orthogonal expansion methods of Han and Niles-Weed. Standalone code reconstructs the finite polynomial certificates using exact arithmetic.

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

Scope of the work

Homogeneous real-stable count laws admit covariance-spectrum chi-square certificates and an exact quadratic likelihood term, sharp small-deficit covariance-matching order, and a full-stable cubic obstruction to a stronger Frobenius estimate. Sharp real coefficient/equality and complex-six bounds yield the stated uniform TV and degree-six complete-homogeneous conclusions.

Reproduction notes

The 45-page manuscript and editable main.tex are supplied in a 12-file package with references.bib, citation metadata and checksum inventories. README.md gives Tectonic/pdfLaTeX build commands and Python 3.10+ standard-library reproduction commands. verify_spectral.py uses verify_scalar_one.py, verify_sharp_two.py and verify_refined.py to reconstruct the exhaustive finite scalar/equality certificates; verify_complex_six.py reconstructs the separate exact complex-six polynomial certificates. Analytic reductions, stable-law transfer and limit arguments are proved in the manuscript.

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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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Cite this edition

Scott Schubr. (2026). Covariance-Spectrum Certificates for Homogeneous Stable Count Laws (Version 7). Luminary Research Archive. https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/7

BibTeX
@techreport{luminary_moment_certificates_homogeneous_stable_count_laws_v7,
  title = {Covariance-Spectrum Certificates for Homogeneous Stable Count Laws},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 7},
  url = {https://research.luminaryoddities.com/moment-certificates-homogeneous-stable-count-laws/v/7}
}

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