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Covariance-Spectrum Certificates for Homogeneous Stable Count Laws

Scott Schubr ORCID

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Abstract

We compare a homogeneous real-stable count law with the multinomial distribution having the same mean. The normalized covariance deficit controls chi-square distance and total variation, with an optimal universal linear coefficient. A spectral refinement retains the squared Frobenius norm, identifies the exact quadratic likelihood component and bounds its higher-order remainder. It yields diffuse-spectrum convergence and a sharp covariance-matching exponent; nonfactorizable stable cubics exhibit the boundary of a stronger product-class estimate. The coefficient input is an optimal centered-real elementary-symmetric inequality in every degree, with equality classification. For six complex coordinates we prove the sharp fifth-coefficient bound and classify every nonzero equality case as a common complex multiple and permutation of (2,2,-1,-1,-1,-1). A weighted-median radial reduction, real-rooted coefficient perturbations and complete rational certificates establish this rigidity statement. Ternary determinantal representation and Gaussian coefficient orthogonality give the optimal factor 3/4 in the degree-six fifth moment at every covariance rank, attained by positive stable product laws. Further complete-homogeneous bounds cover all orders with factor 24, provide sharper fifth-order comparisons and include structured factor-one regions. We also identify the Gaussian total-variation limit along powers of a fixed law. The arguments adapt classical centered-polynomial, tensor and orthogonal-expansion methods; standalone code reconstructs the finite exact certificates.

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  1. Edition 8 · 2026-10-30 · public
  2. Edition 7 · 2026-10-27 · public
  3. Edition 6 · 2026-10-24 · public
  4. Edition 5 · 2026-10-23 · public
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  6. Edition 3 · 2026-10-23 · public
  7. Edition 2 · 2026-10-23 · public
  8. Edition 1 · 2026-10-23 · public

Research details

Subjects
Mathematics · Probability and Statistics
Keywords
real-stable polynomials, covariance spectrum, multinomial approximation, total variation, elementary symmetric polynomials, exact certificates
Manuscript date
2026-10-23
Reader reuse
none

Scope

Covariance-spectrum likelihood certificates and sharp covariance-matching stability for homogeneous stable count laws, with sharp real and complex coefficient bounds, complete complex-six equality classification, full-stable cubic obstructions and fixed-law Gaussian limits.

Disclosures

OpenAI ChatGPT and Codex assisted mathematical exploration, literature comparison, drafting, exact-code development and proof checks. The author assumes responsibility for the manuscript and its mathematical claims.

Reproduction

Complete 41-page author preprint and 15-file source package. Build with pdfLaTeX/BibTeX and the supplied unmodified ALEA class/style. Five standard-library Python programs reconstruct exact finite certificates; analytic reductions and limit proofs appear in the manuscript.

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