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Moment certificates for homogeneous stable count laws

Scott Schubr

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Abstract

For a homogeneous real-stable count law P and its mean-matched multinomial law M, the normalized covariance deficit D gives finite chi-square bounds retaining both x = Tr(D) and s₂ = Tr(D²). The likelihood ratio has a covariance-determined quadratic component with exact squared norm ns₂/[2(n − 1)] and a uniform higher-order remainder. A hybrid Gaussian coefficient envelope gives chi-square convergence when x remains bounded and s₂ tends to zero, without a lower bound on retained category means. Matching mean and covariance gives total-variation agreement of order x³ᐟ², with a cubic example showing the exponent is sharp. Nonfactorizable positive stable cubic laws exclude a dimension-uniform third-chaos bound by a constant times s₂³ᐟ² over the full class. The paper proves the optimal real coefficient inequality |eₖ(z)|² ≤ binomial(n,k) for centered real vectors of squared norm n in every order, classifies equality, and retains TV(P,M) ≤ min(1,x). The complete homogeneous spectral envelope holds with coefficient one at every top order, in orders two through four, and for rank-one deficits; an all-order factor-24 bound covers the remaining orders. A fixed-law Gaussian mutual-total-variation limit is also proved. Classical centered-coefficient, orthogonal-expansion, Banach and determinantal methods are explicitly credited. Standalone standard-library code reconstructs the finite certificates; analytic existence and limit arguments remain manuscript proofs.

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Research details

Subjects
Mathematics · Probability and Statistics
Keywords
real-stable polynomials, multinomial approximation, covariance spectrum, total variation, chi-square divergence, elementary symmetric polynomials, Krawtchouk polynomials
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

The real scalar coefficient-one theorem and its equality classification cover every order. The full-spectrum bound has factor 24 in general, with factor one at top orders, orders two through four and rank-one deficits.

Disclosures

Generative AI systems assisted mathematical exploration, literature discovery, proof drafting, exact-code development and independent adversarial checks. AI-assisted checks support the manuscript; the author retains responsibility for it.

Reproduction

The eleven-member package includes the complete PDF, LaTeX, bibliography, four standard-library Python files and explicit manifests. With Python 3.10 or later, keep the four Python files together and run verify_spectral.py --output certificate.json and python -O verify_spectral.py --output certificate-optimized.json. The complete exact JSON outputs must agree. The code reconstructs finite scalar/equality certificates and spectral, subset, sharpness and cubic controls; analytic tensor, hyperbolicity, Gaussian existence and limit arguments are proved in the manuscript.

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