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Moment certificates for homogeneous stable count laws

Scott Schubr

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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.

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Research details

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

Scope

Sharp scalar coefficient one for centered real vectors at every coefficient order. 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.

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 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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