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Covariance-Spectrum Certificates for Homogeneous Stable Count Laws

Scott Schubr

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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.

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  1. Edition 8 · 2026-10-30 · public
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  3. Edition 6 · 2026-10-24 · public
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  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
No additional permission granted

Scope

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.

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.

Reproduction

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.

References and related resources

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