Monotone center weights and the averaged odd-quintic theta gate
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Abstract
We prove positivity of the full moment matrix through degree five for the normalized Riemann theta product averaged against every nonzero nonnegative nondecreasing center weight with a finite fifth exponential moment, and through degree six under the corresponding sixth-moment hypothesis. This includes the physical weights sinh(2ys)/(2ys) for every finite y>=0. The proof combines the previously published compact fixed-center classification with complete-theta positive matrix tails, separately certified positive means of the odd-quintic and even-sextic local Schur innovations, and a classical single-crossing argument. Both matrix-weighted regression variances are strictly positive; the quintic variance also has an independent two-region lower bound on 0<=y<=1. For physical kernels the full degree-six result implies local positivity on fixed configurations of at most seven points. An exact initial even-sextic obstruction pays a sign interval not settled by the minimum-degree-five classification. A smooth, positive, even, uniformly log-concave density with a negative averaged quintic square shows why generic log-concavity cannot replace the theta-specific argument. All new finite arithmetic is reproducible by outward rational enclosures.
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Research details
- Subjects
- Mathematics · Number Theory
- Keywords
- Riemann theta density, moment matrices, Schur complement, monotone weights, positive definiteness, log-concavity
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
For finite-valued measurable nonnegative nondecreasing W which is not zero almost everywhere, the full theta moment matrix through degree five is positive definite under integral W(1+X^5)s^2 Phi(s)^2 ds < infinity. Under the stronger sixth-X moment hypothesis the full degree-six matrix is positive definite. Both cover every finite physical y>=0 with the normalized y=0 endpoint. Physical kernels have local positivity on fixed configurations of at most seven distinct points at sufficiently small nonzero scale depending on shape and y.
Disclosures
Author preprint. AI-assisted mathematical research and internal proof/code review were used. Classical Schur, parallel-sum, single-crossing, Hermite regression and Vandermonde mechanisms are credited. The published compact fixed-center classification is a cited dependency, reused with its stated normalization.
Reproduction
The 13-member publication package contains the 18-page manuscript PDF, self-contained TeX, eight standalone standard-library Python checkers, README, citation metadata and SHA-256 manifest. Run all README commands normally and with python -O, including verify_separated.py 1024, verify_sextic_initial.py 256 and verify_sextic_mean.py 1024. Analytic whole-halfline and finite-to-global arguments are in the paper. The complete historical compact classification computation is a cited published dependency.