Monotone center weights and the averaged odd-quintic theta gate
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Abstract
We prove positivity of the odd moment block on {z,z^3,z^5} for the normalized Riemann theta product averaged against every nonzero nonnegative nondecreasing center weight with a finite fifth exponential moment. This includes the physical weights sinh(2ys)/(2ys) for every finite y>=0. The proof combines the previously published compact fixed-center classification with a complete-theta positive matrix tail, a separately certified positive mean of the local Schur innovation, and a classical single-crossing argument. The matrix-weighted regression variance is strictly positive and has an independent two-region lower bound on 0<=y<=1. A separate even-block estimate gives full degree-five moment positivity and local six-point positivity for the physical kernels. 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
Full moment matrix through degree five is positive definite for finite-valued measurable nonnegative nondecreasing W which is not zero almost everywhere and satisfies integral W(1+X^5)s^2 Phi(s)^2 ds < infinity. Includes every finite physical y>=0 with normalized derivative limit at y=0. Physical fixed shapes of at most six distinct points have local positivity at sufficiently small scale depending on shape and y.
Disclosures
Author preprint. AI-assisted mathematical research and internal proof/code review were used. Classical Schur, parallel-sum and single-crossing mechanisms are credited. The published compact fixed-center classification is a cited dependency, reused with its stated normalization.
Reproduction
Publication package contains the 13-page PDF, editable TeX, five standalone Python standard-library checkers, reader/build instructions, citation metadata and SHA-256 manifest. Run the five checkers as directed in README, including verify_separated.py 1024, normally and with python -O. All infinite-tail and finite-to-global arguments are in the paper.