Research Article · Preprint Mathematics
Monotone center weights and the averaged odd-quintic theta gate
- Primary subject
- Mathematics
- Subject subfields
- Number Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
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Overview
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. No all-order or Fourier positivity is asserted.
Keywords Riemann theta density · moment matrices · Schur complement · monotone weights · positive definiteness · log-concavity
Scope of the work
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. No arbitrary-spacing six-point, all-order, Fourier positivity or RH result.
Reproduction notes
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.
The manuscript
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Author disclosures
Author preprint. AI-assisted mathematical research and internal proof/code review; not external human refereeing or formal proof-assistant verification. Classical Schur, parallel-sum and single-crossing mechanisms are credited. The published compact fixed-center classification is reused with its stated normalization; its full historical computation is not rerun by the new checkers.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Monotone center weights and the averaged odd-quintic theta gate (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/monotone-center-weights-averaged-odd-quintic-theta-gate/v/1
BibTeX
@techreport{luminary_monotone_center_weights_averaged_odd_quintic_theta_gate_v1,
title = {Monotone center weights and the averaged odd-quintic theta gate},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/monotone-center-weights-averaged-odd-quintic-theta-gate/v/1}
}Loading discussion…