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
- 4 · Permanent link
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Overview
Abstract
We prove positive definiteness of the averaged Riemann theta product moment matrix through degree eight for every nonzero nonnegative nondecreasing center weight under its eighth-moment hypothesis, retaining the earlier fifth-, sixth- and seventh-moment results under their respective hypotheses. Complete-theta upper-tail matrices are positive at every finite cutoff, with separate positive two-column Schur-residual mean certificates. All finite physical parameters are included, and fixed configurations of at most nine points have local positivity. For every finite degree N>=1, the local complete-theta Gram matrix has a 99/100 reserve in normalized moving Hermite coordinates whenever X>=1000 N^2. A complementary exact parity-block algorithm has a computable termination bound. General positive-prefix regression rigidity proves variance strictness without paying a negative residual mean. Classical mechanisms are credited; the complete-theta signs and quantitative regions are the contribution. The all-order initial-center gap remains open; no global Fourier positivity or RH conclusion is asserted.
Keywords Riemann theta density · moment matrices · Schur complement · monotone weights · positive definiteness · log-concavity
Scope of the work
Degrees five through eight retain their respective integrability hypotheses integral W(1+X^d)s^2 Phi(s)^2 ds<infinity, for finite-valued nonnegative nondecreasing W not zero almost everywhere. Every finite physical y>=0 is included with its normalized endpoint. The even-five upper-tail matrix and full two-column residual mean are PD at every finite cutoff. Associated indices n>=1 have degree-eight derivative jets; physical local fixed-shape positivity extends to nine points at sufficiently small scale. For each finite N>=1 the normalized moving Hermite Gram is at least 99/100 I on X>=1000 N^2; this is degree-dependent eventual positivity, not all orders at one center. The terminating parity algorithm certifies order-dependent tails. General variance rigidity assumes an actual globally PD closed-endpoint prefix. No arbitrary-spacing, all-order initial-center, global Fourier or RH theorem is claimed.
Reproduction notes
The 21-member package contains the 33-page PDF, self-contained TeX, sixteen standalone standard-library Python checkers, README, citation metadata and a complete 20-entry SHA-256 manifest. Eleven retained checker byte identities preserve the earlier verified evidence. Fourteen fresh public normal/optimized runs cover the five new octic, Hermite and effective-tail checkers, including compact/residual 1024 cells and finite-order N=0,8,10. Independent cold whole-manuscript and full member content reviews passed. An isolated extracted-source rebuild reproduced all mathematical text with clean diagnostics; all 33 rendered pages were inspected. Finite exact checks, analytic all-order proofs, reused classification and shared arithmetic dependencies are distinguished. The all-order initial-center problem is left open.
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Author disclosures
Author preprint. AI-assisted research and internal proof/code review; not external human refereeing or proof-assistant verification. Classical Schur, layer-cake, parallel-sum, Hermite/Cauchy, Gaussian regression, analytic rigidity, coefficient-shift and confluence mechanisms are credited. The compact classification is a cited reused dependency, not a new historical full-computation replay. Compact septic/octic certificates share an outward interval engine. Originality assessment is bounded by sources actually accessed and is not a worldwide-first claim.
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Scott Schubr. (2026). Monotone center weights and the averaged odd-quintic theta gate (Version 4). Luminary Research Archive. https://research.luminaryoddities.com/monotone-center-weights-averaged-odd-quintic-theta-gate/v/4
BibTeX
@techreport{luminary_monotone_center_weights_averaged_odd_quintic_theta_gate_v4,
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 4},
url = {https://research.luminaryoddities.com/monotone-center-weights-averaged-odd-quintic-theta-gate/v/4}
}Loading discussion…