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
- 6 · Permanent link
Overview
Abstract
We prove averaged Riemann theta moment positivity through degree eight for nonzero nonnegative nondecreasing center weights under the respective finite-moment hypotheses. Every fixed degree has a positive terminal center region X>=1000 N^2, yet the all-order monotone cone fails. At A=23/2 we supply exact rational even and odd polynomial witnesses of degrees 4096 and 4097. Their complete normalized integrals lie strictly between -3e-7 and -2e-7, with all theta products, outward rounding, continuous sampling error and infinite tails paid. Positive, strictly increasing bounded real-analytic center weights also fail. For every sufficiently large real cutoff A, both parities admit rational negative witnesses of degree at most ceil(A^(31/4)), while any such witness has degree exceeding sqrt(A/1000). The quantitative proof combines classical Meixner-Pollaczek projection and Kaiser-Bessel concentration with reference cancellation and quadratic sign transfer. No minimal degree, numerical universal onset, physical-weight counterexample or RH theorem is claimed.
Keywords Riemann theta density · moment matrices · Schur complement · monotone weights · Fourier obstruction · imaginary-order Bessel functions
Scope of the work
Finite positive conclusions through degrees five, six, seven and eight retain their respective moment-integrability hypotheses and include every finite physical y>=0 with its continuous normalized endpoint. Theorem 29 gives exact ordinary rational witnesses at A=23/2 in both parities; Corollary 30 extends failure to suitable positive strictly increasing bounded analytic weights. Theorem 31 proves sqrt(A/1000)<N_epsilon(A)<=ceil(A^(31/4)) for both parities at every sufficiently large cutoff; its onset is not numerically evaluated. These are upper-tail/all-monotone obstructions, not failures of the physical sinh-weight family or RH. No optimal degree or worldwide-first claim is asserted.
Reproduction notes
The 33-member package contains the 47-page PDF, three editable TeX sources, eighteen retained checkers, two additional python-flint==0.9.0 witness programs, exact even/odd rational coefficient lists, four point-enclosure data files, reader instructions, citation metadata and a complete 32-entry SHA-256 manifest. All 4000 point enclosures were freshly regenerated through the public implementation and the complete consumer passed on both supplied and fresh data. Normal and optimized consumer runs agree. An independent backward-Clenshaw implementation supplies higher-precision controls. All final plaintext files and PDF pages were inspected; isolated package-source rebuilding reproduces all mathematical text. Analytic asymptotic proofs and finite outward computations are distinguished. Internal independent AI checks are not external human refereeing or formal proof verification.
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Author disclosures
Author preprint. AI-assisted research and internal independent mathematical/code review; not external human refereeing or proof-assistant verification. Classical Schur, layer-cake, Hermite/Cauchy, Meixner-Pollaczek approximation, Kaiser-Bessel concentration, strip quadrature, analytic rigidity and confluence tools are credited. The compact fixed-center classification remains a reused dependency, and retained compact certificates share an interval engine. Source discovery includes formal and informal material, but originality conclusions are bounded by arguments actually accessed and do not establish worldwide priority.
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Cite this edition
Scott Schubr. (2026). Monotone center weights and the averaged odd-quintic theta gate (Version 6). Luminary Research Archive. https://research.luminaryoddities.com/monotone-center-weights-averaged-odd-quintic-theta-gate/v/6
BibTeX
@techreport{luminary_monotone_center_weights_averaged_odd_quintic_theta_gate_v6,
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 6},
url = {https://research.luminaryoddities.com/monotone-center-weights-averaged-odd-quintic-theta-gate/v/6}
}Loading discussion…