Monotone center weights and the averaged odd-quintic theta gate
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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.
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- Paper.pdf310.2 KiB · pdf
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514a40336d0f953a7a8e23586df7f421124ff320e7ad3749be791933429bbb94 - Publication_Package.zip831.2 KiB · package
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Research details
- Subjects
- Mathematics · Number Theory
- Keywords
- Riemann theta density, moment matrices, Schur complement, monotone weights, Fourier obstruction, imaginary-order Bessel functions
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
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. These are upper-tail/all-monotone obstructions.
Disclosures
Author preprint. AI-assisted research and internal independent mathematical/code review were used. 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, and the inspected arguments are credited.
Reproduction
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.