Degree-eight positivity and polynomial obstructions for monotone Riemann theta averages
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Abstract
Monotone center averaging makes the complete Riemann theta product moment matrix positive definite through degree eight, under the corresponding moment-integrability hypothesis. The lower-degree conclusions retain their weaker hypotheses, and every finite physical parameter is included. The proof separates positive mean Schur residuals from variance reserves, with complete-theta tail and finite-interval certificates. This finite positivity has a quantitative extension but an all-order boundary. At every fixed degree N, the local moment Gram has a 99/100 reserve in normalized Hermite coordinates for X≥1000N^2. In contrast, every sufficiently large center cutoff admits negative rational polynomial squares in both parities, of degree at most ⌈ A^(31/4)⌉. At A=23/2, explicit witnesses have degrees 4096 and 4097, with certified complete-theta moment signs. Positive, strictly increasing bounded real-analytic center weights also yield negative polynomial squares. The local physical consequences include fixed configurations of at most nine points. Classical Schur, approximation and quadrature tools are identified as inputs; the complete-theta estimates, witnesses and quantitative domains are the results studied here.
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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
Full averaged theta moment positivity through degree eight under the corresponding monotone-weight moment hypotheses, explicit degree-dependent local positivity, and constructive both-parity obstructions to all-order monotone-weight positivity.
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.
Reproduction
47-page manuscript and 33 flat package members. Compile main.tex with Tectonic or compatible LaTeX. README.txt gives the standard-library exact polynomial and interval checks. Install python-flint==0.9.0 for the complete Fourier and rational-witness checks; python verify_theta_polynomial.py consumes the supplied exact coefficients and 4000 outward point enclosures. Its --recompute-grid START STOP option regenerates the four 1000-point ranges for independent full replay, and --grid-dir selects the regenerated data. Analytic complete-tail and continuous-integral reserves are retained.