Monotone center weights and the averaged odd-quintic theta gate
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Abstract
We prove positivity of the full normalized Riemann theta product moment matrix through degrees five, six and seven under the corresponding fifth-, sixth- and seventh-moment hypotheses on every nonzero nonnegative nondecreasing center weight. All finite physical parameters are included. The new degree-seven theorem follows from positive upper-tail odd-four matrices at every real cutoff, using a complete-theta unbounded tail and an exact compact matrix certificate. A second certificate proves positive means of the full two-column local Schur residual. We also prove strict regression variance for any fixed finite higher odd output list against the positive lower prefix, even for nonmonotone density weights. Polynomial weights yield positive degree-seven derivative jets for every associated theta-kernel index. Physical kernels have local positivity on fixed configurations of at most eight points. Earlier separated quintic and sextic estimates and a smooth log-concave counterexample are retained. The generic tools are classical; the quantitative signs are theta-specific.
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Research details
- Subjects
- Mathematics · Number Theory
- Keywords
- Riemann theta density, moment matrices, Schur complement, monotone weights, positive definiteness, log-concavity
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
For nonnegative nondecreasing measurable finite-valued W, not zero almost everywhere, the full theta moment matrices through degrees five, six and seven are positive definite under their respective integral W(1+X^d)s^2 Phi(s)^2 ds < infinity hypotheses, d=5,6,7. All finite physical y>=0 are included with the normalized y=0 endpoint. The odd-four upper-tail matrix is positive definite at every finite cutoff; its two-column local residual has positive upper-tail means. A separate strict-variance theorem covers any fixed finite higher odd output list for nonnegative density weights with the required highest moment. Every associated kernel index n>=1 has a positive degree-seven derivative jet at the origin. Physical local positivity holds for fixed shapes of at most eight points at sufficiently small scale.
Disclosures
Author preprint. AI-assisted mathematical research and internal proof/code review were used. Classical Schur, parallel-sum, layer-cake, Hermite regression, analytic rigidity and Vandermonde mechanisms are credited. The published compact classification is a cited dependency, reused with its stated normalization. The two compact septic certificates share one outward interval arithmetic engine.
Reproduction
The 16-member publication package contains the 24-page manuscript PDF, self-contained TeX, eleven standalone standard-library Python checkers, README, citation metadata and complete SHA-256 manifest. Run all README commands normally and with python -O, including verify_separated.py 1024, verify_sextic_initial.py 256, verify_sextic_mean.py 1024, verify_septic_tail.py, verify_septic_compact.py 1024 and verify_septic_residual.py 1024. Whole-halfline estimates, signed arbitrary-cutoff enclosures and finite-to-global proofs are in the paper. The complete historical compact classification is a cited dependency.