Monotone center weights and the averaged odd-quintic theta gate
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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.
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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
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, giving degree-dependent eventual positivity. The terminating parity algorithm certifies order-dependent tails. General variance rigidity assumes an actual globally PD closed-endpoint prefix.
Disclosures
Author preprint. AI-assisted research and internal proof/code review were used. 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. Compact septic/octic certificates share an outward interval engine.
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