Monotone center weights and the averaged odd-quintic theta gate
- Primary subject
- Mathematics
- Subject subfields
- Number Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
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.