Monotone center weights and the averaged odd-quintic theta gate
Loading original first page…
Abstract
We prove averaged Riemann theta moment positivity through degree eight for nonzero nonnegative nondecreasing center weights under the respective fifth- through eighth-moment hypotheses. At these finite degrees the complete upper-tail matrices are positive at every finite cutoff. For each finite degree N>=1, a normalized moving Hermite Gram has a 99/100 reserve whenever X>=1000 N^2, and an exact parity algorithm has a computable termination bound. We then prove the stronger all-order monotone cone is false: every sufficiently large real cutoff A has a negative complete-theta Fourier value at the explicit frequency 2(2 floor(5A/(2d))+1)d, d=pi/log 2. Hence some finite rational polynomial has a negative raw moment square at each such cutoff, with degree exceeding sqrt(A/1000). All atom products and asymptotic remainders are retained. At A=23/2, rigorous outward quadrature and complete infinite-tail bounds certify negativity at frequency 154 and on a closed neighboring interval.
Read the full paper in the original PDF.
Edition files
- Paper.pdf261.6 KiB · pdf
SHA-256
8affa3a7ebf3ecadfe3451398d8d54fe186c46914f677261057647b05fbf626e - Publication_Package.zip345.4 KiB · package
SHA-256
e5a4e18fce5400ad97216f9b127bd47154e09933a737580badb53ab5af886007
Available editions
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-degree positive results retain their respective integrability hypotheses integral W(1+X^d)s^2 Phi(s)^2 ds<infinity for d=5,6,7,8. All finite physical y>=0 are included at those degrees, with the normalized endpoint. The general local region X>=1000 N^2 is degree-dependent. Theorem 26 proves negative total upper-tail Fourier values at every sufficiently large cutoff, and finite rational negative-polynomial existence. Proposition 27 certifies -8e-49 < hatJ_(23/2)(154) < -3e-49 and a closed negative interval. These step weights refute the all-order/all-monotone cone.
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.
Reproduction
The explicit 24-member package contains the 39-page PDF, main.tex and cutoff_obstruction.tex, sixteen retained standard-library arithmetic checkers, two new python-flint==0.9.0 Fourier checkers, README, citation metadata and a complete 23-entry SHA-256 manifest. The new point/interval and independent original-H implementations pass fresh normal and optimized runs with identical outputs. They share Arb arithmetic and proved contour-tail envelopes; the second uses a different integrand. Sixteen retained checker byte identities preserve earlier verified evidence. All new analytic proofs and the merged manuscript received internal independent AI review. All intended plaintext contents and rendered PDF pages were inspected; isolated extracted-source rebuilding reproduced all mathematical text. Finite arithmetic, analytic asymptotics and existential polynomial consequences are distinguished. No ordinary polynomial coefficient list is supplied.