Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem
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Abstract
We give a reproducible method for locating and certifying quintic obstructions for the normalized fixed-center Riemann theta product, and prove a unique transition in its odd quintic moment block on [3.64, 3.66]. The transition parameter lies strictly between 3.64888758188562 and 3.64888758188563. Within this interval, below the transition, degree five is the minimum degree of a polynomial whose square has negative signed-moment value; at the transition the degree-five form is positive semidefinite with a one-dimensional nullspace; above it the form is positive definite. The proof combines a Schur-complement reduction with uniform directed interval bounds, including the complete infinite theta tails. An exact rational arithmetic replay verifies the bounds on the entire interval. An explicit integer-coefficient quintic illustrates why testing one fixed polynomial does not locate the transition of the full quintic block.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions
- Keywords
- Riemann theta density, moment matrices, positive definiteness, quintic obstruction, computer-assisted proof
- Manuscript date
- 2026-10-17
- Reader reuse
- No additional permission granted
Scope
The minimum-degree and transition theorem is restricted to x ∈ [3.64, 3.66] for the normalized fixed-center product. Positive definiteness above the transition refers to the degree-five form. The precursor remains at https://luminaryoddities.com/researcharchive/explicit-quintic-obstruction-riemann-theta-kernel (edition 1).
Disclosures
AI assisted derivation, verification code, and preparation; the author takes responsibility for the manuscript.
Reproduction
The reviewed October 17 publication package includes the exact manuscript, PDF, supplemental verification code, directed interval and exact rational certificates, source-only reproduction entrypoint, review records, and checksum manifest. A fresh source-only reproduction completed successfully before staging.
Publication package corrected to contain publication materials only. The manuscript and edition are unchanged.