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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Available editions
Research details
- Subjects
- Mathematics · Number Theory and Special Functions
- Keywords
- Riemann theta density, moment matrices, positive definiteness, quintic obstruction, computer-assisted proof
- Manuscript date
- 2026-10-18
- 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
This corrected October 18 edition preserves the October 17 local theorem, interval [3.64, 3.66], transition bracket and numerical claims. It clarifies the abstract and Hudson context, makes the conjugate-transpose matrix identity explicit, aligns the standalone verifier with the documented 1000-cell reproduction, records interpreter/platform/source hashes, labels historical scripts, and validates evidence structure before checking printed claims. The package contains the corrected manuscript and PDF, current source-only reproduction scripts, directed interval and exact rational evidence, revision records, and a checksum manifest. Fresh source-only reproduction and revision checks completed before staging. Internal AI review records are not external human peer review.
Publication package corrected to contain publication materials only. The manuscript and edition are unchanged.