Minimum-Degree Obstructions in the Riemann Theta Product: Classification, transition bounds and robust Gram certificates
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Abstract
We classify the least degree of a polynomial whose square has negative signed-moment value for the normalized fixed-center Riemann theta product. On 3 ≤ x ≤ 12 the degree has fourteen transitions, takes values between five and fifteen, and is nonmonotone; every transition and equality case is determined. Jacobi symmetry extends the classification to π²/12 ≤ x ≤ 12. Two- and three-summand theta models approximate all fourteen crossings with respective root errors below 0.00049 and 2.588 × 10⁻¹⁰. Twenty explicit rational Gram certificates include a seven-point example stable under absolute entrywise errors of 1.94 × 10⁻⁷. The paper also establishes fixed-degree Gaussian limits and single-summand degree-five positivity.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions
- Keywords
- Riemann theta density, moment matrices, polynomial-square obstructions, Gram certificates, minimum degree, computer-assisted proof
- Manuscript date
- 2026-10-18
- Reader reuse
- No additional permission granted
Scope
Expanded edition of the existing classification, extending [3,8] to [3,12] with three additional transitions and maximum degree fifteen. Jacobi reflection gives the interval [π²/12,12]. Results concern the normalized fixed-center theta product and the stated finite Gram certificates.
Reproduction
The complete package includes the eleven-page paper, LaTeX source, a self-contained reproduction supplement, 385 directed-cover cells, fourteen transition windows and equality classifications, twenty rational Gram certificates, twenty-eight finite/full root comparisons, verification controls and checksum manifest. The corrected package records a successful fresh thirty-stage source-only replay and manuscript-binding checks. README.txt gives the reproduction instructions.
Publication package corrected to contain publication materials only. The manuscript and edition are unchanged.