Research Article · Preprint Mathematics
Minimum-Degree Obstructions in the Riemann Theta Product: Classification, transition bounds and robust Gram certificates
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords Riemann theta density · moment matrices · polynomial-square obstructions · Gram certificates · minimum degree · computer-assisted proof
Scope of the work
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 notes
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.
The manuscript
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Cite this edition
Scott Schubr. (2026). Minimum-Degree Obstructions in the Riemann Theta Product: Classification, transition bounds and robust Gram certificates (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/minimum-degree-obstructions-riemann-theta-product/v/2
BibTeX
@techreport{luminary_minimum_degree_obstructions_riemann_theta_product_v2,
title = {Minimum-Degree Obstructions in the Riemann Theta Product: Classification, transition bounds and robust Gram certificates},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/minimum-degree-obstructions-riemann-theta-product/v/2}
}Loading discussion…