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Minimum-Degree Obstructions in the Riemann Theta Product: Classification, transition bounds and robust Gram certificates

Scott Schubr

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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. We isolate every transition and determine the degree at equality. Jacobi symmetry extends the complete classification to π^2/12≤ x≤12. We quantify how the two- and three-summand theta models shift all fourteen crossings: the respective root errors are below 0.00049 and 2.588 × 10^(-10). A constructive finite-difference bound converts polynomial obstructions into finite translation Gram witnesses. Twenty explicit rational certificates include a seven-point example stable under absolute entrywise errors of 1.94 × 10^(-7). We also prove fixed-degree Gaussian limits and single-summand degree-five positivity. Infinite tails, directed interval covers and independent rational checks are included in a self-contained reproduction supplement.

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Edition files

Available editions

  1. Edition 3 · 2026-10-27 · public
  2. Edition 2 · 2026-10-19 · public
  3. Edition 1 · 2026-10-19 · public

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

Complete minimum-degree obstruction classification with fourteen transitions on 3 ≤ x ≤ 12 and its Jacobi-symmetric extension, certified finite-summand root shifts and explicit robust finite translation-Gram witnesses for the fixed-center theta product.

Disclosures

AI assisted mathematical derivation, verification code and preparation; the author takes responsibility for the manuscript. This research received no external funding.

Reproduction

The 11-page paper contains analytic arguments and theorems supported by explicit computational certificates. The 56-file package supplies complete TeX sources, the identical PDF, standard-library verification programs, literal rational inputs and a SHA-256 inventory. Compile main.tex. Run python -B -O supplement/reproduce_all.py --workdir /absolute/new/work, then supplement/check_manuscript.py against the generated evidence. The work directory must be new. Full interval covers may take tens of minutes.

References and related resources

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