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
- 3 · 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. 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.
Keywords Riemann theta density · moment matrices · polynomial-square obstructions · Gram certificates · minimum degree · computer-assisted proof
Scope of the work
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.
Reproduction notes
The paper has 11 pages. The editable-source archive contains 56 files. Compile main.tex. Run standard-library supplement/reproduce_all.py with --workdir and a new directory for a complete interval/certificate replay; supplement/check_manuscript.py binds the printed numeric tables to those outputs.
The manuscript
Read the paper
The complete manuscript, as published in this edition.
If the reader is unavailable on your device, open or download the PDF directly.
References and related publications
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Minimum-Degree Obstructions in the Riemann Theta Product: Classification, transition bounds and robust Gram certificates (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/minimum-degree-obstructions-riemann-theta-product/v/3
BibTeX
@techreport{luminary_minimum_degree_obstructions_riemann_theta_product_v3,
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 3},
url = {https://research.luminaryoddities.com/minimum-degree-obstructions-riemann-theta-product/v/3}
}Loading discussion…