Research Article · Preprint Mathematics
Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem
- 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 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. For x∈ I=[3.64,3.66], 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.
Keywords Riemann theta density · moment matrices · positive definiteness · quintic obstruction · computer-assisted proof
Scope of the work
A unique certified odd-quintic transition on [3.64,3.66], exact minimum degree five below it and full degree-five positivity classification at and above it within that interval, with complete infinite theta tails.
Reproduction notes
The paper has 6 pages. The editable-source archive contains 19 files. Compile main.tex. Standard-library supplement/reproduce_all.py regenerates the interval and rational certificates in a new work directory; supplement/locate_quintic.py supports additional fixed-center witnesses.
The manuscript
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Cite this edition
Scott Schubr. (2026). Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/locating-quintic-obstructions-riemann-theta-product/v/3
BibTeX
@techreport{luminary_locating_quintic_obstructions_riemann_theta_product_v3,
title = {Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 3},
url = {https://research.luminaryoddities.com/locating-quintic-obstructions-riemann-theta-product/v/3}
}Loading discussion…