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
- 2 · 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. Within this interval, 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
The minimum-degree and transition theorem is restricted to x ∈ [3.64, 3.66] for the normalized fixed-center product. Positive definiteness above the transition refers to the degree-five form, not the entire kernel. No conclusion about center-averaged kernels, global minimum degree, or the Riemann Hypothesis is drawn. The precursor remains at https://luminaryoddities.com/researcharchive/explicit-quintic-obstruction-riemann-theta-kernel (edition 1).
Reproduction notes
This corrected October 18 edition preserves the October 17 local theorem, interval [3.64, 3.66], transition bracket and numerical claims. It clarifies the abstract and Hudson context, makes the conjugate-transpose matrix identity explicit, aligns the standalone verifier with the documented 1000-cell reproduction, records interpreter/platform/source hashes, labels historical scripts, and validates evidence structure before checking printed claims. The package contains the corrected manuscript and PDF, current source-only reproduction scripts, directed interval and exact rational evidence, revision records, and a checksum manifest. Fresh source-only reproduction and revision checks completed before staging.
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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 2). Luminary Research Archive. https://research.luminaryoddities.com/locating-quintic-obstructions-riemann-theta-product/v/2
BibTeX
@techreport{luminary_locating_quintic_obstructions_riemann_theta_product_v2,
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 2},
url = {https://research.luminaryoddities.com/locating-quintic-obstructions-riemann-theta-product/v/2}
}Loading discussion…