Proof / theorem Mathematics
An Explicit Quintic Obstruction for the Riemann Theta Kernel: An Independent Rational Reconstruction
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Overview
Abstract
We give a complete rational reconstruction of an explicit obstruction to positive definiteness for a fixed-center product of the Riemann theta density. At s₀ = ¼ log(13/(4π)), the signed derivative functional evaluated on the square of P(X) = X⁵ − 264X³ + 10952X lies in the exact rational interval [−1412370791.837791749, −1412370791.837791748]. The proof constructs derivatives by formal power series, retains twelve theta summands, bounds the entire remaining series, and encloses exponentials by range-reduced alternating series. A printed table of rational moment intervals proves the negative contraction. The odd derivative Hankel block is indefinite, and the six-point translation Gram has a negative direction at every sufficiently small positive spacing.
Keywords Riemann theta density · positive definite functions · Hankel matrices · rational interval arithmetic
Scope of the work
This companion independently reconstructs the fixed-center obstruction in the original edition: https://luminaryoddities.com/researcharchive/explicit-quintic-obstruction-riemann-theta-kernel . It does not determine positive definiteness after averaging over the center, identify the smallest possible polynomial degree of failure, give an explicit numerical spacing for the six-point consequence, or resolve the Riemann Hypothesis. The original publication and its edition 1 remain unchanged.
Reproduction notes
The complete publication package contains main.tex and its required proof.tex, the PDF, a Python-standard-library verifier, certificate.json, moment_table.json, and checksum manifest. A fresh local run reconstructed the intervals and matched the retained certificate. Download the complete package to reproduce all dependencies.
The manuscript
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Author disclosures
Generative AI tools, including OpenAI's Codex, assisted with mathematical exploration and review, preparation of verification code, literature searches, and manuscript drafting and formatting. The author takes responsibility for the manuscript and supplementary materials, including all mathematical claims, computations, and references.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). An Explicit Quintic Obstruction for the Riemann Theta Kernel: An Independent Rational Reconstruction (Version 1). Luminary Research Archive. https://luminaryoddities.com/researcharchive/independent-rational-reconstruction-quintic-theta-obstruction/v/1
BibTeX
@techreport{luminary_independent_rational_reconstruction_quintic_theta_obstruction_v1,
title = {An Explicit Quintic Obstruction for the Riemann Theta Kernel: An Independent Rational Reconstruction},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Proof / theorem},
note = {Version 1},
url = {https://luminaryoddities.com/researcharchive/independent-rational-reconstruction-quintic-theta-obstruction/v/1}
}Reader correspondence
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