Proof / theorem Mathematics
An Explicit Quintic Obstruction for the Riemann Theta Kernel
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Overview
Abstract
We exhibit an explicit polynomial square on which the signed derivative functional of a fixed-center Riemann theta product is negative. At the center s₀ = ¼ log(13/(4π)), the polynomial P(X) = X⁵ − 264X³ + 10952X satisfies ℒₛ₀(P²) ≤ −1,351,625,671. The certificate uses rational interval arithmetic for derivatives through order ten and an explicit bound on the complete infinite theta tail. It follows that the normalized product at this center is not positive definite and that its odd derivative Hankel block is indefinite.
Keywords Riemann theta kernel · Positive definite functions · Hankel matrices · Rational interval arithmetic
Scope of the work
This result concerns the normalized theta product at the specified fixed center. It does not determine positive definiteness of kernels obtained by averaging over the center.
Reproduction notes
The complete publication package includes the standalone LaTeX manuscript, two portable verification programs, retained exact rational certificates, integer moment bounds, reproduction instructions, and a SHA-256 manifest. The supplied README records that the PDF was rendered and every page visually inspected; LaTeX compilation was not completed during assembly. The certificates were retained rather than regenerated, and the verification programs were syntax checked and structurally compared but were not newly executed during that preparation. See the package README and supplement/README.txt for the exact reproduction commands.
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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.
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Scott Schubr. (2026). An Explicit Quintic Obstruction for the Riemann Theta Kernel (Version 1). Luminary Research Archive. https://luminaryoddities.com/researcharchive/explicit-quintic-obstruction-riemann-theta-kernel/v/1
BibTeX
@techreport{luminary_explicit_quintic_obstruction_riemann_theta_kernel_v1,
title = {An Explicit Quintic Obstruction for the Riemann Theta Kernel},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Proof / theorem},
note = {Version 1},
url = {https://luminaryoddities.com/researcharchive/explicit-quintic-obstruction-riemann-theta-kernel/v/1}
}Reader correspondence
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