Finite positivity and correction inertia for the Riemann theta form
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Abstract
We study the original averaged Riemann theta form and its decomposition into a positive energy and a correction. A verification of the Riemann hypothesis through height T implies strict positivity, uniformly over the averaging parameter, for polynomials of degree at most ⌊ R/24⌋ whenever 10^8≤ R≤2T-2. The published verification through 3·10^(12) therefore gives degree 200{,}000{,}000{,}000. For the endpoint correction, a simultaneous polynomial approximation theorem in a derivative-and-potential norm transfers disjoint inverse-square channels near any fixed finite collection of real zeros into common polynomial subspaces. The contributions of the centers add. Infinitely many simple critical-line zeros then imply, in each parity, negative correction inertia divided by loglog N tending to infinity. Opposite channels of dimension Ω(log N) have positive physical form but vanishing physical-to-energy ratio. Together these show that a uniform strict contraction obtained by scalar centering and an arbitrary symmetric correction requires rank ω(loglog N). All mixed entries and approximation errors are retained. These are finite-positivity and proof-route results.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- Riemann theta form, polynomial approximation, negative inertia, verified zero height, Meixner–Pollaczek polynomials
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Verified RH height T yields uniform finite-degree physical theta positivity in the stated R/24 range; endpoint correction channels give growing signed inertia and necessary correction-rank bounds with all mixed errors retained.
Disclosures
AI tools assisted mathematical development, internal independent checks, literature comparison, computation and manuscript preparation. Internal checks are not external human refereeing or formal proof verification. The author is responsible for the manuscript. No external research funding is declared.
Reproduction
The 21-page manuscript and complete editable LaTeX sources are supplied in this 13-member package. README.md gives the build and reproduction commands. Reproducibility code: verify_correction_constants.py, verify_physical_positivity.py. The accompanying manuscript and reader instructions specify the mathematical scope and required dependencies.