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 floor(R/24) whenever 10⁸ ≤ R ≤ 2T − 2. The published verification through 3 × 10¹² 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 log log 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(log log 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
Linear finite verified-height-to-degree positivity; additive finite-center endpoint correction inertia and an unconditional super-loglog consequence; necessary rank obstruction for scalar-centered strict contraction.
Disclosures
AI tools assisted mathematical development, internal independent checks, literature comparison, computation and manuscript preparation. The author is responsible for the manuscript. No external research funding is declared.
Reproduction
Complete editable TeX sources and portable verification programs are in the publication ZIP. Python 3.10+; python-flint 0.9.0 for two 256-bit Acb xi sign enclosures. Normal and optimized outputs agree. The Platt–Trudgian verification through height 3 × 10¹² is used as a cited input. Finite constants support the analytic proofs.