Research Article · Preprint Mathematics
Finite-order positivity and growing negative inertia for averaged Riemann theta products
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory · Number Theory and Special Functions · Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We study the contrast between finite-order positivity and all-order failure for averaged Riemann theta products. Our main new theorem proves that, at the exact upper-center cutoff A=23/2, each polynomial parity through degree N has negative inertia at least 3 log(N)/(8 pi)-O(log log N). A fully explicit conservative bound constructs common negative subspaces, with every theta term and approximation error retained. Consequently a correction making the full form positive semidefinite must have rank at least 3 log(N)/(4 pi)-O(log log N), even if the correction changes with N or couples parities. The proof combines classical sinc concentration with gamma-reference cancellation and uniform Meixner-Pollaczek projection. We consolidate, with explicit attribution, previously established positivity through degree eight, degree-dependent terminal positivity, exact rational negative witnesses of degrees 4096 and 4097, and eventual witness degree at most ceil(A^(31/4)) in both parities. A classical lognormal counterexample explains why finite moments and a negative density region alone are insufficient. The physical all-order problem remains open; no zero theorem or optimal inertia asymptotic is claimed.
Keywords Riemann theta density · negative inertia · moment matrices · Meixner-Pollaczek polynomials · time-band concentration · finite-rank repair
Scope of the work
Common negative subspaces and rank obstructions concern the complete upper-tail form and the broader monotone-weight cone. They do not refute the physical sinh-weight family. Constants are conservative; no practical first-failure degree, uniform joint cutoff/degree limit, optimal spectral law or worldwide-priority claim is made. The earlier edition 6 remains a separate attributed predecessor.
Reproduction notes
The 44-member package contains the 60-page PDF, eleven editable TeX sources, twenty-two Python programs, seven exact-data JSON files, reader instructions, citation metadata and a complete 43-entry checksum manifest. The new eight-node band inputs were freshly regenerated from the complete Mellin evaluator, and normal and optimized runs agree for the band and inertia constant checks. Twenty inherited programs and six inherited data files retain their verified published byte identities. The isolated package-source rebuild reproduces all manuscript page text. Complete analytic proofs, outward finite computations, inherited evidence and cited classification inputs are distinguished. The effective inertia count is conservative and can remain vacuous at very large finite degrees.
The manuscript
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Author disclosures
Author preprint. AI-assisted research and internal independent mathematical/code review; not external human refereeing or formal proof-assistant verification.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Finite-order positivity and growing negative inertia for averaged Riemann theta products (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/finite-order-positivity-growing-negative-inertia-averaged-riemann-theta-products/v/1
BibTeX
@techreport{luminary_finite_order_positivity_growing_negative_inertia_averaged_riemann_theta_products_v1,
title = {Finite-order positivity and growing negative inertia for averaged Riemann theta products},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/finite-order-positivity-growing-negative-inertia-averaged-riemann-theta-products/v/1}
}Loading discussion…