Research Article · Preprint Mathematics
Beyond logarithmic powers: critical-tail localization of theta correction inertia
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions · Spectral Theory · Complex Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
For the Riemann theta transform F(z)=xi((1+iz)/2)/4, let E(P)=(4π)⁻¹∫|(FP)′|², Q(P)=(8π)⁻¹∫(F′²−FF″)P² and D=E−Q. We obtain ordinary-polynomial subspaces, in either parity, with opposite signs of D and strictly positive original Q on their entire nonzero spans. Their dimensions are at least C exp(c√(log N))√(log N) for every sufficiently large degree N. A critical Meixner–Pollaczek receiver controls the derivative and absolute-potential norm globally. A simultaneous inverse-square window selection and a relative root-product budget improve the cost of localized targets. A finite-order compact-cutoff family with uniform low-order energies and a fully controlled growing differentiation order gives the stated scale. All local complex roots, multiplicities, exterior roots and mixed Gram entries are retained. The same growth is necessary for repair rank in a specified scalar-centered strict-contraction scheme. No negative original Q, RH conclusion, sharpness, an optimal constant or effective numerical onset is asserted.
Keywords Riemann theta form · correction inertia · critical exponential weight · Meixner-Pollaczek approximation · finite-order cutoffs
Scope of the work
For every fixed 0<θ<1 and all sufficiently large ordinary degree N, each parity has two rational-basis real polynomial spaces of dimension at least C exp(c√(log N))√(log N), with Q/E≥2−θ and separately 0<Q/E≤θ on their entire nonzero spans. The rank consequence applies only to the specified scalar-centered uniform strict-contraction architecture. No RH, original-Q negativity, optimal rate, effective onset or runtime bound.
Reproduction notes
The supplementary package contains the manuscript, complete editable LaTeX sources, build instructions, citation metadata, member checksums and a standard-library Python verifier. Normal and optimized runs each perform 1169 exact finite checks; these corroborate bounded algebra and do not replace the analytic proofs.
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Author disclosures
Research preprint. AI tools assisted with mathematical exploration, drafting, symbolic and exact-arithmetic checks, source comparison and internal proof review. Scott Schubr is the author and assumes responsibility for the manuscript. Internal AI review is not external peer review.
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Scott Schubr. (2026). Beyond logarithmic powers: critical-tail localization of theta correction inertia (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/beyond-logarithmic-powers-critical-tail-theta-correction-inertia/v/1
BibTeX
@techreport{luminary_beyond_logarithmic_powers_critical_tail_theta_correction_inertia_v1,
title = {Beyond logarithmic powers: critical-tail localization of theta correction inertia},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/beyond-logarithmic-powers-critical-tail-theta-correction-inertia/v/1}
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