Research Article · Preprint Mathematics
Weighted polynomial localization and theta correction inertia
- Primary subject
- Mathematics
- Subject subfields
- Number Theory · Spectral Theory · Complex Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
Let F(z)=ξ((1+iz)/2)/4, G=F'^2-FF'', and let D_0=E_0-Q_0 be the original endpoint correction, where E_0(P)=(4π)^(-1)∫ |(FP)'|^2 and Q_0(P)=(8π)^(-1)∫ GP^2. For every fixed 0<θ<1 and A>0, every sufficiently large degree N, and each polynomial parity, we construct a real subspace with a rational basis, dimension at least c_(A,θ)(log N)^A(log log N)^2, and Q_0≥(2-θ)E_0 on its entire span. The proof combines classical unconditional short-interval zero moments with bounded-population microscopic seed selection, a two-scale root-product budget, seed-averaged logarithmic root energy and higher fixed-order weighted approximation in a derivative and absolute-potential norm. All local complex zeros, multiplicities, exterior roots and mixed polynomial Gram entries are retained. Companion spaces have 0<Q_0/E_0≤θ at the same dimension scale, yielding the corresponding necessary repair-rank bounds for scalar-centered fixed strict contractions. A finite Gaussian zero-ideal approximation theorem and an exact generic current certificate delineate the scope of the method. The constructed correction-negative spaces have positive original Q_0.
Keywords Riemann theta form · correction inertia · weighted polynomial approximation · logarithmic root energy · rank obstruction
Scope of the work
Rational polynomial spaces in each parity with correction-negative and original-square-positive full spans of dimension at least a fixed log-degree power times squared log-log degree, with corresponding scoped repair-rank consequences.
Reproduction notes
25-page manuscript and 15 source-package members. Use build.py and Tectonic to compile main.tex and its included sections. Standard-library support_verify_gaussian.py checks the finite Gaussian controls. Install support_requirements.txt for support_verify_current.py and support_verify_current_independent.py, which independently check the generic interval-current certificate.
The manuscript
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Author disclosures
AI assistants were used in mathematical development, checking, source comparison, code and manuscript preparation. This preprint does not claim external human refereeing or formal proof-assistant verification.
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Cite this edition
Scott Schubr. (2026). Weighted polynomial localization and theta correction inertia (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/weighted-polynomial-localization-theta-correction-inertia/v/2
BibTeX
@techreport{luminary_weighted_polynomial_localization_theta_correction_inertia_v2,
title = {Weighted polynomial localization and theta correction inertia},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/weighted-polynomial-localization-theta-correction-inertia/v/2}
}Loading discussion…