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Research Article · Preprint Mathematics

Critical-tail localization and growing theta correction inertia

Scott Schubr

Luminary Oddities

  • Mathematics
  • Number Theory and Special Functions
  • Spectral Theory
  • Complex Analysis
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

For the Riemann theta transform F(z)=ξ((1+iz)/2)/4, let E(P)=(4π)^(-1)∫|(FP)'|^2, Q(P)=(8π)^(-1)∫(F'^2-FF'')P^2 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 Cexp(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.

Keywords Riemann theta form · correction inertia · critical exponential weight · Meixner-Pollaczek approximation · finite-order cutoffs

Scope of the work

Critical-tail joint-norm polynomial localization producing positive-original-form channels of both correction signs with square-root-logarithmic dimension growth in each parity.

Reproduction notes

14-page manuscript and 13 package members. Run python build.py --tectonic tectonic, or compile main.tex with its five included TeX components. Run python verify_weighted.py and python -O verify_weighted.py using the Python 3 standard library; each reports 1169 exact checks of the MP Gram, derivative, weighted-norm, pole-bin and convolution identities.

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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.

References and related publications

  1. Weighted polynomial localization and theta correction inertia
  2. Finite positivity and correction inertia for the Riemann theta form
  3. Cordaro and Fürdös: classical finite-order cutoff ancestry

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Critical-tail localization and growing theta correction inertia (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/beyond-logarithmic-powers-critical-tail-theta-correction-inertia/v/2

BibTeX
@techreport{luminary_beyond_logarithmic_powers_critical_tail_theta_correction_inertia_v2,
  title = {Critical-tail localization and growing theta correction inertia},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/beyond-logarithmic-powers-critical-tail-theta-correction-inertia/v/2}
}

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