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Research Article · Preprint Mathematics

Source-Resolved Resolvent Growth under Reciprocal-Theta Confinement

Scott Schubr

Luminary Oddities

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

Abstract

We prove two-sided bounds for the complete inverse-energy Gram of prescribed theta-derivative sources in the original polar-constrained archimedean realization of the Weil form. For every finite source size d + 1, diagonal normalization by explicit powers of U = log(1/ε) gives positive degree-dependent lower and upper bounds whenever U ≥ 64(d + 2)². For each fixed family the normalized Gram is bounded above and below by constants, and source i has exact logarithmic exponent 4i + 13/2. The upper estimate uses exact half-line polar cancellation and a uniformly priced Fourier primitive, retaining the complete exterior polar tail. The lower estimate combines an exact polynomial coefficient map, all-atom theta bounds and a common-weight shell covariance with exact polar shorting. A two-metric residual theorem receives non-square-integrable divided sources, and explicit finite receiving bounds accompany the analytic estimates.

Keywords resolvent · theta functions · spectral theory · constrained quadratic forms

Scope of the work

Complete constrained source-Gram bounds for all d ≥ 0 and U ≥ 64(d + 2)², constant normalized fixed-family bounds and exact logarithmic exponents, with two-metric residual and finite polar receiving estimates.

Reproduction notes

The publication package includes editable LaTeX, a portable build script and Python finite-control code with python-flint 0.9.0. Directed 512-bit scalar enclosures and exact rational joint/shorted moment checks accompany the analytic proofs. See README.md and REPRODUCING.md.

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Author disclosures

AI assistance was used in mathematical development, manuscript preparation and verification-code review. The author is responsible for the statements and presentation.

References and related publications

  1. Quantitative Null-Space Compression for a Weighted Weil Operator, edition 2
  2. Observation norms under double-exponential confinement, edition 2
  3. Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators, edition 3
  4. Suzuki, Spectral convergence associated with Weil's quadratic form
  5. NIST DLMF, Recurrence Relations and Derivatives
  6. Johansson, Arb interval arithmetic

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Source-Resolved Resolvent Growth under Reciprocal-Theta Confinement (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/source-resolved-resolvent-growth-reciprocal-theta-confinement/v/1

BibTeX
@techreport{luminary_source_resolved_resolvent_growth_reciprocal_theta_confinement_v1,
  title = {Source-Resolved Resolvent Growth under Reciprocal-Theta Confinement},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/source-resolved-resolvent-growth-reciprocal-theta-confinement/v/1}
}

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