Research Article · Preprint Mathematics
Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory · Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 3 · Permanent link
Overview
Abstract
We obtain a joint finite-rank approximation law for the complete Birman--Schwinger family of a constrained theta-weighted Weil operator. At each fixed absolute error tolerance, the required rank is at most C exp(C√(log(1/ε))) as the positive resolvent parameter ε tends to zero. The construction pays the complete paired arithmetic tail, high-frequency block and mixed localization errors. It gives a corresponding bound on the entire negative spectral projection and implies that the negative eigenvalue magnitudes decay as C exp[−c log²(n+1)]. Thus the full negative part belongs to every Schatten class. We separate two arithmetic cutoff regimes. Effective simultaneous prime-phase returns give signed high-frequency saturation when the retained cutoff grows sufficiently slowly. In the faster regime of the rank construction, theta confinement and compressed-shift estimates instead give a strict high-block norm gap: its normalized upper bound is at most three quarters for every positive limiting arithmetic ratio, with a stronger branch above the theta-overlap threshold. Both arguments retain the original constraints, weighted norms and complete localization errors.
Keywords Weil operator · negative spectrum · Schatten classes · joint cutoffs · finite-rank approximation · theta kernel · prime-phase recurrence
Scope of the work
At fixed absolute tolerance the specified complete Birman–Schwinger family has rank at most C exp(C√log(1/ε)); the whole negative part has stretched-log eigenvalue decay and belongs to every Schatten class. Slow-cutoff signed saturation and a strict fast-cutoff high-block gap hold in their stated joint regimes.
Reproduction notes
The 14-page manuscript and complete editable main.tex are supplied with CITATION.bib, README.md, SHA256SUMS.txt and verify_constants.py in this 6-member package. README.md gives Tectonic/pdfLaTeX and Python standard-library commands. The checker supplies finite arithmetic-tail, trigonometric-bump, integer-frequency and error-allocation controls; the joint-rank, full-spectrum and fast/slow regime conclusions are analytic proofs.
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Author disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs. The author declares no competing interests.
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Cite this edition
Scott Schubr. (2026). Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/joint-cutoffs-negative-spectrum-decay-weighted-weil-operators/v/3
BibTeX
@techreport{luminary_joint_cutoffs_negative_spectrum_decay_weighted_weil_operators_v3,
title = {Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 3},
url = {https://research.luminaryoddities.com/joint-cutoffs-negative-spectrum-decay-weighted-weil-operators/v/3}
}Loading discussion…