Frequency Localization for Weil Operator Certificates
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Abstract
We give a frequency-block enclosure for a constrained theta-weighted Weil operator. The retained paired-prime operator is translation invariant before the theta multipliers are applied. Splitting at that intermediate stage eliminates the cross-frequency blocks exactly. A finite-rank approximation of the remaining low-frequency map then has a complete operator error whose high-frequency term is proportional to the reciprocal digamma symbol, rather than its inverse square root. We retain the signed finite interaction matrix and enclose its resolvent Gram by full residuals. We prove matching leading asymptotics for both the raw resolvent tail and the positive spectral edge of the actual signed high-frequency block. The retained arithmetic coefficient is therefore asymptotically exact at every fixed prime cutoff and positive resolvent parameter. A compact-torus recurrence argument supplies the arithmetic peaks; an abstract conditional extension covers valid null-space deflations. An elementary theta estimate makes the multiplier constant self-contained.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- Weil operators, Fourier localization, operator inequalities, resolvent Gram matrices, rigorous spectral enclosures
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Frequency splitting before spatial localization gives a complete reciprocal-digamma high-block error and a signed finite sufficient certificate with full residual covariance. Raw and positive signed high-frequency asymptotics match at fixed arithmetic cutoff and positive resolvent parameter, with the stated conditional null-deflation extension.
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.
Reproduction
The 10-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 verifies eleven exact rational endpoints in the theta multiplier estimate. The source proves the series identities, operator enclosures, full residual and asymptotic statements analytically.