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 also prove that the inverse-square-root bound for the raw resolvent output is asymptotically sharp; an abstract conditional extension covers valid null-space deflations. These results specify improved sufficient localization budgets and a limitation of raw-output approximation. They provide neither a new numerical lower certificate nor positivity at the Riemann-hypothesis threshold.
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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
Complete frequency-block operator enclosure and signed finite residual-Gram certificate for the declared constrained theta-weighted realization; sharp raw high-frequency norm asymptotic. Null-space extension requires the stated domain and compactness hypotheses.
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. This is an author preprint.
Reproduction
The supplementary ZIP contains the exact PDF, complete editable LaTeX source, build instructions, citation metadata and SHA-256 manifest. Compile main.tex with Tectonic or run pdflatex twice. All analytic proofs and explicit bounds are in the manuscript.