Finite certificates for the lower edge of a weighted Weil operator
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Abstract
We characterize the reach of complete finite lower certificates for the original minimal weighted Weil operator. The infimum of their certified negative allowances equals the operator’s actual negative spectral depth, and a computable constrained core supplies rigorous two-sided approximations to its lower edge at every prescribed positive accuracy. The proof combines logarithmic form-core density with compact domination of a monotone frequency limit. Exact derivative-null vectors allow finite deflation without changing the original form or discarding mixed terms. An executed certificate with eight such vectors proves I + H ≥ 0.999999978600004918685 I, with a retained positive potential on the null complement. All arithmetic, frequency, spatial, quadrature and finite-matrix errors are included; directed ball arithmetic verifies both signed matrices of order 4085. We also prove that, at fixed arithmetic cutoff and positive multiplication reference, boundedly many constraints cannot maintain the required cap as the regularization decreases, even with an arbitrary growing frequency band. The quantitative obstruction forces growing constraint dimension and identifies the continuation mechanism that the adaptive framework must control.
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Research details
- Subjects
- Mathematics · Number Theory · Functional Analysis · Spectral Theory
- Keywords
- Weil positivity, Riemann xi-function, spectral lower bounds, exact null-space deflation, signed Gram certificates, rigorous computation
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
The original full-line reciprocal-theta weighted Weil operator on its specified minimal constrained domain. Complete finite certificates characterize its actual negative spectral depth. Eight exact derivative-null vectors give the full-domain bound I + H ≥ 0.999999978600004918685 I with a projected positive potential. The continuation obstruction allows every frequency schedule at fixed arithmetic cutoff and positive reference.
Disclosures
AI assistance was used in mathematical development, literature comparison, manuscript preparation and computational verification.
Reproduction
The source package contains the editable manuscript, pinned dependencies and portable programs reconstructing 60,042 analytic gates, all 11,670,288 primitive conversion enclosures, complete quadrature sums and both 4085-dimensional directed congruences. The finite verifier is the exact program used for the primary execution and a separate optimized reproduction. The general lower-edge dovetail construction is proved in the manuscript; the supplied executable reconstructs the stated M7 certificate.