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
Complete finite certificates characterize and computably approximate the actual lower edge of the specified minimal weighted Weil operator; exact-null deflation gives an executed eight-null bound, and fixed arithmetic cutoff forces constraint growth.
Disclosures
AI assistance was used in mathematical development, literature comparison, manuscript preparation and computational verification.
Reproduction
The 26-page manuscript and editable LaTeX sources are supplied in this 18-member package. README.md gives build instructions. code/verify_bounds.py, code/verify_certificate.py reconstruct the analytic bounds and complete finite signed certificate using NumPy and python-flint. The executable covers the finite eight-null instance; the arbitrary-precision lower-edge search is a mathematical construction.