An explicit global lower certificate for a weighted Weil operator
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Abstract
We prove a global weighted lower bound for the complete Weil form of the Riemann xi-function. For every complex compact smooth test f whose Laplace transform vanishes at ±1/2, the form satisfies Q[f] ≥ −(155299736/10^14) ∫ℝ |f(u)|² cosh(u/2)/ψ(u) du + (33/25000000)||f||₂². where ψ is the normalized Riemann theta kernel. In particular, the specified minimal weighted realization has T≥0.99999844700264 I. Its endpoint T≥ I is exactly Weil's criterion for the Riemann hypothesis. The construction replaces an unevaluated reference resolvent by a known positive multiplication operator. A full exterior-frequency floor, a paired arithmetic remainder, local Legendre approximation and two exact global constraint rows reduce the bound to signed Gram inequalities. The bound includes all omitted prime powers, spatial and polynomial tails, quadrature errors and arithmetic errors. Reflection reduces the executed certificate to two matrices of order 897; directed ball arithmetic verifies both. Portable programs reconstruct the certificate from the displayed formulas.
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- main.pdf132.3 KiB · pdf
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7785073cd904e28c0be9dcd5fbdf15f72d79735192f80b530568c2d916536e5f - Weighted_Weil_signed_certificate_publication.zip158.1 KiB · package
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83cb6b8148a72c8a3455e208003812fdead14944335883545fdaa29a5fa778de
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- Weil positivity, Riemann xi-function, global spectral lower bounds, computer-assisted proof, interval arithmetic
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
A complete global weighted lower bound for compact smooth Weil tests with both polar moments zero gives T ≥ 0.99999844700264 I on the specified minimal form domain, retaining a positive unweighted potential.
Disclosures
AI systems assisted with exploration, literature comparison, mathematical and code checking, and manuscript preparation. The author takes responsibility for the statements, proofs and citations.
Reproduction
The 12-page manuscript, editable source and citation metadata are supplied in this 11-member package. README.md gives build and execution commands. The NumPy 2.3.5/python-flint 0.9.0 programs verify_bounds.py and verify_certificate.py regenerate scalar enclosures, both 897-dimensional signed matrices and directed congruences; the manuscript supplies the continuum and domain arguments.