An explicit global lower certificate for a weighted Weil operator
Loading original first page…
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, 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. All omitted prime powers, spatial and polynomial tails, quadrature errors and arithmetic errors are paid. 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.
Read the full paper in the original PDF.
Edition files
- main.pdf130.8 KiB · pdf
SHA-256
0b37f1fa5959ec5e70410beab8f7209b17ec5780678603d1f44baa1eccaf9096 - Weighted_Weil_signed_certificate_publication.zip156.6 KiB · package
SHA-256
d9e709dfa9ebd71f99799a97c1865d9869fcab0e8b163f2dd4abbd57d86062d7
Available editions
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
Global weighted lower inequality for the complete Weil form on every complex compact smooth two-moment test and the entire specified minimal form domain. Retained positive potential; scalar corollary T ≥ 0.99999844700264 I. The exact positivity endpoint is T ≥ I.
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 source package reconstructs all inputs from formulas using Python, NumPy 2.3.5 and python-flint 0.9.0. The analytic checker verifies 5791 conditions; the arithmetic checker verifies 1232816 primitive conversions and both exact 897-dimensional signed congruences. README instructions include an independent source build with Tectonic.