All Research

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.