More than 79.47% of the zeros of the Riemann zeta function are simple and on the critical line

Abstract

This preprint presents a certified-candidate proof that, with ineffective constants, more than 79.47% of the nontrivial zeros of the Riemann zeta function are simple and lie on the critical line, and more than 89.73% lie on the critical line. Building on the "two-thirds theorem" (C26), the authors extend the Gabor-compressed Weil matrix framework to evaluate the third through sixth trace moments. The fourth moment is proved exactly (m4=13/4), while the fifth and sixth moments are evaluated up to three connected constants proved to be rational. These moments are consumed via an exact Chebyshev–Markov dual certificate (valid on unbounded support) to yield the stated proportions. The claim grade is certified-candidate: the identity, certificate, and pairing-constant layers are machine-verified (kernel-compiled in core Lean 4, zero sorry but the analytic layer remains unformalized and external review is pending. A standalone reproduction package accompanies the paper. The mathematical development was carried out by Claude (Anthropic) under the direction of the authors.