A Lean-Verified Refinement of the Unconditional Simple-Zero Bound for the Riemann Zeta Function (67.251%)

Main result: The unconditional Lean-verified lower bound is 67.251% (exact value 67.25126275289546%) for the proportion of nontrivial zeros of the Riemann zeta function that are simple and lie on the critical line. This raises the pinned Alpöge-Furman Montgomery-Taylor-window value of 67.2500703679...% by about 0.001192 percentage point.

Status: Preprint; not peer reviewed. This result does not prove the Riemann hypothesis and makes no numerical-record claim.

This preprint presents a Lean 4-verified refinement obtained by retaining a nonnegative spectral remainder in the rank-trace argument and lower-bounding it using local Gram interactions between consecutive simple critical-line zeros. A path-forest aggregation and an exact rational gap program yield the constant 0.6725126275289546.

The accompanying Lean development kernel-checks the exact-window statement (0.6725126275289546 - epsilon) N(T, 2T) <= N_0^s(T, 2T) eventually for every epsilon > 0. The most delicate kernel inequalities have two independent Lean proofs: a five-box analytic proof and a fixed-point reflection proof over 170 accepted leaves.

Formal verification establishes the encoded theorem but does not replace independent mathematical review.

Claude Code and OpenAI Codex assisted with proof exploration, code generation, adversarial review, Lean debugging, certificate design, and manuscript drafting. Takayuki Komada is the author and remains responsible for all mathematical claims and for the decision to circulate this manuscript.

This Zenodo record includes the DOI-bearing preprint PDF and the companion Lean 4 source archive. The PDF is licensed under CC BY 4.0; the code archive is licensed under Apache License 2.0 as stated in its LICENSE and NOTICE files.