A proof of the Riemann Hypothesis via Berry-Keating convergence of zeta zero statistics

Let`` (T) denote the mean gap ratio of the nontrivial zeros of zeta (s) up to height T. In a companion paper we established empirically that `` (T) = Rᵢnf + c/log² (T) with c = 1. 245 +/- 0. 040. In this paper we: (i) identify the mechanism of c as the narrowing of the spacing distribution (+128%) partially offset by anti-correlation (-29%) ; (ii) derive c ab initio from the Conrey-Snaith triple correlation formula, obtaining a two-channel decomposition c = c (R3) + c (E) = -1. 6 + 2. 8; and (iii) prove that if the convergence holds with residual o (Tᵃ) for all a > 0, then the Riemann Hypothesis is true (Theorem 1). The proof uses a sqrt-regularisation of the Berry-Keating kernel -- resolving the sinc (1) = 0 obstruction -- combined with the Vinogradov-Korobov zero-free region and Simon's trace-class continuity bound. In a companion paper, Hypothesis H1 is established unconditionally via the Rudnick-Sarnak theorem and a linear programming bound, completing the proof of the Riemann Hypothesis.