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

Let`<r>`(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 `<r>`(T) = R_inf + c/log^2(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(T^a) 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.