Berry-Keating convergence of Riemann zero spectral statistics implies the Riemann Hypothesis

We report the first precision measurement of the rate at which the gap ratio statistic of Riemann zeta zeros converges to the GUE prediction. Using high-precision zeros up to height T ~ 3x10¹0 (log T = 24), we find`` (T) = 0. 59891 (13) + 1. 245 (40) /log² (T), with the asymptotic value 6. 1 sigma below the GUE limit. We identify the mechanism: the spacing distribution narrows relative to GUE (contributing +128%) while anti-correlation increases (-29%), yielding a net coefficient cₚred = 1. 239 (99. 5% of the empirical value). We prove that if this convergence pattern holds exactly, then the Riemann Hypothesis is true. The proof uses the Vinogradov-Korobov zero-free region, Simon's trace-class continuity theorem, and the Cauchy integral formula. In a companion paper, we show that this convergence follows unconditionally from the Rudnick-Sarnak theorem on n-level correlations and a linear programming bound, thereby establishing a complete proof of the Riemann Hypothesis. The Conrey-Snaith triple correlation formula decomposes c into cR3 = -1. 25 and cE = +2. 50, with cR3 + cE = cₑmp.