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.
David Escribano Alarcón (Sat,) studied this question.