An Empirical Note on the Riemann Zero Gap-Ratio Peak

This note reports a clean empirical measurement of the consecutive-spacing gap-ratio statistic on the non-trivial zeros of the Riemann zeta function across four publicly available Odlyzko zero datasets, spanning heights from gamma approximately 7.5 x 10^4 (zeros1, 100,000 zeros) to gamma approximately 1.4 x 10^21 (zeros5, 10,000 zeros). The gap-ratio statistic is taken from Atas, Bogomolny, Giraud, and Roux (2013), used in both restricted r-tilde = min(sₙ₊₁, s_n) / max(sₙ₊₁, s_n) and unrestricted r = sₙ₊₁ / s_n forms after standard local unfolding.Three anchors are tested as candidates for the peak of the unfolded distribution: 1/phi approximately 0.6180 (reciprocal golden ratio), ln 2 approximately 0.6931, and 1/sqrt(2) approximately 0.7071 (the chi greater than or equal to 0.707 chiral-invariant boundary from the author's framework, included for symmetry of testing).A 10,000-resample bootstrap on the highest-T dataset (zeros5) is reported. ln 2 is eliminated as the peak's closest anchor in both statistics (0.38% unrestricted, 0.26% restricted closest-anchor share). On the unrestricted statistic, 1/phi wins decisively at all heights (94.93% closest-anchor share at gamma approximately 10^21). On the restricted statistic the picture is mixed: 1/phi wins closest-anchor share (61.19% versus 38.55%), but the empirical KDE density is higher at 1/sqrt(2) (1.450 plus or minus 0.094) than at 1/phi (1.380 plus or minus 0.090). The corridor [1/phi, 1/sqrt(2)] is load-bearing in a way the standard GUE Wigner surmise alone does not predict.The note explicitly disclaims any claim to prove or disprove the Riemann Hypothesis, any connection to framework-internal claims (E8, Stiefel manifolds, chiral invariants), and any new mathematics. Data and analysis code are released in the same directory under CC-BY-4.0 for full reproducibility. Companion to the kappa-tau genesis note (DOI 10.5281/zenodo.20210868) released in the same batch.