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 10⁴ (zeros1, 100, 000 zeros) to gamma approximately 1. 4 x 10²1 (zeros5, 10, 000 zeros). The gap-ratio statistic is taken from Atas, Bogomolny, Giraud, and Roux (2013), used in both restricted r-tilde = min (s₍+₁, sₙ) / max (s₍+₁, sₙ) and unrestricted r = s₍+₁ / sₙ 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 10²1). 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.