What question did this study set out to answer?

To evaluate the empirical pair correlation of the first 10⁵ non-trivial Riemann zeros and to apply the Bogomolny-Keating correction.

May 4, 2026Open Access

ECI v7-note — Empirical Bogomolny-Keating 2-point correction on 10⁵ Riemann zeros

Key Points

To evaluate the empirical pair correlation of the first 10⁵ non-trivial Riemann zeros and to apply the Bogomolny-Keating correction.
Utilized a pre-registered testing pipeline for ECI v6 differential GSL inequality over 10⁵ Riemann zeros.
Performed empirical analysis of pair correlation, comparing observed data to Montgomery-Dyson predictions and applying a Bogomolny-Keating correction.
The empirical pair correlation deviated from predictions with χ² = 434.19/58 dof.
After applying the Bogomolny-Keating 2-point correction, the fit improved to χ²/dof = 4.17 with the best-fit amplitude.
A companion test indicated a BEC analog Hawking saturation ratio of ρ_meas ≈ 8.29%.

Abstract

Two independent small-scale results from a pre-registered testing pipeline for the ECI v6 differential GSL inequality. (1) On the first 10⁵ non-trivial Riemann zeros of Odlyzko's public tables, the empirical pair correlation R₂ᵉmp (x) deviates from the pure Montgomery-Dyson GUE prediction at χ² = 434. 19/58 dof; the Bogomolny-Keating (1996) arithmetic 2-point correction accounts for this deviation with χ²/dof = 4. 17 at best-fit amplitude. (2) Companion test on the BEC analog Hawking saturation ratio ρₘeas ≈ 8. 29% (Steinhauer 2019). This v7-note is a companion to the ECI v6. 0. 16 framework (DOI 10. 5281/zenodo. 19708889) and v6 formal paper. v6. 0. 16-era retraction context applies (notes/m1ₐudit₂026₀5₀2. md and notes/cosmoconstₐotₕonest₂026₀5₀2. md in the crossed-cosmos repo).

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