Finite-height accuracy of effective matrix size approximations for Riemann zero correlations

At finite height the pair and triple correlations of the Riemann zeros deviate from their sine-kernel limits by terms of order 1/L², with L = log(T/2π). Bogomolny, Bohigas, Leboeuf and Monastra showed that the leading term coincides with the 1/N² correction of the circular unitary ensemble at the effective size N_e = L/√(12Λ). We measure how well this effective-size description, unrescaled and with two rescalings, approximates the full Bogomolny–Keating (R₂) and Conrey–Snaith (R₃) formulas on the same zero data, at three heights covering block heights L = 5.42–22.31 (2×10⁶ zeros from Odlyzko's tables and 1.3×10⁷ from Platt's). On 0 < r ≤ 4 the unrescaled approximation has a larger mean squared error than the sine kernel itself at all three heights, and the full formulas are closer to the data than every effective-size approximation in every segment. The projected coefficients of the 1/N² shapes are 12–30% (R₂) and 4–19% (R₃) of their asymptotic values, and the contrast A₃ + 3a, which vanishes for any unrescaled common-N correction, equals −1.50, −2.09 and −2.14. Both agree with the full-formula projections within bootstrap intervals, conditional on an estimator-bias correction transferred from GUE simulations. Internal kernel rescaling is the best of the three approximations for R₂ but remains worse than the sine kernel for R₃ on the full domain. This record contains the note (PDF) and an archive with the analysis code, the frozen protocol (in Polish) with its reporting annex and a dated English chronology, the six input archives, the stage records, the SHA-256 manifests of inputs and results, the scripts that print every table and draw the new figures, and a script that checks every file against the manifests. The 334 MB results archive is regenerated by the runner (see README.md in the archive). The underlying pair- and triple-correlation pipeline is described in an earlier preprint, doi:10.5281/zenodo.23046590.