Pre-registered pair- and triple-correlation experiments on the first two million Riemann zeros, and a parameter-free validation of the Bogomolny-Keating and Conrey-Snaith formulas at low effective dimension

We report a pre-registered numerical study of the pair correlation of the first 2×10^6 nontrivial zeros of the Riemann zeta function, a height range corresponding to effective spectral dimension L = log(γ/2π) ∈ [8.8, 12.1], where lower-order corrections of order 1/L² are large (about 10% of R₂ locally). With a lag-based block estimator (with an exact-span boundary correction), block and superblock bootstrap confidence intervals, and a calibrated noise floor from matched CUE surrogates, we find that (i) the residual of the data against the GUE sine kernel exhibits strong structure (deficit −0.038 near r ≈ 0.38, all 20 windows above the 97.5% surrogate quantile, a departure that survives a protocol-matched block-estimator null: all 20 windows above even its maximum, minimum margin +9.9%); (ii) a finite-N form-factor model with N_eff = log(γ/2π) improves the fit by an order of magnitude less than the observed residual; and (iii) the zero-parameter Bogomolny–Keating (1996) prime-correction formula, identified post hoc but fixed by the 1996 literature and fitted to nothing, explains 95.2% of the global residual power (RMSE 0.01488 to 0.00327), wins in 20/20 windows under both bootstrap schemes, and leaves a residual consistent with white noise (Ljung–Box p = 0.48); its numerical evaluation is sweep-stable to at most 7.1×10^-5 under grid and cutoff upgrades. We then extend the entire protocol to the triple correlation R₃: a pre-registered form-factor test fails (1/20 windows, with a null global effect), the residual against the determinantal sine kernel exhibits an arithmetic lattice in the two gap variables (a departure that survives a protocol-matched block-estimator null: all 20 windows, minimum margin +11%), and the zero-parameter triple-correlation formula of Conrey and Snaith (2008), evaluated with its random-matrix backend validated against the exact CUE determinantal identity to 1.8×10^-14, explains 93.2% of the global R₃ residual power (RMSE 0.02319 to 0.00603), winning in 20/20 windows (18/18 on windows untouched by a pre-run smoke test). Both matched-null verdicts are reproduced in a fresh-draw null rerun. Finally, the entire analysis replicates out of sample on two new segments of Platt's zero tables at L ≈ 17.9 and L ≈ 22.3 (about 6.0 and 7.5 million zeros): all eight pre-registered hypotheses pass, and the inferred structural amplitude tracks the parameter-free predictions to within about 1–8% at heights where the predicted signal is 2–4.4× smaller. An independent full rebuild in the pinned Python 3.10.6 environment reproduced every numeric field of all nine confirmatory archives within rtol = 10^-12 and atol = 10^-14 (largest absolute difference 1.14×10^-13; four of the nine archives were byte-identical and five differed at that bit level), while all fourteen pipeline figures (fig7 to fig20, the internal figure set from which the six figures of this note are drawn) were byte-identical. Along the way we document a misprint in the printed form of the arithmetic factor P(x,y) in the Conrey–Snaith theorem, whose printed prime sum diverges; the form from their derivation converges absolutely. None of this is new theory: the formulas date to 1996 and 2008 and earlier numerical comparisons exist at much greater heights. The contribution is an independent, statistically rigorous validation in the low-to-moderate-L regime, with pre-registered hypotheses, post-hoc steps explicitly flagged, and code, data hashes, and convergence sweeps included.