Topological Convergence of Riemann Zero Spacings to GUE: A Height Scan from γ ≈ 10⁴ to 10²¹

Version 2 (August 2026) adds five intermediate heights (γ = 10⁷ to 3×10¹⁰) from the LMFDB zeros database, filling the gap left in version 1 and closing its stated open questions. Version-1 conclusions are unchanged; the error analysis is strengthened and one statistical lesson is added. We answer the open question left in our previous study, "Persistent Homology of Riemann Zero Spacings: an Exploratory Study in 1D and 3D Time-delay Embeddings" (Zenodo, doi:10.5281/zenodo.19693831). That study found that the total persistence of the Riemann zero gap sequence sits about 8–9% below a proper GUE random-matrix control, an offset that survived three robustness checks. Its origin was left open, with three candidate explanations: (a) a real finite-height effect of the slow Montgomery–Odlyzko convergence, (b) a mismatch between the two unfolding schemes, or (c) a sensitivity to the fraction of the GUE spectrum used as the bulk. This paper answers that question by repeating the same measurement at fourteen heights spanning 39 units of log γ, using the first 2 million zeros (Odlyzko), five stitched windows of ~39,000 zeros each at γ = 10⁷, 10⁸, 10⁹, 10¹⁰, and 3×10¹⁰ drawn from the LMFDB zeros database (103,800,788,359 zeros computed by D. J. Platt to precision 2.5×10⁻³¹), and Odlyzko's tabulations around zeros number 10^12, 10^21, and 10^22. The methodology is locked to the first paper: local Riemann–von Mangoldt unfolding, 3D time-delay embedding at τ = 3, Alpha-complex persistence through dimension 2 on matched 3,000-point clouds, and the same GUE control (M = 800 Hermitian matrices, semicircle-CDF unfolding, bulk fraction 0.60, fifteen pooled replicates). The LMFDB windows are stitched from overlapping downloads by exact zero-index continuity, with audit logs included; the LMFDB and Odlyzko sources were cross-validated at an overlapping height to a worst case of 3×10⁻⁹ per zero, and yield the same offset through the full pipeline. The offset shrinks steadily and monotonically with height: −8.7% at γ ≈ 5×10³, −6.0% at 1.8×10⁴, −2.8% at 10⁷, −1.3% at 3×10¹⁰, −1.4% at 2.7×10¹¹, and consistent with zero at γ ~ 10²⁰–10²¹. Both a 1/log and a 1/log² decay law fit the full fourteen-point curve well (χ²/dof = 3.3/13 and 4.5/13, generalized least squares with the shared GUE-reference error of ±0.41% treated as a fully correlated systematic and verified with independent seed families). Under either law the asymptotic offset is consistent with zero: |a| < 2.2% at two sigma in every model considered. The leading coefficient is −0.41 ± 0.05 under 1/log(γ/2π), or −3.5 ± 0.4 under 1/log². Distinguishing the two functional forms is shown to be beyond the reach of this observable at achievable statistics — the model separation (~1.1%) equals the intrinsic block-to-block noise at our cloud size — and would require ~4×10⁵ zeros per height; we state the requirement rather than pursue it. A paired scan of the GUE bulk fraction (0.4 / 0.6 / 0.8) rules out explanation (c); the asymptote bound constrains explanation (b) to the percent level. We conclude that the residual reported in the first paper was a finite-height effect: the slow convergence of the zeros to their asymptotic GUE statistics, seen through a topological lens. To our knowledge this is the first measurement of a convergence law for the Montgomery–Odlyzko phenomenon using persistent homology. Version 2 also documents a methodological lesson: three-block error bars underestimated the true block variance by factors of 5–8, and an intermediate analysis based on them spuriously preferred the 1/log² law; with honest twelve-block bars the preference evaporates. The remaining open item is theoretical: deriving the measured coefficient, and the correct power of 1/log, from the known lower-order terms of the two-point statistics of the zeros. We make no analytic claims. Supplementary material includes the analysis, stitching, validation, and GUE-reference-check scripts, the per-height CSV outputs, the stitch audit logs, and full download provenance for independent reproduction.