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

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 nine heights, using the first 2 million zeros and Odlyzko's tabulations around zeros number 10^12, 10^21, and 10^22 — a span of 39 units of log γ. 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 offset shrinks steadily with height: from −8.7% at γ ≈ 5×10^3, to −6.0% at γ ≈ 1.8×10^4, to −1.4% at γ ≈ 2.7×10^11, and is consistent with zero at γ ~ 10^20–10^21 (pooled value −1.4% ± 1.7%). A single decay law, offset ≈ −0.40 / log(γ/2π), fits all nine points well (χ²/dof = 2.83/8); adding a free asymptote does not improve the fit (a = +0.5% ± 0.9%, Δχ² = 0.34). A separate paired scan of the GUE bulk fraction (0.4 / 0.6 / 0.8) shows sub-percentage-point sensitivity, ruling out explanation (c); the bound on the asymptote constrains explanation (b) to at most ~1%. 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. Open questions are stated plainly: the data cannot yet separate a 1/log from a 1/log² decay (data around zero #10^8 would decide this), the error bars at the two highest points rest on the minimum possible three blocks, and a theoretical derivation of the coefficient b ≈ −0.40 from the known lower-order terms of the two-point statistics of the zeros remains open. We make no analytic claims. Supplementary material includes the analysis script and the per-height CSV outputs.