Persistent Homology of Riemann Zero Spacings: an Exploratory Study in 1D and 3D Time-delay Embeddings

We explore what persistent homology, applied to the nearest-neighbour gap sequence of the first 10, 000 non-trivial Riemann zeros, can and cannot see about the Montgomery–Odlyzko law. Two main observations emerge. First, in a one-dimensional filtration on locally unfolded gaps, the β₀ statistic — essentially the single-linkage component count at threshold ε — cleanly separates the Riemann sequence and a proper GUE bulk eigenvalue control from an exponential (Poisson-gap) reference; a maximum discrimination ratio of 3. 44× is reached around ε ≈ 0. 20. Riemann and proper GUE agree with each other to within 5% across the entire ε range, while both differ from i. i. d. samples of the Wigner surmise by about 11σ, showing that the statistic is sensitive to multi-point correlations rather than to the marginal spacing distribution alone. Second, we lift the analysis into 3D via time-delay embeddings (sₙ, s₍+⏣, s₍+₂⏣) and compute Alpha-complex persistence through dimension 2. Wasserstein-1 distances on the resulting diagrams place Riemann 3. 3–6. 9× closer to proper GUE than to the exponential reference, in each of H₀, H₁, and H₂, at time-delay τ=3. A small residual offset persists — Riemann total persistence sits about 8–9% below proper GUE — and we subject it to three robustness checks: (i) a matrix-size scan over M ∈ 800, 1600, 2400 of the GUE generator, which rules out finite-N of the control as the explanation; (ii) a matched-scale comparison of three non-overlapping Riemann sub-samples against fifteen pooled GUE replicates, which shows the offset is not within sub-sample variance (permutation p = 0. 0012) ; and (iii) a scan of the time-delay parameter τ ∈ 1, 2, 3, which shows that a portion of the apparent τ=1 offset (roughly 3–4 percentage points) is an artifact of the fact that consecutive triples at τ=1 share two of three coordinates, and that the remaining 8–9% is stable across τ. We report this stable residual as an open observation rather than a claim. As a cautionary methodological note, we also show that a naive global normalization of Riemann gaps (instead of the standard local Riemann–von Mangoldt unfolding) produces a dramatic but entirely artifactual monotone trend in all Hₖ as a function of height γ. We apply the 1D filtration to confirm the Lehmer close-pair near γ ≈ 7005. We make no analytic claims; this is an exploratory note intended to characterize what a topological eye sees when looking at the zeta zeros and to document the methodological traps that appear along the way.