Spacing Statistics in Thresholded Even-Sector Nested Prime Pair Sets

This paper complements "Parity-Resolved Nested Prime Pair Distributions." This paper studies the spacing geometry of thresholded even-sector nested prime pair sets Sᵉᵛ_k(M;R), where Nᵉᵛ(m;R) counts odd radii r for which m − r and m + r are both prime. Using sieve-theoretic admissibility analysis and exact computations up to (M,R) = (10⁹, 4096), it shows that the sparse tail emerges only near k = 40–45, that median raw gaps are strongly locked to arithmetic scales, and that the extreme tail is spatially reconcentrated rather than uniformly thinned. A residue-level analysis at moduli 30, 210, and 2310 shows hierarchical admissibility ordering, leading to an arithmetic-geometric picture of high-threshold sets as arithmetically localized subsets of the even sector.