An exact-census spectral characterization of squarefree almost-prime dyadic-window homology

We study finite cell complexes Dω on the squarefree ω-almost-primes in the dyadic window (N/2, N], whose codimension-one Betti number is an integer arithmetic function of N. For ω = 4 we give an explicit identity for b₂(N) as a signed combination of exact prime-counting combs whose only non-comb component is an O(1) topological boundary term, validated to unit precision against an independent boundary reduction. Writing each term as a signed union of N-intervals yields a support-interval census C₄ of 193,979 members. Evaluated at the first Riemann-zero ordinates (read-only sampling points; no new zero information), the √N-normalised fluctuation of b₂ has an amplitude envelope with an excess in the 30–34 band that the census reproduces at Pearson 0.9992. A marginal-preserving width-shuffle collapses the band to the generic floor (+52.9σ global, +218.9σ within sign class), locating it in the centre–width correlation of the census. A generalization-budgeted wall-comb search finds no fitted compact predictor beyond held-out r ≈ 0.65, while a single derived comb {4,6,2} generalizes to held-out 0.73 and out-generalizes a 62× larger fitted dictionary. We introduce a shuffle-fragility statistic separating a compact derived object (robust) from an exact-census residue (fragile). We claim a characterization of one complex's coupling to the ordinates; we make no claim about the zeros, no correspondence, and no implication for the Riemann hypothesis. Reproducibility. Every number regenerates from the attached archive (fargame_bundle_v1.tar.gz); a fresh-shell replay verified the manifest (26/26 sha256 OK) and the headline numbers. AI assistance: human-directed and AI-executed; see the disclosure in the manuscript (Appendix B), meeting the Leiden Declaration on AI and Mathematics (2026).