A ladder of phase transitions in the homology of squarefree window complexes: results record, certificates, and adjudication trail

For an integer parameter N and a rational window ratio ρ > 1, let R_ω(N, ρ) be the simplicial complex whose (ω−1)-dimensional cells are the squarefree products of ω distinct primes lying in the window (N/ρ, N], with all their subsets as faces. We prove that the codimension-one Betti number of this complex is an exact prime-counting expression, computable without linear algebra at any scale, and that its density in the cell count is a piecewise linear function of 1/ρ whose breakpoints and root are exact rationals built from prime-advance ratios. THEOREM TRACK (proved, unconditional): the exact identity h1 = pi(N/4) − pi(N/6) + cap − D_E for all rational rho ≥ 2, N ≥ 154; the closed form K(rho), piecewise linear in 1/rho; rho_c(3) = 55/21 = (5/3)(11/7); the death curve N/(21 r*(N/55)); the eternal critical cell (3, 7, r*(N/55)) with h1 = 1 for all N > 8,580,385 at criticality; the K/6 coefficient identity (denominator = the primorial of 29). The same transition repeats at every ω: the critical ratio is the product of the top ω−1 prime-advance ratios (a ladder of phase transitions, rungs ω = 2..10 confirmed under sealed preregistration). The finite-N correction families are certified by an exact-integer algorithm at every rational rho > 5/2, hash-verified identical across two independent clean-room implementations over 79,772,002 member rows (externally adjudicated: BANKED), with an explicit analytic N0(rho) bound via Dusart's prime-gap estimate. Separately, four sealed preregistered instruments show the Betti number's fluctuation across eleven decades is the Riemann zeros' wave (residual chain 0.0372 → 0.0041, controls silent, per-zero phase lock), yielding a homological placement bound |beta_k − 1/2| ≲ 0.01–0.04 for the first eighteen zeros from prime counting alone. The archive contains the full paper, a plain-language guide, the tiered results record, both lemmas, sealed preregistrations with SHA-256 hashes, the certifier source code and gap-record tables, verbatim external adjudication rulings, and the complete correction ledger. Human-directed, AI-executed (Claude, Anthropic; independent AI clean-room reimplementation; external AI adjudication), disclosed per the Leiden Declaration on AI and Mathematics (2026). Sphenic window complex; the transition at 55/21.