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 − DE for all rational rho ≥ 2, N ≥ 154; the closed form K (rho), piecewise linear in 1/rho; rhoc (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 |betaₖ − 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.