Fix a threshold N and window W = (N/2, N]. The sphenic window complex is the 2-complex whose triangles are the sphenics (squarefree integers pqr with three distinct prime factors) in W, whose edges are the co-occurring prime pairs, and whose vertices are the primes. We reduce its entire homology to prime-counting functions. A purely combinatorial slab identity, proved over every field with an explicit ±1 integral basis, gives rank ∂₂ = F − elem; a finite-parts cancellation collapses the first Betti number to a signed sum of fourteen prime counts, with exact rational limiting density b₁/π(N/6) → 6H − 3/22 = 97360699/1078282205 ≈ 0.0902924, unconditional on the prime number theorem, Bertrand's postulate, Dusart's explicit bounds, and a finite computation. The top Betti number is b₂ = elem = F − D(N/2) + cap, so the full profile is a triple of counting functions. An honest negative delimits the appearance of the non-trivial zeros of ζ as exactly the classical explicit-formula duality and nothing more, with no claim toward the Riemann hypothesis. This is the third paper in a series on the −1-eigenspace of the coprime graph (Papers 1 and 2: doi:10.5281/zenodo.20933665 and doi:10.5281/zenodo.21169867); it determines the three-prime layer of that decomposition and pins the signed density underlying S. Banerjee's eigenvalue −1 multiplicity question. Machine-verifiable certificates are in the linked repository (concept DOI 10.5281/zenodo.21135221).
Lee Rich (Fri,) studied this question.