The Closing Prime: Projective Closure and Collatz Convergence on the Poincaré Homology Sphere

We demonstrate that the Collatz conjecture is a consequence of the geometry of the Poincaré homology sphere S³/2I. The odd integers are identified with the twelve vertices of the icosahedral lattice modulo 24, and the Collatz operator T = sᵏ ∘ (t + 1) is derived from the generators of the binary icosahedral group 2I. The spectral gap of the Laplacian on S³/2I is λ₁ = 168 = |GL(3,𝔽₂)|, the order of the automorphism group of the Fano plane PG(2,𝔽₂), in which no parallel lines exist. The prime 7 = 2³ − 1, the first prime outside the construction set {2, 3, 5}, acts as the closing prime that supplies projective completion. The transfer operator of T on odd residues mod 2ᵐ is nilpotent (unique eigenvalue λ = 1)…