The Universal Cover: Why the Integers Are the Covering Space of S³/2I

The Collatz series of papers has established that the odd integers behave as states on the Poincaré homology sphere S³/2I, with the Collatz operator T = sᵏ ∘ (t + 1) derived from the generators of the binary icosahedral group 2I. The convergence proof is conditional on one identification: that the integer tower is the covering space of S³/2I. This paper proves that the identification is not a free assumption but a conditional theorem. The argument has three steps. First, the Collatz generators s = ÷2 and t = ×3 are type-compatible with the 2I generators in quaternion algebra: the group presentation ⟨s, t | s² = t³ = (st)⁵ = −1⟩ is verified explicitly with unit quaternions s = (0, φ⁻¹, −φ, 1)/2 and t = (1, −1,…