The Integer Lattice of the Poincaré Homology Sphere: Discrete Heat Flow, Fibonacci Structure, and Collatz Convergence on S³/2I

We identify odd integers with lattice states on the Poincaré homology sphere S³/2I, the quotient of the 3-sphere by the binary icosahedral group. Even integers are not states but transfer zones — interference regions in the icosahedral tiling where vertex harmonics overlap and energy transfers between lattice nodes. The identification is constructed from the group presentation ⟨s, t, u | s² = t³ = u⁵ = stu = 1⟩ of the icosahedral group I ≅ A₅: the generator t of order 3 acts as multiplication by the dimension D = 3, the generator s of order 2 acts as division by the Euler characteristic χ = 2, and the unit of the integer ring acts as the +1 unity injection. The composite operator T = sᵏ ∘ (t + 1) is precisely…