Spectral Gap of the Poincaré Homology Sphere and the Proton Radius: Deriving the Collatz Decay Rate from 2I Representation Theory

The spectral gap of the Poincaré homology sphere S³/2I is derived from the representation theory of the binary icosahedral group 2I. The Laplacian on S³ has eigenvalues λₗ = l(l+2) with multiplicity (l+1)². On the quotient S³/2I, eigenspace l survives if and only if the restriction of the (l+1)-dimensional irreducible representation of SU(2) to 2I contains the trivial representation. By computing the character inner product for all l, we find that the first eleven eigenspaces above the ground state (l = 1 through l = 11) are entirely killed by the 2I quotient — the manifold is spectrally rigid below the icosahedral vertex scale (spectral confinement). The first surviving excited eigenspace is l = 12 = V (the vertex count of the icosahedron), giving spectral gap λ₁ = 12 × 14 = 168 = V × (V + χ). The surviving eigenspace at l = 12 decomposes into 2I irreducible representations of dimensions 1 + 2 + 3 + 5 = 11 = L₅, the fifth Lucas number — the icosahedral self-reference cost. As a physical correspondence: if one Collatz step corresponds to one 120° colour rotation at the proton's Compton frequency, then the distance per step is exactly (2π/3) times the reduced Compton wavelength ƛ_C. The ratio of the proton charge radius to ƛ_C is observed to equal L₃ = 4 (the Lucas number at dimension D = 3, the base of the Collatz arithmetic) to 0.08% with zero free parameters. Hadronization timescales at RHIC and LHC energies are estimated from the Collatz stopping time T(n) = 7.23 log₂(n) and match experimental measurements to order of magnitude.