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…