The Icosahedral Spectrum: Eigenvalues of S³/2I as Combinations of Vertices, Faces, and Edges

We compute the spectrum of the Laplacian on the Poincaré homology sphere S³/2I and prove that the surviving eigenvalues have a simple characterisation: the harmonic degree k contributes a non-trivial eigenspace if and only if k lies in the numerical semigroup generated by {12, 20, 30} — the vertex count, face count, and edge count of the regular icosahedron. The proof combines Klein's classification of the invariant ring ℂ[x,y]^{2I} (generated by invariants of degrees 12, 20, 30 with one relation at degree 60) with the Molien series P(t) = (1 − t⁶⁰)/((1 − t¹²)(1 − t²⁰)(1 − t³⁰)), verified independently by direct character computation for all k ≤ 120 with zero mismatches. The filter is finite: exactly 15 even…