The vibrational mode spectrum of an icosahedral shell: forced degeneracies, the fivefold ceiling, and selection rules

Icosahedral symmetry caps irreducible representation dimension at five. That one fact decides the shape of the vibrational spectrum of any shell carrying the full group I_h, before a single material parameter is named: the sphere's clean (2l+1) tower cannot survive intact, and the first place it breaks is exactly l=3, where seven exceeds the ceiling. Forced, by group theory alone: the degeneracy pattern through l=5, obtained by subducing each SO(3) representation to I_h; the location of the first split; the fivefold ceiling itself, so no mode of such a shell is ever more than fivefold degenerate; and the Raman/infrared mutual exclusion following from the centre of symmetry. Not forced: the absolute frequencies, which need an elastic speed and a radius and are therefore imported, so no frequency comparison is made in either direction. The prediction is accordingly a symmetry prediction — degeneracies and activities, not pitches — and it is falsifiable sharply: a sevenfold-degenerate mode, or one mode active in both Raman and infrared, refutes the I_h assignment outright, and no material parameter can repair it. C60 is the worked instance, where 2A_g+8H_g = 10 Raman-active and 4T_1u = 4 infrared-active species are forced and are what experiment finds, with nothing fitted. A forward reading is offered: the degeneracy pattern is a symmetry assay that tests an icosahedral assignment independently of the imaging that produced it, and l=3 is the lowest shell at which icosahedral symmetry becomes visible in a spectrum at all. Includes a verification script that derives every row of the subduction table from character projection without assuming it.