Six orbit-polarization classes on the twelve-station icosahedral topology, and a gapped scalar mode

The twelve pentagonal faces of C60, equivalently the twelve icosahedral vertices, carry a station topology whose 144 ordered pairs fall into four orbits of I_h: diagonal, adjacent, distance-two and antipodal. Adding polarization refines four into exactly six, and the refinement is forced rather than assigned.

The engine is each orbit's own stabilizer. A mode labelled by a pair orbit is invariant under the stabilizer of its pair, and that single requirement decides which polarizations survive. For the antipodal pair the stabilizer is C5v: the axis is invariant so longitudinal survives, while the transverse plane carries the two-dimensional irreducible E1 and therefore contains no invariant vector at all — the antipodal orbit is longitudinal-only, not by stipulation but because there is nothing for a transverse mode to be. The diagonal pair is a single point with no separation vector, so it is scalar-only. Adjacent and distance-two pairs have stabilizer of order two, a single mirror containing the axis, leaving one invariant transverse direction and admitting both. 1+2+2+1 = 6.

We also give both twelve-station permutation representations as induced representations from the site group, Ind(A1) = A_g+H_g+T1_u+T2_u and Ind(A2) = A_u+H_u+T1_g+T2_g, exact parity partners spanning 24 dimensions. Linearising a forced cubic potential about equilibrium gives a single massive Klein-Gordon mode of scalar polarization, gapped and causal; the gap's existence is forced by the curvature at equilibrium, its numerical value is not, and it is stated symbolically only.

One correction to earlier drafts of this material: the longitudinal electric field is the non-radiating Coulomb near-field fixed by Gauss's law. Nothing was dropped from Maxwell's equations, and any claim that a physical radiating mode was discarded is withdrawn.