One parity per Galois orbit on the C60 cage, and the two golden points where it fails

Parity under the central inversion is constant on every Galois orbit of the C60 adjacency spectrum. The even and odd characteristic polynomials are (x - 3)(x + 2)⁴(x - 1)⁹(x² + 3x + 1)³(x² - x - 3)⁵ and (x² + x - 4)⁴(x² + x - 1)⁵(x⁴ - 3x³ - 2x² + 7x + 1)³, coprime over Q, so each of the eight rational irreducible factors lies wholly in one sector and parity is a property of the minimal polynomial rather than of the individual eigenvalue. Two things follow. The golden levels are two orbits and not one family with an exception: x² + x - 1, with roots 1/φ and -φ at multiplicity five, is odd, while x² + 3x + 1, with roots -1/φ² and -φ² at multiplicity three, is even, and no inversion-even operator couples them at any order. And parity is the finest label that survives conjugation, because the two three-dimensional icosahedral characters are irrational and every orbit carrying one carries the other in equal number. Here x² + 3x + 1 splits over Q(√5) into the T₁ branch x + 1/φ² and the T₂ branch x + φ², and the quartic splits as (x² - φ²x - 1/φ²)(x² - 1/φ²x - φ²), carrying T₁ twice and T₂ twice, with exact five-fold characters 1 + √5 and 1 - √5. Constancy is a fact about this graph and not a theorem about free involutions. Along the inversion-symmetric weighting A_ph + tA_hh it fails at exactly two positive values, 5φ/2 and 5φ³/3, where one level is sixfold with three even and three odd dimensions and another sevenfold with four even and three odd. It fails on 13 of 300 random inversion-symmetric cubic cages of the same size.