The preferred magnetic flux on the C60 cage, and an exact golden gap

Lieb's flux-phase theorem settles the energy-minimising magnetic flux for a half-filled tight-binding model on a bipartite planar graph. The fullerene C60 is planar — it is the graph of a convex polyhedron — but it is not bipartite, because its twelve pentagons are odd cycles. The theorem is therefore silent about the most studied graph in its neighbourhood, and the preferred flux on the cage has been open. We answer it. Threading the canonical quarter-turn π/2 through every pentagon and nothing through the hexagons minimises the half-filled ground-state energy over the full 90-dimensional space of bond phases: all 40 random starts of a gradient search converge to it, recovering the pentagon fluxes as 0.5π and the hexagon fluxes as zero. This is the flux Lieb and Loss prove optimal for a single odd ring, acting twelve times over. Two exact statements follow. First, at quarter-turn flux every odd spectral moment of the Hamiltonian vanishes — tr H^5 = 120 cosΦ, tr H^7 = 1680 cosΦ, and so on — so the spectrum becomes symmetric about zero and half filling sits exactly at the gap centre. The quarter-turn restores, by flux alone, the particle-hole symmetry that bipartiteness would have supplied. Second, the whole spectrum is annihilated by one integer polynomial, (x^2+x-1)^8 (x^2-x-1)^8 (x^2-5)^6 (x^4-9x^2+10)^4, with the frontier levels eightfold degenerate at ±1/φ. Exactly thirty levels are negative, so the gap is not merely close to a golden number: it is 2/φ = √5 - 1 = 1.2360679774997896... exactly. Includes complete verification scripts reproducing every number.