Fermion Modes on the C60 Cage Carry a Forced Scale Profile

The hopping operator on the truncated icosahedron, H0 = -A, admits an exact wavelet decomposition. In a multi-scale basis the operator is a radial-hopping rewriting of itself, with the same spectrum, so nothing is approximated. In that basis every mode carries a forced per-scale retention, the fraction of its weight that sits at each resolution. The smooth band edge keeps all of its weight at a single scale. The most staggered mode keeps a quarter of its weight per scale. The modes at the Fermi level, the Dirac point where the matter sector lives, keep only one sixteenth per scale. That number is forced by the honeycomb Dirac-point condition: the corner sum vanishes, so a mode's wavelet coefficient is half its central value, and the centers are one child in four, which gives a quarter of a quarter. The matter sector is therefore the most ultraviolet-localized sector on the cage, shedding fifteen sixteenths of its weight at each coarser scale. This is a computed, forced structural fact about where fermions sit in the scale hierarchy of the geometry.