The C60 Geometry Forces Chirality to Be an Undetermined, Unobservable Input

The truncated icosahedron forces almost everything about its own structure. There is one thing it cannot fix, and it cannot fix it for a reason that is itself a theorem. The central inversion of the cage has no fixed point: it pairs the sixty vertices into thirty antipodal pairs and sends no vertex to itself. From that single fact two conclusions follow at once. First, there is no global orientation the geometry can carry, so a handedness has to be put in from the outside, by whatever vantage reads the cage. Chirality must be an input. Second, because the adjacency matrix is real, the system is time-reversal symmetric, and the net Chern-Simons index is zero independent of the geometry, so the two choices of hand are not distinguished by any observable. Which hand is chosen carries no measurable consequence. So the one degree of freedom the geometry leaves open is both forced to exist, by the absence of a fixed point, and forced to be invisible, by the reality of the operator. Handedness is an inserted posit the cage can neither determine nor detect.