The Minimal SU(5) Single-VEV Route Is Blocked on the C60 Code, and the Only Escape Is a Wedge of Two VEVs That Lands on the Pentagon Axis

The binary code carried by the C60 cage embeds the icosahedral rotation group A5 into SU(5). On that embedding, no single Standard-Model-preserving Higgs vacuum value can produce the color triplet that minimal SU(5) breaking needs. The adjoint representation splits so that the triplet lives entirely off the diagonal, where it breaks color and charge, while the traceless diagonal, the only place a Standard-Model-preserving value can sit, is a pure four-dimensional piece with zero projection onto the triplet. The anti-fundamental Higgs has no triplet at all. The protection is exact, by explicit projector, and it survives the whole minimal breaking layer. That is the no-go. It has exactly one escape, and the escape is forced too: the wedge of a pair of values, the hypercharge direction and the doublet value, does reach a nonzero triplet, and the vector it produces lands exactly on a five-fold sub-axis of the cage. So the Standard Model cannot be reached from one minimal-SU(5) value on this geometry. It can be reached from two, and the two are tied to the pentagon structure of the cage.