Why quasicrystals conduct poorly: a combinatorial selection principle on K5

Aluminium, copper, and iron are excellent conductors. Alloy them into a quasicrystal and the resistivity rises by a factor of 10⁶. Anneal the alloy to perfection and it conducts worse. Remove the Drude peak: there is none. Forty years after Shechtman's discovery, no model explains these three facts together. We show that one equation does: C(n,2) = C(n,3), whose unique solution is n = 5. Five vertices force a lattice in four dimensions, a five-fold rotation forbidden in periodic 3D, and quasiperiodic order in any three-dimensional observation. The 4D lattice is a crystal — periodic, fully conducting, with ten open channels per cell. Projected into 3D, six of those channels become invisible: the observer sees only four, and those are the shallowest. The quasicrystal does not conduct poorly. We observe it poorly.