You're claiming: > the argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope

You're claiming: > the argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope to show is that the theorem is true for all n > n_0, where n_0 is some large constant. but there is no mention of this fact. the theorem is claimed for all n. They're claiming: > Theorem 7. If there is a map L which cannot be 4-coloured then only an exponentially small fraction of the maps with n edges can be 4-coloured. These claims appear mutually exclusive.