 CKM parameter counting

An $N\times N$ unitary matrix has $N^2$ real parameters. Its real rotation subgroup has $N(N-1)/2$ mixing angles, leaving $N(N+1)/2$ phases. The $N$ up-type and $N$ down-type quark fields allow $2N$ quantum field rephasings, but their common phase acts trivially on the CKM matrix. Removing $2N-1$ phases therefore leavesFor two quark generations, there is one mixing angle and no physical CP-violating phase. For three quark generations, there are three mixing angles and one physical CP-violating phase. The count describes generic, nondegenerate quark masses and generic mixing; a vanishing mixing angle can make the remaining phase unobservable, and mass degeneracies enlarge the allowed basis changes. Three quark generations permit CP violation but do not force it.