[SU2×U1]×Snflavor dynamics and a bound on the number of flavors

We unify the $n$ quark and $n$ lepton generations within the standard S${\mathrm{U}}_2$ × ${\mathrm{U}}_1$ gauge model by means of the generation symmetry group ${\mathrm{S}}_n$. We show that no more than five generations of quarks and leptons can be incorporated into the theory. The resultant model always has either one or two exactly conserved multiplicative quantum numbers with eigenvalue ±1 carried by each generation. These constrain the number of nonzero weak mixing angles between generations so that, even if there exist six or eight quarks, exact Cabibbo universality can hold naturally in the light-four-quark sector. The $b$ quark must then always decay semileptonically with lepton-flavor violation, a key test.