Arithmetic Admissibility and the Number of Quark-Lepton Generations

I show that if the structural integers of the Standard Model have no prime factor exceeding 3 (arithmetic admissibility), then the number of quark-lepton generations is at most 4 unconditionally, and exactly 3 if neutrinos are Majorana. The proof uses Størmer's theorem (1897) on consecutive 3-smooth numbers and the Kobayashi-Maskawa mechanism for CP violation (1973). The same arithmetic obstruction (the prime 5) that excludes a fourth generation also excludes SU(5) grand unification and the minimal supersymmetric standard model.