Three fermion families

Fermion families and Pontryagin class: Topological field theory via colour symmetry extension

We show that a colour symmetry extension cancels the Standard Model’s discrete baryon-plus-lepton anomaly without sterile neutrinos. Four-dimensional anomalies classified by five-dimensional spin bordism and Pontryagin-class constraints are used to identify which anomalies admit topological-field-theory cancellation, and to explain why the minimal odd-colour, odd-family solution has three colours and three families.