Cubic U(1) gauge anomaly (${\mathcal{A}}_{U(1{)}^3}={\sum }_a{Q}_a^3$)

In four spacetime dimensions, the pure local gauge anomaly for a circle group is proportional to the sum of cubed U(1) charges over left-handed Weyl fermions, counting a right-handed Weyl fermion through its left-handed charge conjugate. A consistent anomaly-free gauge theory requires this sum to vanish, absent an additional anomaly-cancelling mechanism. Cancellation of the mixed gauge-gravitational anomaly additionally requires ${\sum }_a{Q}_a=0$. If the same spectrum is restricted to a finite cyclic group, $q_a=Q_a\mathrm{mod}n$ implies ${\sum }_a{q}_a^3=0\mathrm{mod}n$ and, when imposed, ${\sum }_a{q}_a=0\mathrm{mod}n$. These elementary consequences alone do not classify intrinsic discrete gauge anomalies or guarantee an anomaly-free integer lift.