= Cubic U(1) gauge anomaly
{title2=$\mathcal A_{U(1)^3}=\sum_a Q_a^3$}
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 $\sum_aQ_a=0$. If the same spectrum is restricted to a <finite cyclic group>, $q_a=Q_a\bmod n$ implies $\sum_aq_a^3=0\bmod n$ and, when imposed, $\sum_aq_a=0\bmod n$. These elementary consequences alone do not classify intrinsic discrete <gauge anomalies> or guarantee an anomaly-free integer lift.
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