= One-generation hypercharge anomaly traces
{title2=$\operatorname{tr}_LY^3=\operatorname{tr}_RY^3=-2/9$}
In normalization $Q=T_3+Y$, one <Standard Model> generation has left quark and lepton doublet hypercharges $1/6$ and $-1/2$, and right singlet hypercharges $2/3,-1/3,-1$. Including colour multiplicity, $\operatorname{tr}_LY=6/6-2/2=0$. On every left doublet $T_3^2=I/4$, and all right fields have $T_3=0$, so the mixed weak-hypercharge <gauge anomaly> vanishes. The cubic left trace is $6(1/6)^3+2(-1/2)^3=-2/9$, and the right trace is $3(2/3)^3+3(-1/3)^3+(-1)^3=-2/9$. Their difference is zero, proving generation-by-generation cancellation of these two anomalies.
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