One-generation hypercharge anomaly traces (${\mathrm{tr}}_L{Y}^3={\mathrm{tr}}_R{Y}^3=-2/9$)

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, ${\mathrm{tr}}_{L}Y=6/6-2/2=0$. 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.