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Count every right-handed field as a left-handed conjugate, which reverses its $U(1)$ charge and conjugates its non-Abelian representation. The nontrivial local anomaly-cancellation conditions areThe pure $SU(3{)}^3$ anomaly cancels because the two triplets in $Q_L$ balance the conjugates of $u_R$ and $d_R$. There is no perturbative $SU(2{)}^3$ anomaly, and the Witten SU(2) anomaly also vanishes because the three colored quark doublets plus one lepton doublet make four. The mixed gauge-gravitational anomaly is excluded at this stage as requested.

There is no perturbative $SU(2)$ gauge anomaly because both the fundamental and adjoint representations are real or pseudoreal and have vanishing cubic gauge index. The Witten SU(2) anomaly also vanishes: the $N_f$ Dirac fermions give $2N_f$ fundamental Weyl doublets, an even number, while the adjoint Weyl fermion is in an integer-isospin real representation.