What coupling order requires

The factor $\kappa {\mathsf{X}}_1[\gamma ]$ has one metric leg. This observation alone does not eliminate a cubic Einstein vertex: that vertex has coefficient $\kappa$, so the two factors have total coupling degree two before any additional graph requirements are specified. No rooted-tree topology is selected or calculated here.

More importantly, an action identity is not a graph selection rule. The nonlinear classical identities require the relevant Einstein coefficients. The Fierz–Pauli-only alternative is explicitly audited in Section sec:quadraticfailure.