The metric Euler equation

The switch $g$ multiplies the matter source and nothing else; $\partial g$ does not appear in $\mathsf{E}$. It appears in the divergence of $\mathsf{E}$, which is the content of E1's identity. The scalar Euler operator ${\mathsf{E}}_{\varphi }:=\delta S/\delta \varphi =\delta {S^{\mathrm{sc}}}_g/\delta \varphi$ is X-A's switched operator, with ${\mathsf{E}}_{\varphi }=-P_m\varphi +O(\kappa )$.