Proposition: Vacuum and the limits of emptiness

The unrestricted jet family of the selected action is not empty. The family with $\gamma =0$ and $\varphi =0$ is a solution.

Proof. The flat Einstein tensor and scalar stress vanish. Both $J_3$ and ${{\mathcal{D}}_{\sigma }}_{\epsilon }$ are quadratic in the metric field, so their values at $\gamma =0$ vanish. The scalar Euler expression also vanishes.