Lemma: Relative action symmetry

On $U$, $sI=0$ as a local variational class.

Proof. The curved densities are natural under $sq=\kappa {\mathcal{L}}_{c}q$ and $s\varphi =\kappa c\partial \varphi$:

The background-only subtraction is $s$-inert. The resulting variation is a divergence. Theorem thm:kinematic gives $s(s{\Psi }_{\bar{q}})=0$.