Lemma: Derivative characterization

On the unrestricted affine configuration space, ${\mathrm{supp}}_{\mathrm{fun}}F$ is the closure of the union of the supports of all first functional derivatives, over all smooth configurations and species.

Proof. If every derivative vanishes on perturbations in an open set, integration along $\Psi +\lambda k$, $0\le \lambda \le 1$, shows that every such perturbation leaves $F$ unchanged. Conversely, if a derivative pairs nontrivially with a variation, the directional function has nonzero derivative and is not locally constant. Taking all configurations, local variations, and closures gives the assertion.