Lemma: Support through first derivatives

For a differentiable functional on the unrestricted affine smooth configuration space, its functional support is the closure of the union of the supports of its first-derivative distributions over all configurations and all field components.

Proof. If the functional is independent of perturbations supported in an open set $U$, every first derivative annihilates such perturbations. Conversely, if every first derivative annihilates them at every configuration, then for a perturbation $\eta$ supported in $U$,

This is exactly absence of functional dependence in $U$. Taking the complement of the union of such open sets proves the support characterization.