Proposition: Compact Ward criterion

If

then ${\mathcal{O}}_{J,q}$ is invariant under the coupled transformation for every smooth descriptor for which that transformation is considered. It descends continuously to ${{\mathcal{S}}_{r_{\ast }}}^{\mathrm{red}}$.

If invariance is required for every compact descriptor, the local identity (eq:Ward) is also necessary.

Proof. Integration by parts, with no boundary term, gives

This proves sufficiency and, by compact test separation, the stated necessity. Compact smooth pairings are continuous in the solution topology, so quotient continuity follows.