Proposition: Coupled Ward identity

If

then ${\mathcal{O}}_{J,c}$ is invariant under the coupled transformation for every smooth $\xi$ for which that transformation is considered. It defines a continuous linear functional on ${{\mathcal{S}}_{\mathrm{C}}}^{\mathrm{red}}$.

Proof. Compactness permits integration by parts:

The compact pairings are continuous on every solution stage and therefore on its inductive limit. Invariance gives descent through the quotient topology.