A coupled Ward identity and an on-shell-zero candidate

For comparison, consider the triangular classical system

It is preserved by

but arbitrary such transformations do not preserve zero response data at the initial surface.

For compact smooth $J,c$, the joint pairing

is invariant under all compact gauge parameters if and only if

This follows directly by integration by parts.

For any $\psi \in {C_c}^{\infty }(M)$, set

The divergence identity proves (eq:jointWard), but

on every solution of the triangular system. Thus this immediate compact joint construction is on-shell zero. A nontrivial coupled observable would require a separately specified solution space and quotient; the Ward identity alone does not supply one.