Corollary: Finite probes are not automatically a field identity

Let $P:V\to {\mathbb{C}}^m$ be a finite probe map. The equation $P(B+LC)=0$ implies the complete Ward identity only if the remaining residual is known to lie in a subspace on which $P$ is injective.

Proof. Without injectivity, a nonzero residual in $\ker P$ satisfies all the finite equations. Under the stated restriction, injectivity forces that residual to vanish.