Proposition: Finite probes do not certify exact repair

Let $P$ be a finite probe map on the combined defect space. A tested residual $P(R)=0$ implies $R=0$ only with an independently proved injectivity or coverage condition on the allowed residual space.

Proof. Without that condition a nonzero element of $\ker P$ is invisible. For example, $P(x,y)=x$ misses $(0,1)$. With injectivity on the allowed residual space, the implication follows immediately.