The relevant distinction

A witness response is an observation of a distribution. A joint-graph construction is a distribution, or a map producing one, on a specified full test space. These coincide only if the observation is proved to determine the object.

Let $W$ be a smooth manifold and $V$ a finite-rank residual bundle. A distributional section is a continuous functional on the corresponding compactly supported dual tests. For a linear test subspace $\mathcal{H}$, define