Attainability and robust propagation

Thus “no reachable intermediate event on the same ray” cannot define null propagation.

A viable geometric operational test instead requires a topology and a continuum of controllable endpoints. Let $\mathcal{R}\subset V$ be the reachable displacement set, including zero. Call $v\ne 0$ robustly reachable when an open neighborhood of $v$ is contained in $\mathcal{R}$, and sharp when it is reachable but not robustly reachable. If $\mathcal{R}=K$ is proper, the sharp set is exactly $\partial K\setminus \{0\}$. Applying this definition directly to a sparse event sample would generally make every displacement nonrobust; passing to a completion is an additional assumption.