A Euclidean-ball one-step section

The exact $n$-step displacement set is

One inclusion follows because each step has ${\ell }^1$-length at most one. For the other, use coordinate steps and pad with zero steps.

The closed cone generated by the spacetime increments $(1,s)$ is

Indeed, the convex hull of $S$ is the ${\ell }^1$ unit ball. For the standard embedding, reachability is exactly membership of the event difference in $K_{\diamond }$.

Finite support rules out a finite target set being a positive-radius ball in continuous ${\mathbb{R}}^2$, but not a metric ball on a discrete slice. Moreover, Euclidean spatial balls are themselves sections of a Lorentz cone. The obstruction here is the generated polyhedral cone, not the word “ball.”