Proposition: Fixed-stencil directional obstruction

Suppose the available one-tick velocities at every refinement lie in one fixed finite set $V\subset c{\overline{B}}_2(0,1)$. If all positions are obtained by concatenating those steps, then every positive-time endpoint velocity lies in

For $d\ge 2$, these permissions cannot approximate every point of the Euclidean speed-$c$ sphere at its null travel time.

Proof. The endpoint velocity of an $n$-step route is the arithmetic mean of its $n$ step velocities, hence belongs to $P$.

The finite convex hull $P$ is a polytope contained in the Euclidean ball. It cannot contain the entire sphere when $d\ge 2$, since that would force it to contain the ball and hence equal the ball. Choose a unit vector $u$ with $cu\notin P$. If endpoints and positive arrival times approached $(ru,r/c)$, their endpoint velocities would approach $cu$, contradicting closedness of $P$.