Proposition: A fixed cardinal stencil does not satisfy the theorem

In $D=2$, consider only cardinal hops $\pm a{e}_1,\pm a{e}_2$, each taking time $a/c$. Such a family cannot approximate every Euclidean future-null segment with vanishing macroscopic time error.

Proof. In elapsed time $T$, a cardinal path has

The Euclidean null displacement

has ${\ell }^1$-norm $\sqrt{2}cT$. Approximating that displacement requires elapsed time at least $\sqrt{2}T+o(1)$, rather than $T+o(1)$. The discrepancy is macroscopic.