Theorem: Fixed-stencil metric obstruction

Under a fixed positive physical length-to-time normalization, the large-scale word-metric cone of a fixed finite $S$ has a cross-section proportional to $C_S$. For $D\ge 2$, it is not a nonzero Euclidean ball.

Proof. Divide Lemma lem:filling by $k$ and let $k\to \infty$. The error tends to zero. The limiting body is a finite polytope; a Euclidean ball in $D\ge 2$ has infinitely many extreme points.