Local-to-global lemma for simplicial grid order

If initial segments of the simplicial order minimize vertex boundary in $[k{]}^2$, then they do so in every $[k{]}^d$. Inductively compress all coordinate sections. The section formula for a grid neighbourhood shows that no compression enlarges the boundary. Among boundary-minimizing compressed sets choose one with minimum coordinate-sum weight. Any departure from an initial simplicial segment produces an inversion in a two-coordinate face; the two-dimensional theorem replaces that face by its initial segment without enlarging the boundary and strictly lowers the weight. Hence no inversion remains and the set is an initial simplicial segment.