= 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.
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