Proposition: Closure and collar holonomy

Every collar or mixed core–collar vertex word is the affine identity. All cells close and all matching edges have identical induced lengths.

Proof. Every transition is $A_i{A}_j^{-1}$ for globally defined affine isometries of the patch, so closed words telescope in the affine group.

Core cells are intersections of their affine dominance half-spaces with ${\mathcal{R}}_s$. The collar polygons are the explicitly listed simple polygons and their reflections. Adjacent cells share the same developed endpoints and segment. Affine isometries preserve its induced length.