Proposition: Collar words and polygon closure

Collar and mixed vertex words are the affine identity. All polygons close, and matching edges have equal induced lengths.

Proof. All cell charts are globally defined affine isometries $A_i$ of the same patch; transitions $A_i{A}_j^{-1}$ telescope. The core polygons are intersections of their declared half-spaces with ${\mathcal{R}}_s$, and the collar polygons have the listed boundaries. Shared endpoints and segments are identical.