Lemma: Flattening and segment matching

The nonresting faces have chart time zero. The identity charts retain resting chart time $2+s$. Canonical transitions match all paired endpoints and segments, preserve reverse-map consistency, and give closed developed face boundaries.

Proof. In sector coordinates,

For the new faces,

and the reflected formula holds with $\xi$. The resting charts are deliberately not recentered.

For a nonseam edge, its endpoints are the same developed points on the two adjacent planes. Their chart coordinates therefore match under ${\mathcal{A}}_g{{\mathcal{A}}_f}^{-1}$. For a seam edge, $J_0$ identifies its upper developed endpoints with the corresponding lower endpoints, with time unchanged. Inserting ${J_0}^{n_{gf}}$ gives the same matching conclusion. Affine endpoint matching matches the full segment.

Reversing the crossing component changes its sign, so A:eq:canonical gives inverse transitions. Each boundary is the ordered boundary of the supplied closed simple polygon.