Realization before matching

A one-sided incidence germ here means the restriction of a realized family to an open parameter interval adjacent to the event, together with its labelled limiting data. Positivity is required on that open epoch, not for a collapsed outgoing face at the event itself.

For each face $f$, let ${\mathcal{A}}_f\in {\mathrm{ISO}}^{+}(2,1)$ be its adapted chart change from the appropriate cut-space development. For an oriented overlap component from $f$ to $g$, let $n_{gf}\in \{-1,0,1\}$ denote its signed seam crossing. Chart paths retain their individual overlap components. The prescribed transition is

Holonomy alone does not determine these individual maps: the realization, adapted charts, and specific overlap component are also required.