Proposition: Boundary matching and particle word

Incoming and outgoing descriptions use the same boundary labels, constant spatial boundary vertices and lengths, resting boundary plane $T-H=s$, and fixed exterior transitions. The cone metric, exterior incidence, and resting particle neighbourhood are not rewritten. The particle word remains

Proof. The cap is resting near $\partial K$; all its boundary data and charts are common. A particle loop can be chosen in the unchanged axis neighbourhood. A path through the patch instead has a telescoping word determined by its endpoint charts, independent of its internal incidence.