Proposition: Square certificate

The update is a certified $(G)$-rule preserving the metric, marked resting slice, and vertex monodromies. Particle deficit and rapidity remain $\pi /2$ and $0$.

Proof. Both triangulations cover the same square and boundary. Every internal transition word is $(I,0)$. The particle word remains $(R(\pi /2),0)(I,0)$, with axis and resting normal $e_0$.