Theorem: One supported four-face certificate

The preceding family supplies a local certificate containing:

1. piecewise spacelike ordered markings and transverse incoming/outgoing central edge laws;

2. complete transverse edge data, affine charts, and exact ordered products;

3. a compact transverse collar and a compactly supported $3+1$-dimensional lift matching a fixed horizontal exterior;

4. a transition stratum with exact guard and fixed abstract location transport;

5. a decoder-correct reversible finite event implementation, with separately charged history and spectator reserves.

It is a type-A transverse incidence illustration only. It changes neither $\mu$, $\mathcal{W}$, nor $J_{\mu }$.

Proof. Lemma lem:central supplies (i). The explicit charts and Proposition prop:edge-data supply (ii). The collar construction and Proposition prop:lift supply (iii). The event definitions and Lemma lem:event-code supply (iv)–(v). The construction lies in one regular developing patch and is horizontal near its boundary. It changes no global gluing map or worldsheet data.