Theorem: One complete four-face certificate

In the fixed conical spacetime of deficit $\pi /2$, the capped slicing eq:cap has exactly one central adjacency exchange in $[-1,1]$. Its certificate contains:

1. spacelike, time-ordered slices and transverse incoming/outgoing edge laws;

2. explicit adapted affine charts and exact ordered local products;

3. an admitted event stratum and an exact parameter guard;

4. a compact collar matching one fixed exterior cellulation;

5. fixed abstract location transport on declared cell labels;

6. spectator-blind control under separately reserved spectator location/history storage;

7. a decoder-correct finite event permutation and exact reversible circuit with clean work uncomputed and matter unchanged.

The particle data are those in eq:frozen-observables throughout.

Proof. Items 1–4 follow from prop:core,prop:products,prop:cap,prop:collar,prop:edge-certificate,prop:exterior. The germ and transport construction and prop:geometric-rule give items 3, 5, and 6. Item 7 is prop:decoder,prop:circuit. The particle statement is prop:exterior.