Lemma: Complete branch certificate

Every successor in (eq:example-successors) belongs to ${\mathsf{G}}_B$. Its slice is embedded and spacelike, and advances in spacetime time by $\tau +a\delta >0$.

The complete affine peripheral word is

on every label, with the same marked stationary axis. Location transport is the identity on the persistent cell labels. The resource decreases by exactly one on every active edge.

Proof. The inequalities and parity in (eq:example-labels) are preserved by $k\mapsto k+1$, $q\mapsto q+a$. A horizontal graph has induced metric equal to the spatial cone metric and is embedded. Formula (eq:slice-height) gives the time increment.

The upper seam maps to the lower seam by the rotation $R(\pi /2)$, which fixes the time axis pointwise. A positively oriented primitive meridian crosses that seam once; its angular lift advances by $3\pi /2$. Its affine word is therefore $J$. Moving the adapted horizontal chart by $-T(k,q)e_0$ commutes with $J$, so there is no induced affine time translation. The cellulation and collar are unchanged spatially, giving identity cell-label transport. Finally, $B-(k+1)=B-k-1$.