Proposition 8.5

Proposition 8.5 (Carrier-surgery CAT(0) geometry). For a fixed unary distance-one final figure and history length, the temporal slide graph on strong/garbage chambers is the undirected Hasse graph of the order-ideal lattice of the rectangle above. Filling every Boolean family of independent slides gives the CAT(0) cube complex $X(P)$. If the rectangle has $r$ rows and $c$ columns, then the complex has $rc$ hyperplanes, dimension $\min(r,c)$, edge distance $$d(t,s)=\sum_{i=1}^{r}|t_i-s_i|,$$ and coordinatewise-median carrier tuples. In particular, its diamond-filled two-skeleton is simply connected.