Theorem 3.2

Theorem 3.2 (Carrier cube). Filling every Boolean family of independent temporal slides gives the rooted CAT(0) order-ideal cube of Proposition 3.1. For a rectangle with $r$ rows and $c$ columns it 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.