Proposition: Geometric separation of propagated supports

Let two initial regions $X_0,Y_0$ satisfy $d(X_0,Y_0)>2r\tau$. Propagate each region forward by adjoining all outputs of every block meeting its current support. Up to tick $t_0+\tau$, the two propagated families are disjoint, and no update before that terminal tick meets both families.

Proof. After $j$ ticks, each family lies within distance $rj$ of its initial region. If a block at step $j<\tau$ met both families, the triangle inequality would give

a contradiction. At the terminal tick, a common site would give $d(X_0,Y_0)\le 2r\tau$, also a contradiction.