Partitioned channels and scheduled support

Let $d$ be an integer-valued graph metric with finite balls. At tick $s$, a partition ${\mathcal{P}}_s$ consists of finite blocks of diameter at most one. The update is

All outputs remain inside their block. On a countable substrate this product is understood through its action on finite observables and compatible finite-region states.

For finite terminal region $Y$ at tick $T$, set