A finite local execution model

Let $\Lambda$ be a finite set of static sites in $U$, with finite-dimensional site spaces $H_a$. Fix times

At each tick $m$, a partition ${\mathcal{P}}_m$ of $\Lambda$ is supplied. Each block obeys

A tick is one complete product channel

All quantum ports, records, controller memories, and resource wires that are available during execution must be included in the finite network. An external reference has no execution wire.

The access contract is stroboscopic: input and output access is specified at tick boundaries. No continuous-time Hamiltonian or within-tick apparatus dynamics is asserted. A two-site gate is not treated as an instantaneous equal-time interaction. Its dependencies run from input ports at $T_m$ to output ports at $T_{m+1}$.

For $Y\subseteq \Lambda$, write

For a terminal cut $n$, define backward sets by