Lemma: Restriction in a fixed experimental library

If $N\le N^{\prime }$ and $\mathsf{S}\in {\mathcal{A}}_{N^{\prime }}$, restricting its laws and residuals to inputs and experiments of peak occupancy at most $N$ gives a member of ${\mathcal{A}}_N$.

Proof. Every capacity-$N$ experiment is a capacity-$N^{\prime }$ experiment. Its state spaces, fixed gates, and observation interface are unchanged. Every required equation is therefore inherited.