Lemma: Marginals and remote invariance

For $\mathsf{S}\in {\mathcal{A}}_N$:

1. composability implies

   $${{\mathsf{P}}^{\mathsf{S}}}_{\psi }={\mathrm{marg}}_S{{\mathsf{P}}^{\mathsf{S}\cup \mathsf{T}}}_{\psi };$$

2. applying a unitary outside $S$ does not change ${{\mathsf{P}}^{\mathsf{S}}}_{\psi }$;

3. adjoining fresh zero-state spectators outside $S$ does not change ${{\mathsf{P}}^{\mathsf{S}}}_{\psi }$, provided the enlarged configuration fits.

Proof. For (i), sum (eq:C) over $\lambda$ and use normalization. For (ii), compare the local readout of $S$ against a remote gate followed by return, and against remote return alone. For (iii), replace that remote gate experiment by allocation followed by return. These are capacity-expressible no-signaling comparisons.