Theorem: Capture certificate on the stated joint input

For the radiation two-mode marginal and every admitted spectator–reference extension,

Here $\mathcal{M}$ is the capture-plus-storage channel obtained from Specification spec:capture.

Proof. Replacing the independently prepared memory by vacuum costs $p_c$, including references, by tensor-product equality and contractivity. For that preparation, capture and storage reduce to ${\mathcal{L}}_{{\tau }_{\ast }}$ on the $f$-mode, so Lemma lem:loss costs $\ell (E_f,{\tau }_{\ast })$. The comparison between the $f$-mode and the target $h$-mode costs ${\mu }_L$, by Lemma lem:modeerror. The ideal target traces out the other participating mode; no independence between that mode and $hSR$ is used. Apply the triangle inequality.