Theorem: Derived joint transfer budget

Capture, storage, joint preparation, implemented extraction, and memory discard have referenced transfer error at most

against $({\mathcal{B}}_{\mathbf{a}}\otimes {\mathrm{id}}_{SR}){\mathcal{I}}_H$.

Proof. Apply the same implemented extraction first to the actual captured state and to the ideal target-in-memory state. Their difference costs ${\xi }_{\mathrm{cap}}$. Replace the independent register preparation at cost $p_Q$. The remaining ideal memory input is the baseline packet pair, so Lemma lem:stability and Proposition prop:extraction apply with its own tail. This ordering does not assume a geometric tail for an imperfectly captured state.