Theorem: Finite-code endpoint continuity

If ${\overline{V}}_t|i⟩\to V_{\infty }|i⟩$ on a code basis, then $V_{\infty }$ is an isometry and

Complete and reduced channels converge in half-diamond distance with allowance $a_t$, and

Thus a proved limit ${\delta }_{\mathrm{full}}(t)\to 0$ gives exact endpoint recovery from radiation.

Proof. Finite-dimensionality of the code gives

Inner products pass to the limit. Lemma lem:distances gives channel convergence. Local embeddings preserve leakage by the CPTP retraction used in Theorem thm:certificate. Zero endpoint leakage implies exact constancy by Lemma lem:witness; apply Theorem thm:exact.