Theorem: Endpoint continuity with the access condition retained

Suppose

for every code basis vector under access-preserving identifications. Then convergence is in operator norm, $V_{\infty }$ is an isometry, and, with

Thus a proved limit ${\delta }_{\mathrm{full}}(t)\to 0$ implies ${\mathcal{C}}_{\infty }={\mathcal{T}}_{\sigma }$ and exact endpoint radiation recovery.

Proof. For a finite code,

Inner products pass to the limit, proving the isometry property. Lemma lem:isometry bounds the complementary channel difference by $a_t$. Distance to a fixed set of replacement channels is one-Lipschitz. Lemma lem:embedding identifies the finite-time leakage.

If ${\delta }_{\infty }=0$, choose replacement states ${\sigma }_k$ with channel distance tending to zero. Evaluation on ${\tau }_C$ gives ${\sigma }_k\to {\mathcal{C}}_{\infty }({\tau }_C)$ in trace distance. The replacement channels converge as well, so ${\mathcal{C}}_{\infty }$ is exactly constant. Apply Theorem thm:exact.