Theorem: Finite-code endpoint continuity

Suppose, on a code basis,

Then $V_{\infty }$ is an isometry and

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

If ${\delta }_{\mathrm{full}}(t)\to 0$, the endpoint complete complement is constant and the endpoint radiation exactly recovers the code.

Proof. Finite-dimensionality of $C$ gives

Inner products pass to the limit, so ${V_{\infty }}^{†}{V}_{\infty }=I$. Lemma lem:bounds gives channel convergence and stability. Lemma lem:dictionaries identifies the embedded tasks with the original local tasks.

It remains to justify exact constancy when endpoint leakage is zero, even if the endpoint output is infinite dimensional. Choose states ${\sigma }_n$ with $d_{\diamond }({\mathcal{C}}_{\infty },{\mathcal{T}}_{{\sigma }_n})\to 0$. Evaluation at ${\tau }_C$ gives

Replacement-channel distance equals the corresponding state distance. The triangle inequality therefore implies

Apply Theorem thm:exact.