Theorem: Access-preserving endpoint certificate

Assume the package in Definition def:four. Then $V_{\infty }$ is an isometry.

If the endpoint complement is exactly constant, there is a deterministic decoder ${\mathcal{D}}_{\infty }:R_{\infty }\to C_{\mathrm{out}}$ such that

The finite-time decoder

uses only $R_t$ and satisfies

If the endpoint complement has the approximate allowance $\zeta$, one may choose ${\mathcal{D}}_{\infty }$ so that

and then

All conclusions preserve external references in the stated diamond-error sense.

Proof. Each ${\hat{V}}_t$ is an isometry. Operator-norm convergence implies

Under exact endpoint constancy, Theorem thm:exact gives an exact endpoint decoder, including its deterministic extension to the unused output subspace.

Under approximate constancy, ${\delta }_{\mathrm{full}}(V_{\infty })\le \zeta$, and Theorem thm:approx, together with attainment of the decoder minimum, gives the asserted ${\mathcal{D}}_{\infty }$.

The map ${\mathcal{D}}_t$ is CPTP and has domain $R_t$. By Lemma lem:stability,

Therefore

For the exact case take $e_{\infty }=0$. Cap the bounds at one. Every step uses complete channel distance, so the reference statement is already included.