Theorem: Fixed-decoder endpoint certificate

Under Definition def:endpoint, $V_{\ast }$ is an isometry and the embedded complete and reduced channels are within $\min \{1,{\nu }_t\}$ of their endpoint channels.

If ${\mathcal{E}}_{\ast }$ is exactly constant, one fixed endpoint decoder ${\mathcal{D}}_{\ast }:R_{\ast }\to C$ gives local finite-time decoders ${\mathcal{D}}_t={\mathcal{D}}_{\ast }{\mathrm{Ad}}_{j_t}$ with

For a finite endpoint with allowance $\zeta$, the same construction has error at most

For the finite-proxy endpoint option, a fixed extended proxy decoder has error at most

A specified endpoint proxy decoder instead gives $\min \{1,{\nu }_t+b_{\ast }+d{r}_{\mathrm{dec},\ast }\}$.

If the exact complete residual is pure, the endpoint radiation is an isometric encoding. If it is mixed, the syndrome normal form applies.

Proof. Operator-norm convergence preserves $V_{\ast }^{†}{V}_{\ast }=I_C$. Lemma lem:stability gives channel convergence. For exact constancy, Theorem thm:exact supplies ${\mathcal{D}}_{\ast }$. For a finite approximate endpoint, (eq:KSW) and finite attainment supply an endpoint decoder. For the proxy option, use Theorem thm:certificate. In each case, postprocessing contraction adds at most ${\nu }_t$. The decoder ${\mathcal{D}}_t$ acts only on $R_t$. Rank one makes the syndrome space one dimensional.