Theorem: Fixed-code limit with an explicit rate

Fix $C$, common separable output spaces $R_{\ast },B_{\ast }$, and factorwise embeddings of a sequence $V_{\ell }$. Suppose the embedded isometries ${\overline{V}}_{\ell }$ converge in operator norm to $V_{\ast }$, with

Then $V_{\ast }$ is an isometry, and

The recovery and leakage diagnostics differ from their limit values by at most $\min \{1,{\beta }_{\ell }\}$.

Proof. Inner products pass to the limit, so $V_{\ast }^{†}{V}_{\ast }=I_C$. Apply Lemma lem:stability, including invariance under the local embeddings.