Corollary: A window requires a verified strict certificate

Suppose the compared family is operator-norm continuous near $t_0$, the same-model geometry and access construction are valid on an open interval $I$ containing $t_0$, and a verified endpoint decoder satisfies

where ${\kappa }_t=\Vert {\hat{V}}_t-V_{\infty }\Vert$. Then some open interval $J\subseteq I$ containing $t_0$ has the deterministic decoder errors

Proof. Continuity of ${\hat{V}}_t$ implies continuity of ${\kappa }_t$. The strict inequality therefore persists on a sufficiently small neighborhood of $t_0$ inside $I$. Apply Theorem thm:endpoint.