Corollary: Uniform concentration on controlled input families

Let $\mathcal{C}$ be any family of data in Specification spec:data with

Let $L_{\ast }=CFH(1+X)$, with $C$ from Theorem thm:estimate. For a common residual state family, all bounds in (eq:concentration-budget) hold uniformly over $\mathcal{C}$ with $\Vert v\Vert$ replaced by $L_{\ast }$. The classical targets are the corresponding $o(f,h,\chi ,z)$.

Proof. Theorem thm:estimate gives ${\sup }_{\mathcal{C}}\Vert v\Vert \le L_{\ast }$. Apply Theorem thm:states.