Corollary: One combined preparation–input–fluctuation ledger

Suppose data $(f_{ϵ},h_{ϵ},{\chi }_{ϵ},z_{ϵ})$ converge to a fixed data set in the controls just specified and stay in a bounded input family. Let $v_{ϵ},o_{ϵ}$ be their corresponding direction and symbol, let $o_0$ be the limiting symbol, and let ${\mu }_{ϵ}$ be the law in the displaced state built from $z_{ϵ},{\rho }_{ϵ}$.

Define

Then

Hence $ϵn_{ϵ}\to 0$ proves the acceptance pair for this fixed limiting target as well.

Proof. Equation (eq:symbol-stability) bounds $|o_{ϵ}-o_0|$ by ${\beta }_{ϵ}$. Use Theorem thm:states and the Wasserstein triangle inequality through ${\delta }_{o_{ϵ}}$. Bounded inputs bound $\Vert v_{ϵ}\Vert$, while Theorem thm:input-stability gives ${\beta }_{ϵ}\to 0$.