Theorem: Conditional outgoing recovery

Under Definition def:endpoint, if $D_0$ is chosen with the information-theoretic bound furnished by Theorem thm:leakage, then

For a named effect or bounded observable, the corresponding bounds are those of Theorem thm:effect. If the inputs hold on a joint confidence event, the conclusions hold on that event.

Proof. The complete complementary bound gives a model decoder with error at most $\min \{1,d\sqrt{2\delta }\}$. Contraction and the triangle inequality give

Apply the channel cap one and Theorem thm:effect.