Theorem: Conditional transport certificate

Suppose uniformly over a nonempty declared target set:

where $b_{n,\zeta }-a_{n,\zeta }=({\tau }_{n,\zeta },x_{n,\zeta })$. Suppose the complete channel has design, encoder, decoder, and per-stage allowances ${\mu }_n,{\alpha }_n,{\beta }_n,{\eta }_n$, with at most $H_n$ stages. Set

Then the logical channel ${\Lambda }_{n,\zeta }={\mathrm{Tr}}_F{\Gamma }_{n,\zeta }$ obeys

for any declared comparison length $\ell >0$.

The same channel allowance controls all message-reference outputs. If the ideal complete channel has no failure or out-of-window arrival, the actual probability of that event is at most ${\epsilon }_n$.

Proof. Lemma lem:telescoping bounds the complete channel by ${\epsilon }_n$. Partial trace contracts that bound and gives the logical estimate. The geometric estimate is substitution. Finally apply the complete-channel allowance to the declared failure or arrival-record effect.