Theorem: Conditional endpoint approximation certificate

Suppose a compatible transport realization supplies, for every $\zeta \in \mathcal{T}$ and $n$:

1. a finite protocol and complete decoded channel ${\Gamma }_{n,\zeta }:L(M)\to L(M\otimes F_n)$;

2. an ideal complete target ${\Theta }_{n,\zeta }(X)=X\otimes {\sigma }_{n,\zeta }$;

3. matching ideal and implemented channel factorizations with at most $H_n$ steps, uniform design allowance ${\mu }_n$, encoder allowance ${\alpha }_n$, decoder allowance ${\beta }_n$, and complete step allowance ${ϵ}_n$;

4. uniform endpoint estimates

   $$\Vert a_{n,\zeta }-T_{D}a{\Vert }_{\ast }+\Vert x_{n,\zeta }-x{\Vert }_2\le {\delta }_n,|{\tau }_{n,\zeta }-\tau |\le {\omega }_n.$$

Then, for the logical reduction ${\Lambda }_{n,\zeta }={\mathrm{Tr}}_{F_n}{\Gamma }_{n,\zeta }$,

In particular, uniform anchored endpoint approximation holds if

The same channel allowance bounds all message–reference output errors. If the ideal arrival records satisfy the window condition, it also bounds the failure and out-of-window probability.

Proof. Corollary cor:assembly gives the uniform complete-channel allowance. Lemma lem:full-logical transfers it to logical channels and referenced inputs. Substituting the endpoint estimates into (eq:endpoint-error) gives (eq:certificate). Equation (eq:certificate-convergence) makes its right side tend to zero. Proposition prop:arrival gives the last statement.