Theorem: Soundness of the certificate

Suppose the finite data satisfy Definition def:certificate, with exact edge containment and target leakage ${ϵ}_{\mathsf{Phys}}(A,B)$. Then:

1. the complete implementation has the scheduled support of Theorem thm:schedule;

2. exact causal edge containment gives (eq:main-containment);

3. if the reduced full-diamond simulation error is at most ${ϵ}_B$, then

   $${\mathrm{Sig}}_{\Phi }(A\to B)\le \min \{1,{ϵ}_{\mathsf{Phys}}(A,B)+{ϵ}_B\};$$

4. certified pre-event, transition, and post-event stages compose without a new instantaneous remapping assumption;

5. a routing-and-local-gate realization satisfying Contract contract:routing also meets Target B.

Proof. The assertions are Theorems thm:schedule, thm:physical, and thm:signal-stability, Proposition prop:eventcomposition, and the elementary routing construction. Each ingredient is proved for complete channels, so entangled input references are retained.