Proposition: Accuracy is not exact causal safety

For channels on the same comparison interfaces,

Thus a total half-diamond error $\epsilon$ changes a signalling value by at most $2\epsilon$. An implemented causally embedded network has exact forbidden-pair no-signalling independently of its simulation accuracy. Accuracy alone supplies only the stated leakage bound.

Proof. For fixed initial state, reference, and interventions, compare the two output pairs by the triangle inequality. Each of the two state-comparison errors has trace distance at most the channel half-diamond distance. Take suprema and exchange $\Phi ,\Psi$. Exact containment follows separately from Theorem thm:physical.