Proposition: Channel commutation counterexample

Pairwise commutation of CPTP maps, even together with a valid finite support ledger, does not imply algebraic microcausality of arbitrarily assigned observable regions.

Proof. On one qubit let

Since $XZ=-ZX$, their conjugation channels commute: ${\Phi }_X{\Phi }_Z={\Phi }_Z{\Phi }_X$. Nevertheless the observables $X$ and $Z$ do not commute.

The one-site schedule satisfies Proposition prop:schedule. If two proposed region names both expose this same site algebra, channel commutation does not make the region algebras commute. The missing requirement is an observable localization compatible with the region relation.

Even genuine disjoint-site tensor locality is insufficient to supply circle covariance, a vacuum representation, positive-energy rotations, interval isotony, or the other conformal-net axioms. Those data are absent from the finite schedule.