Theorem: State-independent encoded partner

The maps ${\pi }_A,{\pi }_B,{\pi }_M$ are unital faithful $\ast$-homomorphisms and exactly intertwine the structural unitary $J$. Their logical-factor images commute pairwise. Moreover,

The partner does not commute with the whole radiation algebra: for $d\ge 2$ its image is noncommutative.

Proof. Unitary conjugation preserves multiplication, adjoints, units, and injectivity. Operators acting on different logical factors commute before conjugation and therefore after conjugation. Since ${\pi }_A(O)$ acts only on $FD$, its image lies in the radiation algebra. Choose two noncommuting matrices on $A$; faithfulness preserves their nonzero commutator.