Proposition: Shared boundary data cannot be independently changed

Suppose $I,J$ are disjoint regions and $z\in \mathfrak{A}(I)\cap \mathfrak{A}(J)$. Every internal Kraus device localized in $I$ fixes $z$. In the theta graph, no device that fixes $\mathfrak{A}(\{3\})$ can induce the sector-swap automorphism

on $\mathfrak{A}(\{1,2\})$.

Proof. Local commutativity implies that each Kraus operator in $\mathfrak{A}(I)$ commutes with $z\in \mathfrak{A}(J)$. Hence

For the theta graph, the proposed swap sends $F$ to $-F$, whereas exterior fixation requires $F$ to remain $F$.