Example: A concrete failed containment test

Let

It is gauge invariant but anticommutes with $F$. Thus $W\notin \mathfrak{A}(\{1,2\})$. Define a binary readout by

The global density matrices

are positive and normalized. Every element of $\mathfrak{A}(\{1,2\})$ is block diagonal in $f$, whereas $W$ is off diagonal, so ${\rho }_{+}$ and ${\rho }_{-}$ have equal regional snapshots. Their output laws under $T_W$ are respectively ${\delta }_{+}$ and ${\delta }_{-}$. This is an exact witness for the failure of containment.