Theorem: Exterior-fixing no-signalling

If $T({\mathfrak{A}}_{\mathrm{out}}(Y))\subseteq {\mathfrak{A}}_{\mathrm{in}}(I)$ and $L$ is exterior fixing, then

Thus $\omega$ and $\omega \circ L$ give the same unconditioned output snapshot for every global input state. The conclusion holds jointly with every independent untouched matrix reference.

Proof. Every $T(b)$ belongs to the fixed algebra of $L$. Amplify that pointwise identity for the reference conclusion.