Definition: Perturbation pairs

For a finite horizon, two runs form an $x$-perturbation pair when they share the schedule, their initial snapshots agree on every finite region not containing $x$, and their channels agree on every block not containing $x$.

The site $x$ signals to $(A,\tau )$, where $\tau \ge 1$, if such a pair produces different laws for a common readout at $(A,t_0+\tau )$. Define

with supremum $0$ if there is no positive-distance signalling.