Definition V.1 — Causal influence

Let $\delta\Psi_{\mathcal{A}}$ be a localized perturbation introduced in region $\mathcal{A}$ at time $t_0$. We say $\mathcal{A}$ causally influences region $\mathcal{B}$ at time $t>t_0$ if the following holds:

1. A statistically reproducible response $\Delta \mathcal{O}_{\mathcal{B}}(t)$ occurs in $\mathcal{B}$, for a fixed diagnostic observable $\mathcal{O}$ used in phase classification.
2. The response is phase-coherent in the sense that it survives:

• repeat trials (seed variation),
• grid refinement,
• horizon extension $T\mapsto kT$ with $k\in\{1,2,4\}$.

If any of these tests fail, the apparent influence is classified as transient and is not counted as causal transmission.