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<math>\Delta_{t, t + 1}(V):=\max _j\left(I\left(V_t ; X_{t + 1}^j\right)-\sum_i I\left(X_t^i ; X_{t + 1}^j\right)\right)</math>

<math>\Delta_{t, t + 1}(V):=\max _j\left(I\left(V_t ; X_{t + 1}^j\right)-\sum_i I\left(X_t^i ; X_{t + 1}^j\right)\right)</math>

When <math>\mathrm{\Delta}>0</math>, there is downward causation from macroscopic variable <math>V</math> to microscopic variable <math>X</math>.

When <math>\mathrm{\Delta}>0</math>, there exists [[downward causation]] from macroscopic variable <math>V</math> to microscopic variable <math>X</math>.

* Indicator for judging [[causal decoupling]]:

* Indicator for judging [[causal decoupling]]:

<math>\Gamma_{t, t + 1}(V):=\max _j I\left(V_t ; X_{t + 1}^j\right)</math>

<math>\Gamma_{t, t + 1}(V):=\max _j I\left(V_t ; X_{t + 1}^j\right)</math>

When <math>\mathrm{\Delta}>0</math> and <math>\mathrm{\Gamma}=0</math>, causal emergence occurs in the system and there is causal decoupling.

When <math>\mathrm{\Delta}>0</math> and <math>\mathrm{\Gamma}=0</math>, causal emergence occurs in the system and there exists [[causal decoupling]].

The reason why we can use <math>\mathrm{\Psi}</math> to identify the occurrence of causal emergence is that <math>\mathrm{\Psi}</math> is also the lower bound of unique information. We have the following relationship:

The reason why we can use <math>\mathrm{\Psi}</math> to identify the occurrence of causal emergence is that <math>\mathrm{\Psi}</math> is also the lower bound of unique information. We have the following relationship: