Extended Data Fig. 1: Integrated information from information decomposition.

(a) Partial information decomposition distinguishes information that is uniquely provided by variable X or by variable Y; redundant information that is equally available from either variable; and synergistic information, which is only available when considering both variables jointly, but not either of them in isolation. (b) Information decomposition enables tracking how information from the system’s past to its future is carried by its constituent elements, corresponding to each of the 4×4 = 16 combinations of redundancy, X-unique information, Y-unique information, and synergy. (c) Naïve sum of the parts. Some information can be obtained by considering element X alone, without reference to any other parts of the system. This information corresponds to the 4 possible combinations of redundant information and X-unique information across past and future. Likewise, some information can be obtained by considering element Y alone, given by the combinations of redundancy and Y-unique information. However, if we simply sum the information that can be found in X without reference to Y, and the information that can be found in Y without reference to X, then this ‘naïve sum of the parts’ will double-count the information that is redundantly present in both X and Y across past and future (persistent redundancy). (d) Integrated information is the information that is present in the system as a whole, over and above the sum of the information provided by each of the parts. However, attempting to quantify this ‘whole minus sum’ by subtracting the naïve sum of the parts from the total information flowing between past and future of the system, yields the original measure of integrated information from Balduzzi and Tononi (Φ₂₀₀₈), which has well-known conceptual difficulties including negative values for redundancy-dominated systems (Mediano et al., 2025). If instead the proper sum of the parts is used (that is, without double-counting the persistent redundancy), we obtain the revised measure of integrated information from (Mediano et al., 2025), Φ_R. In turn, this means that we can re-express Φ₂₀₀₈ as the balance (difference) between integrated information (Φ_R) and the persistent redundancy.