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Glossary

System integrated information (φs)

φs is system integrated information, which quantifies the intrinsic, specific, and irreducible cause–effect power a system has over itself in its current state. It is what the exclusion postulate maximizes when it picks out a complex.

It accounts for three requirements of existence, and a system fails to exist as a whole if it fails any one of them.

A repertoire of alternatives, measured by intrinsic differentiation: the system must make alternative cause–effect states available to itself. A system whose past and future are fully determined by its present has no alternatives among which to make a difference — from its own perspective there is only the one option, so there is no difference to be made.

A specific state among them, measured by intrinsic specification: the system must raise the probability of one particular cause–effect state relative to the alternatives. A system whose states are all equally likely specifies nothing.

The two are measured against opposite extremes with the same intrinsic difference measure. Intrinsic differentiation is the distance from a maximally specific distribution; intrinsic specification is the distance from a maximally differentiated one. Neither can be traded for the other: by the principle of minimal existence, the intrinsic information of the system on each side is the smaller of the two, and the system's intrinsic information ii is in turn the smaller of its cause and effect values. So determinism and randomness both drive φs to zero, and φs is maximized somewhere between them.

Irreducibility, measured by integration: the system must specify its state as one whole rather than as independent parts. This is evaluated over the system's minimum partition — the directional partition that makes the least difference to the system's cause–effect state, found by minimizing integrated information relative to the maximum value that partition could take. That normalization is what lets the search find a system's fault lines, a bridge between two large subsets say, instead of defaulting to cuts between single units and the rest.

Taking all three together, φs is the minimum of the integrated cause information φc, the integrated effect information φe, and the system's intrinsic information ii.

For a single indivisible unit — a monad — there is nothing to partition, so all of its intrinsic information is integrated information and φs = ii. A unit that holds its state with probability p and flips with probability 1 − p makes the tradeoff concrete: intrinsic specification rises with p while intrinsic differentiation falls, and φs peaks at 0.427 where they cross, at p = 0.744. A perfect copy gate scores zero.

φs, φd, and Φ are different quantities. φs belongs to a candidate system and decides whether it exists as one whole; IIT 4.0 also calls it "small phi". φd belongs to a distinction, a single irreducible mechanism, and φr belongs to a relation. Φ ("big Phi", structure integrated information) is the sum of the φ values of the distinctions and relations composing a Φ-structure. Bare φ is used generically across all of these, so the subscript is what carries the meaning.

A purely feedforward system necessarily has φs = 0, so it cannot support a cause–effect structure and cannot be conscious. The reason is structural rather than a matter of what its units do: integrated information is defined so that φs = 0 whenever the system is not strongly connected in the graph-theoretic sense, because in that case there is necessarily a minimum partition that does not lose any information (i.e., cutting edges in the feedback direction; since there are no causal influences in that direction, the partition does not affect the system).

Before 2026

The 2023 formulation defined φs as the minimum of φc and φe alone, without the intrinsic-information requirement. Intrinsic specification was present under the name intrinsic information; intrinsic differentiation had no counterpart, so nothing in the measure required a system to provide itself with alternatives. A deterministic system could therefore carry φs > 0.

This is the sole point on which the 2026 formalism departs from 2023 — the rest of the framework, including everything above about partitions and the minimum partition, is unchanged. See what is current.

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