---
identifier: iit:measure/phi-s
title: System integrated information (φ_s)
vetted_by: "William G. P. Mayner"
formalism: iit-4.0-2026
see_also:
  - iit:postulate/exclusion
sources:
  - ref: iit:ref/mayner-2026
    at: >-
      Section 2 for intrinsic differentiation, intrinsic specification, and the
      updated φ_s, Eqs (12), (13) and (23); Section 3.1 for the single-unit
      example
  - ref: iit:ref/albantakis-2023b
    at: >-
      "Integration: Determining the irreducibility of a candidate system",
      Eqs (22) and (23), for the definition without the intrinsic-information
      requirement
  - ref: iit:ref/marshall-2023
---

φ_s 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 φ_s to zero, and φ_s 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, φ_s is the minimum of the integrated cause
information φ_c, the integrated effect information φ_e, 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 φ_s = 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 φ_s peaks at 0.427 where they cross, at *p* = 0.744.
A perfect copy gate scores zero.

**φ_s, φ_d, and Φ are different quantities**. φ_s belongs to a candidate system
and decides whether it exists as one whole; IIT 4.0 also calls it "small phi".
φ_d belongs to a distinction, a single irreducible mechanism, and φ_r 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 φ_s = 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
φ_s = 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 φ_s as the minimum of φ_c and φ_e 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 φ_s > 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](/current).