Source
Mediano et al. (2022), LIT-025, “Greater than the parts”, read in full in NOTE-010. It restates Rosas et al. (2020), not held, and credits Mediano et al. (2019) with the ΦID lattice. Every formal result here comes to the record second-hand.
What was actually shown
The account (§III). A macro feature V_t is a possibly noisy function of the present microstate X_t, so it carries nothing about the future that X_t lacks jointly. It can still predict the future in a way no single part, or group of at most k parts, does on its own. V_t is causally emergent of order k iff Un⁽ᵏ⁾(V_t; X_t′ | X_t) > 0 (Eq. 3).
The theorem, restated. X_t has causally emergent features of order k iff Syn⁽ᵏ⁾(X_t; X_t′) > 0, its “emergence capacity” ([7, Thm 1]). That is a property of the dynamics alone, and needs no candidate V to be named. The capacity splits as Syn⁽ᵏ⁾ = D⁽ᵏ⁾ + G⁽ᵏ⁾ (Eq. 4): downward causation, where the feature predicts particular parts, and causal decoupling, where it predicts itself beyond the parts. A feature emergent at order k is emergent at every lower order (§IV C).
Applications (§V), reported without numbers. Game of Life particles and a flock’s centre of mass pass the practical criteria. In human fMRI, pairwise G⁽¹⁾ rises from sensory toward association cortex and is higher than in macaques. G is computed only for pairs, because no efficient estimator exists for three or more time series (fn. 5).
What could have come out otherwise. The theorem could have failed, so that a system with synergistic dynamics hosted no emergent feature. The record cannot check it, because the review states it “under relatively general assumptions” without giving them.
What this does not say
- That the measures are causal. They are neither Granger nor Pearl in themselves. They are interventional only if p(X_t′ | X_t) is a do()-distribution and faithfulness and the causal Markov condition hold, and predictive otherwise (§IV A, fn. 3).
- That emergence is independent of the observer’s choice of parts. Recoding XOR’s inputs as (X₁⊕X₂, X₁) turns 1 bit of synergy into 1 bit of unique information (fn. 4). The review calls coordinate dependence a feature of a mereological theory. It does not say which partition to choose.
- That the verdicts are robust. Syn⁽ᵏ⁾, D⁽ᵏ⁾ and G⁽ᵏ⁾ need a redundancy function (fn. 2), and the review does not say which one its applications use. The redundancy-free criteria are said to be sufficient, and are not stated.
- That it compares two descriptions of a system at different resolutions. It relates one macro feature to the parts of one description. Varley and Hoel say it lacks a macro–micro comparison; the review calls that a misunderstanding. NOTE-010 finds each right about a different thing.
- That synergy in association cortex supports abstract reasoning. The review says the evidence “suggests that it may”. The brain results were preprints at the time.
Connections
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For three or more parts the redundancy function cannot be fully principled. On the Williams–Beer lattice no redundancy function has both the identity property and nonnegative atoms from three sources up (THEORY-713), so a three-part verdict depends on which axiom is dropped (CLAIM-343). For two parts the bivariate measures are consistent.
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Effective-information emergence (THEORY-222) is a different formal object under the same name: interventional, and comparing two models of a known mechanism. NOTE-010 gives a system on which the two can part: the two-bit permutation (X¹, X²) → (X¹⊕X², X¹) has no EI gain at any macroscale, but under the minimum-MI redundancy Syn⁽¹⁾ = 1 bit. With a redundancy function that has the identity property, Syn = 0 and they agree. So whether they disagree depends on the redundancy function, and no
rivalsedge is declared. -
Synergy bias across scales (THEORY-581) is Varley and Hoel’s cross-scale use of the same ingredient. For two elements, Syn⁽¹⁾ is the top atom of the lattice their B_syn summarises.
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Local values (THEORY-268). Varley shows that the expected ΦID synergy can be positive while its local value is negative at particular transitions, so emergence on this account belongs to the ensemble.
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Applied to consciousness (THEORY-240). Luppi et al. use a synergy-defined workspace, built on ΦID, to locate where integrated information falls under propofol.
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Dynamical independence (THEORY-257) is the record’s other criterion for a macro variable fixed by the dynamics. Both papers name it. The relation is now worked out for one-step criteria (THEORY-764, from the primary paper, read 2026-10-20 in NOTE-987). Ψ is a whole-minus-sum term minus the one-step dependence, whose vanishing implies Barnett and Seth’s independence. Neither condition implies the other, and both can hold with no interaction between the parts.
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Interventional individuality (THEORY-424). Bourrat makes the same associational-versus-interventional point about Krakauer’s measures, uncited here.
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The primary paper (2026-10-20). Rosas et al. 2020 (LIT-1244, NOTE-987). Three of its findings bear on this account. Its applications compute no ΦID atom: they use only the redundancy-free sufficient criteria Ψ and Γ. Syn = D + G holds by definition. And its Proposition 1, that Ψ > 0 shows a feature emergent, proves only that the system has synergy. The reading gives a counterexample under MMI, a parity of two self-copying bits, with Ψ = 1 bit and no unique information.