NOTE-987: Rosas, Mediano et al. — Reconciling emergences

Contribution

It is the source the record has so far known only through the authors’ review (LIT-025, NOTE-010). It defines causal emergence of a supervenient feature by partial information decomposition, proves that a system hosts such a feature exactly when its parts carry synergy about their own future, and splits that synergy, through ΦID atoms, into downward causation and causal decoupling. It states the axioms a redundancy function must meet for the theory to hold, which the review did not, and it gives three redundancy-free practical criteria, Ψ, Δ and Γ. It applies Ψ (and twice Γ) to the Game of Life, Reynolds boids and macaque ECoG.

Key insight

Supervenience is a relation at one time and predictive power runs across time, so a feature fully fixed by the present microstate can still carry information about the future that no small group of parts carries separately. “Emergence” here means that kind of irreducibility, relative to a fixed partition into parts. It does not mean that a coarse description predicts better than the fine one. By the data-processing inequality I(V_t; V_t′) ≤ I(X_t; X_t′) it never can, which is the paper’s own argument against reading observational mutual information Hoel’s way.

Assumptions

• A fixed partition X_t = (X¹_t, …, Xⁿ_t), fully observed, with Markovian dynamics p(X_t′ | X_t) and two time points t < t′. “Emergence in the context of our theory always refers to ‘emergence with respect to a given microscopic partition.’” The Markov and full-observation conditions are named as limitations in the discussion.
• Supervenience (Definition 1). V_t − X_t − X_t″ is a Markov chain for every t″ ≠ t. Deterministic V_t = F(X_t) and F(X_t) plus independent noise qualify; features computed over several time points generally do not.
• Axioms on the PID (S1 Appendix §2). The theory holds “for any PID” that meets them:
  • deterministic equality (if one source is a function of another, the decomposition reduces to that of the smaller set);
  • non-negativity, Syn⁽ᵏ⁾ ≥ 0 and min{I(Z;Y), I(Z;Y|X)} ≥ Un⁽ᵏ⁾(Z;Y|X) ≥ 0 (see Limitations: the second half is misstated);
  • source data-processing for Un⁽ᵏ⁾ along W − Z − (X, Y);
  • for D⁽ᵏ⁾, a ΦID bound Σ_{|α|=k} Syn⁽ᵏ⁾(X_t; X^α_t′) ≥ D⁽ᵏ⁾ ≥ Syn⁽ᵏ⁾(X_t; X^β_t′) for every |β| = k, assumed, not derived;
  • for the practical criteria, whole-minus-sum, Syn⁽ᵏ⁾(X;Y) ≥ I(X;Y) − Σ_{|α|=k} I(X^α;Y), and target data-processing, Syn⁽ᵏ⁾(X;U) ≤ Syn⁽ᵏ⁾(X;Y) along X − Y − U.
  • The appendix says the minimum-mutual-information (MMI) PID and its ΦID meet them all (“direct to check”; not shown).
• For an interventional reading, p(X_t, X_t′) must come from interventions; otherwise the quantities are Granger-type predictive ones. The paper takes no stand (“without taking a rigid stance on a theory of causality”).

Key results

• k-th-order atoms (Lemma 2, S1 §1). With S⁽ᵏ⁾ the antichains all of whose sets have more than k sources, U⁽ᵏ⁾(β) those containing β with every other set of size > k, and R⁽ᵏ⁾ those with two or more sets of size ≤ k, I(X;Y) = Red⁽ᵏ⁾ + Syn⁽ᵏ⁾ + Σ_{|β|≤k} Un⁽ᵏ⁾(X^β; Y | X^{−β}). The proof shows the three index sets partition the lattice, so it is exact for any PID. Checked.
• Definition 2. V_t is causally emergent of order k iff Un⁽ᵏ⁾(V_t; X_t′ | X_t) > 0, the unique information computed in the PID with V_t added as an (n+1)-th source.
• Lemma 1. An order-1 emergent feature needs n ≥ 2 and is no deterministic function of a single part.
• Theorem 1. X_t has a causally emergent feature of order k iff Syn⁽ᵏ⁾(X_t; X_t′) > 0. Corollary 1: Un⁽ᵏ⁾(V_t; X_t′ | X_t) ≤ Syn⁽ᵏ⁾(X_t; X_t′) for every supervenient V_t, the “emergence capacity”.
  • The forward direction takes V_t = X_t: by Lemma 2 (the microstate as an extra source), Un⁽ᵏ⁾(X_t; X_t′ | X_t) = Syn⁽ᵏ⁾(X_t; X_t′). The authors say so: the system has emergent behaviour “if and only if the system as a whole seen as a feature (i.e. V_t = X_t) is causally emergent”, and the microstate “in most cases also carries non-interesting information”.
  • So the bound of Corollary 1 is attained by the microstate. Maximizing emergence over features selects no coarse-graining.
• Taxonomy (Eqs. 4–7). Each PID atom splits into ΦID atoms I∂^{α→β}.
  • D⁽ᵏ⁾ := Σ_{α∈S⁽ᵏ⁾, β∈A∖S⁽ᵏ⁾} I∂^{α→β}: synergistic sources to targets that some group of ≤ k future parts holds.
  • G⁽ᵏ⁾ := Σ_{α,β∈S⁽ᵏ⁾} I∂^{α→β}: synergistic sources to synergistic targets.
  • Syn⁽ᵏ⁾ = D⁽ᵏ⁾ + G⁽ᵏ⁾ (Eq. 7). This holds by the definitions, since every target node lies in S⁽ᵏ⁾ or outside it. The review’s “exhaustive” is a tautology of the bookkeeping, not a finding.
• Downward causation (Definition 3, Theorem 2). V_t exerts it at order k iff Un⁽ᵏ⁾(V_t; X^α_t′ | X_t) > 0 for some |α| = k. A system admits such features iff D⁽ᵏ⁾ > 0. The proof rests on the assumed ΦID bound above.
• Causal decoupling (Definition 4, Theorem 3). V_t exhibits it at order k iff Un⁽ᵏ⁾(V_t; V_t′ | X_t, X_t′) > 0: information V_t and V_t′ share that no group of ≤ k parts at t or t′ has. Pure decoupling: emergent and Un⁽ᵏ⁾(V_t; X^α_t′ | X_t) = 0 for every |α| = k. A system admits decoupled features iff G⁽ᵏ⁾ > 0 (via Corollary 2, which needs “a natural multi-target extension” of deterministic equality), and is perfectly decoupled if G⁽ᵏ⁾ > 0 and D⁽ᵏ⁾ = 0.
• Practical criteria (Eq. 10, S1 Eq. D1), k = 1:
  • Ψ(V) := I(V_t; V_t′) − Σ_j I(X^j_t; V_t′);
  • Δ(V) := max_j [ I(V_t; X^j_t′) − Σ_i I(X^i_t; X^j_t′) ];
  • Γ(V) := max_j I(V_t; X
  • For general k, the sums and maxima run over groups of k parts.
• Proposition 1 as stated: Ψ > 0 is sufficient for V_t to be causally emergent, Δ > 0 for V_t to exert downward causation, and Ψ > 0 with Γ = 0 for causal decoupling.
  • What the proof shows is weaker (see Limitations). Ψ(V) ≤ I(X_t; V_t′) − Σ I(X^α_t; V_t′) ≤ Syn⁽ᵏ⁾(X_t; V_t′) ≤ Syn⁽ᵏ⁾(X_t; X_t′). So Ψ > 0 gives the system emergence capacity, and with Theorem 1 the existence of some emergent feature.
  • These are sufficient conditions only. Ψ ≤ 0 is “inconclusive”, and redundancy among the parts is counted up to n times.
• Synergistic channels (Eqs. 11–14). Syn⁽ᵏ⁾⋆ := sup I(V; X_t′) over channels p(V | X_t) whose output is independent of every k-group of parts. G⁽ᵏ⁾⋆ uses such channels at both ends, and D⁽ᵏ⁾⋆ := Syn⁽ᵏ⁾⋆ − G⁽ᵏ⁾_⋆ ≥ 0. This is the one specific decomposition the paper names, from Rosas, Mediano, Rassouli and Barrett’s synergistic-disclosure PID. Proposition 2: for stationary X_t, every autocorrelated k-synergistic observable is k-th-order emergent. Nothing in the case studies computes these quantities.
• Case studies. Only Ψ and Γ are computed, never an atom, so no redundancy function enters any reported number.
  • Game of Life. A 15 × 15 board (n = 225) set up as a “particle collider” with randomised particle types and positions. V_t is the indicator vector of particle types, and t′ − t = 1000 steps. 5 × 10⁴ runs, Archer et al.’s quasi-Bayesian estimator.
    • Ψ = 0.58 ± 0.02, from I(V_t; V_t′) = 0.99 and Σ_i I(Xⁱ_t; V_t′) = 0.41 (Fig 4).
    • Γ = 0.009 ± 0.0002, which the authors say “suggest[s]” decoupling.
  • Boids. N = 10 on a torus, a₁ = 0.15 and a₃ = 0.25 fixed, avoidance a₂ swept from 0 to 0.2. V is the centre of mass. Trajectories were preprocessed as in Seth (2010) and the KSG estimator used.
    • Read off Fig 5a: Ψ ≈ −0.3 with a very wide error bar at a₂ = 0, ≈ 0.8 at 0.05, falling to ≈ 0.1 at 0.2.
    • The low-a₂ failure is attributed to redundancy, not to a lack of synergy. Γ is not reported.
  • Macaque ECoG. 64 channels and wrist motion capture from Neurotycho. V_t = F(X_t) ∈ ℝ³ is a PLS (20 components) plus SVM decoder trained on 60% of the data. Ψ was computed on the rest with JIDT’s Gaussian estimator.
    • At t′ − t = 8 ms, Ψ = 1.275 ± 0.002 and Γ = 0.049 ± 0.002. Ψ stays positive out to about 0.2 s.
    • A decoder trained on time-shuffled wrist positions gives a lower Ψ. Read off S1 Fig B, that surrogate is about 0.95 at 8 ms, against 1.275.

Claims

id	claim	strength	support
C1	I(X;Y) splits exactly into Red⁽ᵏ⁾, Syn⁽ᵏ⁾ and the Un⁽ᵏ⁾ terms for any PID	strong	Lemma 2, a partition of the lattice; checked
C2	A system has a k-th-order causally emergent feature iff Syn⁽ᵏ⁾(X_t;X_t′) > 0, and Syn⁽ᵏ⁾ bounds every feature’s emergence	strong, given the axioms	Theorem 1 and Corollary 1, proved; the witness is V = X_t
C3	Syn⁽ᵏ⁾ = D⁽ᵏ⁾ + G⁽ᵏ⁾, and D⁽ᵏ⁾ > 0 (G⁽ᵏ⁾ > 0) iff the system admits downward-causing (decoupled) features	strong for the identity (definitional); moderate for the iffs	Eq. 7; Theorems 2–3 rest on an assumed ΦID bound and an assumed multi-target deterministic equality
C4	Ψ > 0 suffices for the candidate V to be causally emergent; Δ > 0 for V to exert downward causation; Ψ > 0 and Γ = 0 for V to be causally decoupled	weak as stated; strong for the system-level reading	Proposition 1’s proof reaches Syn⁽ᵏ⁾(X_t;X_t′) > 0 only. Under MMI the claim about V fails (Limitations)
C5	The practical criteria hold for every PID that meets the axioms	strong for the system-level reading	each step uses only whole-minus-sum and the two data-processing axioms
C6	Particle collisions in the Game of Life, the boids’ centre of mass and the decoded wrist position are causally emergent; GoL particles and the ECoG feature may be causally decoupled	weak to moderate	Ψ > 0 in simulation and data. That gives the systems capacity, not these features emergence; Γ is never 0; decoupling is hedged as “suggest”
C7	Hoel’s effective-information emergence cannot be read off observational mutual information, because I(V_t;V_t′) ≤ I(X_t;X_t′); its maximum-entropy interventions count transitions the system never makes	strong for the inequality; argued for the rest	data-processing inequality; §Relationship with other theories
C8	Seth’s G-emergence autonomy I(V_t;V_t′	X_t) > 0 conflates unique and synergistic information	weak as written

Concepts

• Supervenient feature: V_t with V_t − X_t − X_t″ Markov for all t″ ≠ t.
• k-th-order synergy Syn⁽ᵏ⁾(X;Y): the information about Y that the whole provides and no group of ≤ k parts holds, considered separately from the rest.
• Emergence capacity: Syn⁽ᵏ⁾(X_t; X_t′).
• Causal emergence of V (order k): Un⁽ᵏ⁾(V_t; X_t′ | X_t) > 0.
• Downward causation: Un⁽ᵏ⁾(V_t; X^α_t′ | X_t) > 0 for some group α of k future parts. System index D⁽ᵏ⁾: synergistic-source → non-synergistic target atoms.
• Causal decoupling: Un⁽ᵏ⁾(V_t; V_t′ | X_t, X_t′) > 0. System index G⁽ᵏ⁾: synergistic-source → synergistic-target atoms. Pure decoupling adds that V predicts no k-group of future parts uniquely, a “statistical ghost”. The parity of Example 1 is the paper’s case of it.
• k-synergistic observable: the output of a channel independent of every group of k parts.
• Order k: the size of the groups of parts excluded, the paper’s “scale”. It is not a coarse-graining.

Connections

The paper places itself after Seth’s G-emergence (2010), Hoel, Albantakis and Tononi’s effective information (LIT-1038, THEORY-222) and Chang et al.’s non-trivial informational closure. It builds on Williams and Beer’s PID and on Mediano et al.’s ΦID (“Beyond integrated information”), and it names Rosas et al.’s synergistic-disclosure PID as one concrete decomposition. The review that restates it is LIT-025. Barnett and Seth’s dynamical independence (LIT-979), a paper with a shared author, came later and is not cited here. The paper’s closest relative to it is the NTIC discussion: “NTIC requires features that are sufficient statistics for their own dynamics, which is akin to our concept of causal decoupling.” That kinship is loose; see Bearing on the record.

Bearing on the record

• THEORY-497 gets its first-hand source. What this reading establishes for it:
  • The conditions of the iff theorem are the axioms above. The witness is the microstate itself.
  • Exhaustiveness. The split into D and G is exhaustive by definition.
  • The case studies use no redundancy function. They use Ψ and Γ only. The review’s silence on which function the applications used (NOTE-010) has that answer.
  • What the axioms exclude. Whole-minus-sum together with target data-processing forces Syn⁽¹⁾ ≥ 1 bit for two independent bits that each copy themselves. Take Y = X¹′ ⊕ X²′: then I(X;Y) = 1 and each I(Xʲ;Y) = 0. A measure with the identity property (Bertschinger et al., Harder et al.) gives 0 synergy for that copy system, and 1 bit for the XOR target. So it breaks target data-processing, and the paper’s admissible class excludes it. MMI is in that class.
  • NOTE-010’s permutation example. The same argument with Y = X¹′ = X¹⊕X² gives that example Syn⁽¹⁾ ≥ 1 bit under every admissible PID. So, inside the theory’s own class, the disagreement with effective information that NOTE-010 found does not depend on the redundancy function. It vanishes only for measures the axioms rule out.
  • The coordinator may want THEORY-497’s source, Connections and promote_when revised accordingly (see the report).
• THEORY-257, THEORY-733: causal decoupling against dynamical independence.
  • They are different conditions. Decoupling is about irreducibility: V’s self-information is not held by any small group of parts. Barnett and Seth’s independence is about sufficiency: the micro history adds nothing to V’s self-prediction.
  • The link the record can prove. For any deterministic supervenient V and a Markov X, Ψ(V) = [I(X_t; V_t′) − Σ_j I(Xʲ_t; V_t′)] − I(V_t′; X_t | V_t). The last term is the one-step dynamical dependence, and its vanishing implies Barnett–Seth independence. Filed as THEORY-764, with examples both ways.
  • Example 1’s “statistical ghost”. The parity is exactly dynamically independent (one-step dependence 0, checked). There the two accounts agree.
  • A parity that is not independent. If its persistence probability depends on one bit, the parity passes the paper’s decoupling test (Ψ = 0.231 bits, Γ = 0, n = 3). It is not dynamically independent: its one-step dependence is 0.141 bits.
  • Independence without emergence. A single part that copies itself is dynamically independent. By Lemma 1 it is not emergent.
• QUESTION-115 (do retention, dynamical independence, effective information and an Occam price pick the same level?).
  • ΦID emergence picks no level. By Corollary 1 and the proof of Theorem 1, Un⁽ᵏ⁾(V; X_t′ | X_t) is maximized by V = X_t, the microstate. Only a price on description length would select a coarse feature. The authors gesture at one (H(V_t) < H(X_t)) and do not impose it. So this criterion needs the Occam price the question names as much as Krakauer et al.’s measures do (LIT-1137).
  • It answers a different question from effective information. Effective information (THEORY-222) ranks levels by how much they say. Ψ and Un⁽ᵏ⁾ ask whether a feature’s information is held by small groups of parts. The paper’s own data-processing argument rules out a “macro beats micro” reading of observational information.
  • Ψ and dynamical independence. Ψ rewards dynamical independence and penalizes redundancy among the parts (THEORY-764).
  • A pair of predictions on which the criteria part. A dynamically independent single part fails ΦID emergence. A non-independent parity passes the decoupling test.
  • Where they agree, it need not be integration. The sum of n ≥ 2 independent, identical Gaussian AR(1) channels is exactly dynamically independent, and it passes Ψ > 0 for every coefficient 0 < |a| < 1 (re-derived; for a = 0.9 and n = 2, Ψ = 0.449 bits, checked by simulation). Persistence of an aggregate of non-interacting parts is enough for both. That bears on A12’s guard that “integration licenses a module”: Ψ > 0 does not license one.
• QUESTION-218 (what is the unit when there is closure without a block?). ΦID’s emergent features are functions of the whole state. Lemma 1 forbids a function of one part, and the theorem’s witness is the whole microstate. Example 1’s parity is a closed collective mode of the kind CLAIM-388 names: it is dynamically independent, no part predicts the future of any part, and it is no block. It adds a fourth candidate unit beside those NOTE-979 lists: a feature whose self-prediction is synergistic across the parts.
• QUESTION-222 (memory). The theory assumes a Markov, fully observed X. The ECoG case violates both, as the authors say, and their suggested remedy is delay embedding. Ψ’s dependence term conditions on V_t alone, not on V’s history. A non-Markov reduction would separate the two (see THEORY-764).
• No bearing on QUESTION-217, QUESTION-220, QUESTION-221 or QUESTION-219 beyond the above. The paper has no coupling strength, no nesting of abstractions and no fluxes.
• No instruction for machine-learning practice. The ECoG decoder is a tool, not a recommendation. Nothing for the anthology.

Limitations

• Proposition 1 claims more than its proof shows.
  • The chain (D2)–(D4) ends at Syn⁽ᵏ⁾(X_t; X_t′) > 0. Combined with Theorem 1, that says some feature, possibly the microstate, is emergent. It does not say the candidate V is.
  • Counterexample under MMI, which the appendix names as admissible. Take two independent bits that each copy themselves, with V = X¹ ⊕ X². Then Ψ(V) = 1 bit. But in the three-source MMI PID of (X¹, X², V) about X_t′, all of V’s information is redundant with X¹, so Un⁽¹⁾(V; X_t′ | X_t) = 0 (computed).
  • The same gap affects Δ: its proof reaches D⁽ᵏ⁾(X_t; X_t′) > 0 and then cites “Theorem 1” where Theorem 2 is meant.
  • The same gap affects Γ. Its proof shows only that Γ = 0 rules out V’s downward causation. It never shows Un⁽ᵏ⁾(V_t; V_t′ | X_t, X_t′) > 0, which Definition 4 requires.
  • So the conclusions that GoL particles, the flock’s centre of mass and the decoded wrist position “are” emergent rest on an unproved step. What the data support is that each system has emergence capacity, under every admissible PID.
• Γ = 0 is never met. GoL gives Γ = 0.009 and ECoG 0.049. The criterion has no continuity bound, so a small Γ licenses nothing.
  • For ECoG, “orders of magnitude smaller” compares Γ with Ψ, and the ratio is about 26.
  • The authors’ own hedges, “suggest” and “may”, are the right strength.
• The ECoG surrogate is itself strongly positive. A decoder trained on shuffled wrist positions gives Ψ ≈ 0.95 at 8 ms, about three-quarters of the observed 1.275 (S1 Fig B, read off the plot).
  • The main text says only that the surrogate is “significantly lower”. It does not say that the surrogate is itself well above zero.
  • That fits the AR(1) result above: a smooth projection of many autocorrelated channels passes Ψ > 0 without any motor content or interaction.
  • It is also unclear whether the sum over parts ran over the 64 electrodes, as the main text says (X_t ∈ ℝ⁶⁴), or over the 20 PLS components the appendix projects onto. The partition matters by the paper’s own account.
• A misstated axiom. Read literally, non-negativity’s bound Un⁽ᵏ⁾(Z; Y | X) ≤ I(Z; Y | X) with X the whole system makes Definition 2 vacuous. Every supervenient V has I(V_t; X_t′ | X_t) = 0. It also contradicts Lemma 2, where Un⁽ᵏ⁾(X_t; X_t′ | X_t) = Syn⁽ᵏ⁾ > 0 while I(X_t; X_t′ | X_t) = 0. The intended bound is presumably by I(Z; Y | X^α) for each group α of ≤ k parts. Lemma 1’s proof goes through under that reading. Both copies carry the literal form.
• Theorems 2 and 3 rest on assumptions about ΦID, not on its axioms: the D⁽ᵏ⁾ sandwich bound and a multi-target deterministic equality. The paper does not show that any ΦID other than MMI’s satisfies them, and it shows MMI’s only by assertion.
• The axioms exclude identity-respecting redundancy measures, as shown above. So the theory is not “agnostic” across PIDs. It is agnostic across a class that includes MMI and excludes Bertschinger et al.’s and Harder et al.’s measures. For three or more parts, THEORY-713 and CLAIM-343 already show those cannot keep non-negativity anyway.
• Emergence without interaction. By those axioms, two non-interacting self-copying bits have emergence capacity, and their parity passes Ψ. This is the record’s inference, not the paper’s. Example 1’s narrative (“no individual variable has any predictive power”) makes emergence look like a property of coupled collectives. The axioms do not require that.
• Seth’s criterion misrendered. For supervenient V_t, I(V_t; V_t′ | X_t) = 0 (by the chain rule, because V_t carries nothing beyond X_t). So the paper’s example of Seth’s autonomy ”> 0 while I(V; V_t′) = 0” cannot occur for the features the theory admits. Seth’s own measure conditions on histories.
• Partition dependence, which the authors state openly and leave open.
• Equilibrium systems are out of scope: a Gibbs distribution does not fix the dynamics (Glauber against Kawasaki).

Open questions

• Is Proposition 1’s claim about the candidate feature true under some admissible PID, and which? A proof that Ψ(V) > 0 implies Un⁽ᵏ⁾(V; X_t′ | X_t) > 0 under a named decomposition would settle it. The MMI counterexample shows it cannot hold for all of them.
• A continuity bound for Γ: how small must Γ be, relative to what, for decoupling to follow?
• With a description-length price on V, which feature maximizes Un⁽ᵏ⁾(V; X_t′ | X_t)? Does it coincide with a dynamically independent one? That is QUESTION-115’s comparison made concrete.
• Does the ECoG verdict survive a surrogate that keeps each channel’s autocorrelation but destroys cross-channel structure? Per-channel phase randomization would test interaction rather than persistence.
• How do emergence verdicts move under an invertible remixing of parts, such as EEG source mixing? The authors pose this one themselves.