Contribution
CE 2.0 (LIT-1136) credited causal contribution along one chosen path of coarse-grainings. This paper removes the choice: every partition of the state space is scored against every finer partition, so a scale counts only if it beats all scales below it on any path. The surviving partitions form a sublattice the authors call the emergent hierarchy. It then defines two entropies over that hierarchy, ties their product to preferential attachment, and builds chains whose hierarchy has exactly one member.
Key insight
Among the Bell-number many partitions of a state space, almost all coarse-grainings make transitions noisier; the few that make them more deterministic or less degenerate than everything beneath them are a small lattice. Where that lattice puts its weight (near the microscale, near the top, or spread across levels and within them) is a property of the dynamics, and it can be designed.
Assumptions
- A known, Markovian, ground-truth TPM with no hidden confounders, so p(X_{t+1} | do(X_t = x)) = p(X_{t+1} | X_t = x) (S1).
- Uniform intervention distribution at every scale (§4), as in CE 2.0.
- Partitions of states only, not of variables or mechanisms (S1).
- No consistency check. Unlike CE 1.0 and 2.0, partitions that are not dynamically consistent are kept, because filtering “did not alter our results in a significant way (data not shown)” (§4).
Key results
- Causal primitives (§2). determinism = 1 − H(E | C)/log₂ n, degeneracy = 1 − H(E)/log₂ n, CP = determinism + specificity − 1 = determinism − degeneracy, “equivalent to the effectiveness” of Hoel et al. 2013.
- ∆CP over the lattice (§3.1, Algorithm 1). ∆CP(P) = CP(P) − max CP over the partitions refining P; the emergent hierarchy is {P : ∆CP > ε}. On a 5-state source–cycle–sink chain, 5 of the 52 partitions survive and the intuitive (source)(cycle)(sinks) one has the largest ∆CP. Summed ∆CP is “generally not” 1.
- A taxonomy (Fig. 4). Five 8-state chains give top-heavy, bottom-heavy, mesoscale-peaked and single-scale (“balloon”) hierarchies.
- Two complexity terms (§3.2). S_path, the mean entropy of normalized ∆CP along paths through the hierarchy (CE 2.0’s emergent complexity, averaged), and the row negentropy log₂ L − S_row, high when ∆CP differs between partitions of the same size.
- Preferential attachment (§3.3, Fig. 5). 40-node trees grown with attachment exponent α and m = 1, made into TPMs by normalizing out-links, five networks per α, 100 sampled paths: bottom-heavy for α < 1, top-heavy for α > 1.5, with the product of the two terms highest near α = 1. The authors call this “literal scale-freeness” and say it overlaps with, but is likely not identical to, the degree-distribution sense.
- Pinpoint emergence (§3.4, Fig. 6). A cycle has no intermediate simplifying scale, so a level-l chain built from one c-cycle (c ≤ l) plus l − c permutations, or from l disjoint cycles, has a single contributing macroscale among 877 partitions (7 states; stay 0.2, advance 0.8).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Scoring each partition against all finer ones leaves a small “emergent hierarchy” | moderate | definition; enumeration on 5- and 8-state chains |
| C2 | Hierarchies differ in shape (top-heavy, bottom-heavy, mesoscale, single) | moderate | five hand-built 8-state chains |
| C3 | Emergent complexity peaks near linear preferential attachment | weak | 40-state networks, five per α, greedy sampling, no error model for the sampler |
| C4 | Cycles and permutations place a single emergent scale at any chosen level | moderate | construction, Fig. 6 |
| C5 | Dropping dynamical consistency does not change the results | weak | ”data not shown” |
Method
Algorithm 1 enumerates all partitions, coarse-grains the TPM by each, builds the refinement lattice and takes ∆CP relative to the best ancestor (finer partition). Algorithms 2–3, for larger chains, start at the microscale, try every pairwise merge of blocks, keep the best few, complete each greedily to the top, and score only the partitions visited. The authors say it runs in subexponential time; no bound is given.
Concepts
- Emergent hierarchy: the partitions with positive ∆CP, ordered by refinement.
- Ancestor: in Algorithm 1, a finer partition, i.e. one below P on some micro→macro path.
- Literal scale-freeness: no one scale of description dominating the causal workings, as measured by the two entropies.
- Pinpoint emergence: a hierarchy with one member.
Connections
It extends CE 2.0 (LIT-1136) and through it Hoel, Albantakis and Tononi 2013 (LIT-1038). It cites Zhang et al.’s dynamical reversibility (LIT-1129) as an alternative to CE 1.0 that “will likely need to be expanded”, Mediano et al.’s information-decomposition review (LIT-025), Rosas et al.’s “software in the natural world” (LIT-1306) and Barnett and Seth’s dynamical independence (THEORY-257) as other approaches to macroscales, and Jansma’s Möbius-inversion mereology and Forré and Jansma’s Shapley values as possible apportioning schemes that would conserve total CP (neither registered).
Bearing on the record
- QUESTION-115. This is the candidate the record filed for A10’s “2024/2025” paper “on searching for coarse-grainings rather than assuming them”. It does search: every partition is scored, and the output is a set of scales rather than one. Read against the question, it searches partitions of one system’s states, not system–environment boundaries, and it prices nothing: there is no Occam term, and a partition survives on any positive margin over ε. The question’s “profile across criteria” is matched here by a profile across scales under one criterion.
No instruction for machine-learning practice; nothing for the anthology. The discussion’s remarks on brain–computer interfaces and human autonomy are not argued.
Limitations
- Small systems. 5, 7 and 8 states by enumeration; 40 by the greedy sampler, whose coverage of the lattice is not measured.
- Not an apportioning. ∆CP values do not sum to the total CP, which the authors concede, and the text then says “in practice the sum of ∆CP reflects the total possible CP” without showing it.
- Inconsistent macroscales are admitted, so some scored scales do not describe the microscale’s dynamics.
- Errors and slips. Figure 2’s left-hand macroscale is given determinism 0.24 and degeneracy 0.09, which make CP 0.15 (0.14–0.16 allowing for rounding), but the text gives 0.13; the other two panels agree with CP = determinism − degeneracy (0.74 − 0.11 = 0.63, 0.84 − 0.00 = 0.84). S_path is called the average of the path entropies but written as their sum. The row negentropy subtracts S_row from log₂ L, the log of the number of levels, though the entropy of a row is bounded by the log of the number of partitions in that row, so the “negentropy” is not bounded below by zero by construction. Eq. (1) writes p(e|c) with T_ce = p(e|c), and eq. (3) then writes T_ec.
Open questions
- Would an apportioning that conserves total CP (Möbius or Shapley, as the authors suggest) keep the same hierarchy?
- Does the α ≈ 1 peak survive larger networks, m > 1, and the consistency filter?
- Can the emergent hierarchy pick out a boundary of a system embedded in an environment, which is what the record’s individuation questions need?