Contribution
It reworks Hoel’s causal emergence (CE 1.0, LIT-1038) in three ways. The measure is no longer effective information but “causal primitives” (CP): a normalized determinism and a normalized degeneracy, the information-theoretic forms of sufficiency and necessity. A system is analysed along a micro→macro path of successive coarse-grainings, and each step’s gain ∆CP is credited to that scale, so several scales can contribute. The spread of the gains along the path is a new number, “emergent complexity”. It is one paper, sole-authored, on three toy chains.
Key insight
EI = (determinism − degeneracy) · log₂ n, so EI rewards a scale for having many states as well as for clean transitions. Drop the size factor and measure only the bracket (CP, called the effectiveness in CE 1.0); then compare each scale with the one just below it. A coarse-graining “contributes” exactly when it makes transitions more deterministic or less degenerate than its refinement, and the contributions along a path form a profile rather than a single winner.
Assumptions
- A known Markov chain. The microscale is a given TPM; macroscales are coarse-grains of its states, built as in Hoel et al. 2013 and Klein and Hoel 2020 (cited, not restated).
- Uniform interventions at each scale. P(C) is uniform over the states of the scale being scored, not a coarse-graining of the micro P(C). §6.1 argues gains survive other choices, citing Comolatti and Hoel, and Fig. 4 shows one case with observational P(C) at both scales.
- Dynamical consistency. S2: a macroscale is kept only if random walkers started at every state give the same coarse-grained trajectories for 5 steps (summed KL divergence zero); others are discarded.
- One chosen path. The endpoint is the macroscale of greatest CP gain (ties to the least reduced), and the path is the longest consistent one up to it (§3.3).
Key results
- Definitions (§2.2, §3.2). determinism = 1 − ⟨H(E|c)⟩/log₂ n; degeneracy = 1 − H(E)/log₂ n with H(E) the entropy of the effect distribution under uniform P(C); specificity = 1 − degeneracy; CP = determinism + specificity − 1. CE = total ∆CP along the path.
- Fig. 2. An 8-state chain with an equivalence class (4,5,6,7): CP rises from 0.66 at the microscale to 1 at the macroscale that lumps the class, then gains stop. CE = 0.33.
- Emergent complexity (§4). EC = −Σ pᵢ log₂ pᵢ with pᵢ = ∆CPᵢ/Σ∆CP over the path’s steps. A “top-heavy” system with two pure equivalence classes scores 1.67 bits; a near-copy whose states 0 and 4 differ from their classmates scores 2.07 bits, because ∆CP peaks before the endpoint.
- CE 1.0 misses cases (§6.2, Fig. 4). In a two-block chain whose within-block probability is moved step by step onto self-loops, EI shows no causal emergence at any step while ∆CP starts positive and falls to zero as the chain becomes a permutation.
- An SVD heuristic (S3). Dropping σ₁ from Zhang et al.’s dynamical-reversibility average (LIT-1129) gives γ*, which tracks CP under added noise and common causes; σ₂ − γ* estimates total CE, and singular values above the mean of the rest flag contributing “directionalities”: one (0.61) for the top-heavy system, three (0.489, 0.06, 0.06) for the mesoscale one.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | CP is EI with the size term removed | strong | the decomposition EI = effectiveness · log₂ n of Hoel et al. 2013 |
| C2 | Along a chosen path, ∆CP apportions causal contribution to scales | moderate | definition; three 8-state examples |
| C3 | EC separates a system with mesoscale structure from one without | weak | two hand-built chains, 1.67 against 2.07 bits |
| C4 | CE 2.0 “captures all macroscale causation” that CE 1.0 misses | weak | one family of chains (Fig. 4); no proof of “all” |
| C5 | CE 2.0 does not depend on the choice of P(C) | weak | asserted from a cited paper and one figure |
| C6 | γ* and σ₂ − γ* estimate CP and CE without search | weak | qualitative agreement on the paper’s own examples |
Method
Enumerate coarse-grains of the TPM, discard inconsistent ones, compute CP at each, pick the endpoint and the longest path to it, record ∆CP at each step, normalize the gains and take their entropy. For large systems a diminishing-returns stop (∆CP below ε, or a run of shrinking ratios) replaces enumeration.
Concepts
- Causal primitives (CP): here both the pair (sufficiency, necessity) and the scalar determinism + specificity − 1; the paper uses the word for both.
- Micro→macro path: a chain π⁽¹⁾ → … → π⁽ᵏ⁾ of consistent partitions, each a coarse-grain of the last.
- Causal apportioning schema: crediting each step its ∆CP over the previous step.
- Emergent complexity: entropy of the normalized gains along a path.
Connections
It is the successor of Hoel, Albantakis and Tononi 2013 (LIT-1038, THEORY-222), whose EI and effectiveness it reuses. It answers the criticism of CE 1.0’s uniform-intervention assumption by Eberhardt and Lee (2022, cited as [25], not registered) and Aaronson. Jansma and Hoel’s “Engineering emergence” (LIT-1123) extends it to all paths at once and, in doing so, drops the consistency filter this paper relies on. Its SVD heuristic adapts Zhang et al. (LIT-1129). It contrasts itself with Rosas et al.’s information decomposition (LIT-1244) and Barnett and Seth’s dynamical independence (THEORY-257), on the ground that both rest on observational mutual information, and with Rosas et al.’s “software in the natural world” (LIT-1306), on the ground that consistency or closure alone does not measure a contribution. Flack’s coarse-graining as downward causation is cited for the thermostat example.
Bearing on the record
- QUESTION-115. The question lists effective information among its criteria. CE 2.0 is the same family with the size factor removed and a per-scale credit added, and it reports a profile along a path, which is what A12 asked of the joint search. It does not search over boundaries of a system (which variables are in it), only over partitions of one chain’s states, so it bears on the macrostate leg of the question and not on the boundary leg.
- THEORY-222. Unaffected: CE 2.0 keeps uniform interventions at each scale and gives the same macroscale as CE 1.0 where the two overlap (§6.2). It adds a case CE 1.0 scores zero.
- The candidate it was filed as. The paper fits A10’s description (“searching for coarse-grainings rather than assuming them”) only in part: it searches along one path, chosen by maximum gain, and calls the full search future work.
No instruction for machine-learning practice; nothing for the anthology. The closing paragraph’s suggestion that CE 2.0 may serve interpretability of deep networks is a remark, not a method.
Limitations
- Three toy chains. No system above 8 states is analysed; no TPM entries are printed, so the examples cannot be re-run from the text.
- One path. §5 concedes that integrating “the full set of non-commensurate micro→macro paths” is open, and the answer depends on the path chosen.
- “All macroscale causation” (abstract, §6.2) rests on one manipulated family, not on an argument.
- Errors and slips. The degeneracy is printed as 1 − H(E | C)/log₂ n (§2.2), but the entropy then written out is that of P(e | C) summed over causes, i.e. H(E); with H(E | C) the formula would be the determinism again. §1 says macroscales are “reducible to their underlying macroscales” for microscales. The S1 text cites “Fig. 2D” for the 100-state run that is Fig. S2D. Fig. S3’s caption says “γ* is used instead of γ*”. §3.2 calls CP both “sufficiency and necessity added together” and “determinism and specificity”, which differ.
- The author’s acknowledged funder (Softmax) is noted, nothing more.
Open questions
- What does the apportioning give when the path is not chosen, and are the multi-path answers of LIT-1123 consistent with this paper’s single-path ones on the same chains?
- Does any ∆CP profile recover a known boundary, not just a known lumping, in a system whose variables are split between agent and environment?
Note of 2026-10-29: the criticism and the reply it cites, read
Eberhardt and Lee 2022, which Connections gives as cited [25] and not registered, is now registered and read: LIT-1839, NOTE-1413. So is Comolatti and Hoel 2022, which §6.1 cites for gains surviving other choices of P(C): LIT-1843, NOTE-1416. Comolatti and Hoel concede that EI under the stationary distribution at both levels shows no emergence. The record’s THEORY-1167 shows why, in general: under any one distribution over micro states at both levels, macro EI cannot exceed micro EI, by data processing. That bound is for EI. This paper’s CP is normalized by the size of the state space, and the bound does not by itself decide Fig. 4’s case with observational P(C) at both scales, which was read from its caption only. Whether that case’s gain survives as a gain of EI was not checked.