Source
Hoel, Albantakis & Tononi (2013), LIT-1038, read in NOTE-726: Theory (definitions, the decomposition CE = ΔI_Eff + ΔI_Size), Results (four examples), Discussion (intrinsicness, exclusion, rules of thumb).
What was actually shown
A system of binary logic gates with a known transition matrix is set into each of its states with equal probability, and the effective information (EI) of its next state is computed. EI equals effectiveness, Eff, times log₂ of the number of states, and Eff is the average determinism minus the average degeneracy. A macro description groups micro elements, micro time steps or both, under mappings that hide which micro element is in which state. The macro is then scored the same way, with each macro state set by setting its micro realizers with equal probability.
The size term is always a loss for the macro. The effectiveness term can be a gain, because grouping averages out noise and merges states that deterministically lead to the same place. In four constructed systems the gain wins: 0.40 bits (noise), 0.57 bits (degeneracy), 0.62 bits (time) and 2.92 bits (space and time). The paper also proves rules of thumb: if the micro is already at least as effective as the macro, no macro can win.
What could have come out the other way: the decomposition shows exactly when the macro can win, and the rules show that systems already near maximal effectiveness cannot display emergence at all. The examples were chosen to display it.
What this does not say
- It does not show that the macro is where causation happens, as opposed to where EI is highest. The paper calls the EI-optimal level “intrinsic” and says the macro then excludes the micro. Both depend on imposing a uniform distribution over states separately at each level, which weights the micro states differently at the two levels. The authors note that this is what allows the macro to win and present it as fairness. It is a choice, and the paper does not test another.
- It does not measure any real system. The search over coarse-grainings is exhaustive and, by the authors’ account, infeasible beyond toy sizes. The claims about brains and about the hierarchy of the sciences are conjecture.
- It does not individuate anything. The coarse-graining is chosen by an analyst’s search. The paper’s “self-definition” at the optimal level means the EI maximum, not a boundary the system draws.
- It is not a model-selection rule from data. The mechanism is known exactly. There is no sample, prior or posterior, so no question of evidence or of how far the search inflates the winning score.
Connections
- QUESTION-093 (which criteria a higher-level agent needs). This is a measure of interventional autonomy: whether macro variables support more informative predictions of the effects of interventions. It also claims predictive autonomy, but only for predictions from imposed states. None of the paper’s systems maintains itself, and the grouping is an analyst’s, so positive CE needs neither organisational closure nor self-individuation. That answers half the question: the first criteria can be had without the last two. It says nothing on whether closure or self-individuation can be had without them, or on whether any of them makes an agent.
- CLAIM-154 (the reification rule). The same shape of trade-off, a gain
against a size cost that is always paid, for a different question.
CLAIM-154 asks whether data justify keeping a structure; CE asks which
description of a known mechanism is the more effective causal model.
Neither implies the other, and a coarse block could pass one and fail
the other. No
supportsis declared: CLAIM-154 rests on the Bayes-factor identity, not on this. QUESTION-040’s worry about search applies to any application of CE with an estimated mechanism, and the paper does not address it. - THEORY-133 (proportional causation). A complement. List & Menzies reach downward exclusion when an effect would survive a change of realizer; Hoel et al. reach it when averaging over realizers raises EI. Degeneracy reduction is realization insensitivity in information terms. Both make the answer system by system. The record’s promote_when for THEORY-133 asks for evidence of realization insensitivity in a particular system, and CE is one way to measure it, but only under this account’s choice of intervention distribution. No relation is declared, since neither depends on the other and they do not conflict.
- LIT-025 and LIT-027 (information-decomposition emergence). A different formal object under the same name: interventional and comparing two models here, observational and asking what a macro feature predicts beyond the parts there. The ΦID account is THEORY-497. Its reading (NOTE-010) gives a two-bit permutation with no EI gain at any macroscale that hosts an emergent feature under the minimum-MI redundancy and none under one with the identity property, so whether the two disagree turns on the redundancy function, and no rivalry is declared. Whether they pick the same level on a shared system is untested in the record.
- THEORY-071 (minimal agency). Its individuality condition, a boundary the system makes for itself, is exactly what CE does not supply.
Note of 2026-10-20: a four-state construction
For the promotion audit (batch 049, ADR-047 v2). Take micro states a, b, c and d. Each of a, b and c moves to one of a, b, c uniformly at random, and d moves to itself. With interventions uniform over the four states, P(Y = a) = P(Y = b) = P(Y = c) = (3/4)(1/3) = 1/4 and P(Y = d) = 1/4, so H(Y) = 2 bits and H(Y | X) = (3/4) log₂ 3 ≈ 1.189 bits, and EI ≈ 0.811 bits. Group A = {a, b, c} and B = {d}. A moves to A and B to B, so with interventions uniform over the two macro states EI = 1 bit. The macro gains about 0.189 bits, and the level with the highest EI is the macro one. The construction depends on the uniform distribution being imposed again at the macro level, which weights d at 1/2 instead of 1/4.
Note of 2026-10-20: the wider claim filed
The promotion audit narrowed this entry to what was shown. Under ADR-047 v4 the claim the narrowing dropped, that causation itself, not only effective information, can be most effective at a macro scale, is now filed as THEORY-787, Proposed, which extends this entry and takes it as evidence. This entry’s title and status are unchanged.
Note of 2026-10-29: what the gain is a gain of
THEORY-1167 (Active), filed from the readings of Comolatti and Hoel (LIT-1843, NOTE-1416) and Eberhardt and Lee (LIT-1839, NOTE-1413), extends this entry. Nothing here changes: the arithmetic holds, and the title already names the measure and the intervention distribution. What the extension adds is what the advantage is. Uniform interventions at each level amount to one distribution over micro states, uniform over blocks and then within each block, which is not the micro uniform whenever blocks differ in size. Under any one distribution at both levels, macro EI cannot exceed micro EI. So a positive CE equals the micro mechanism’s gain from that distribution over the uniform, less a non-negative loss. That answers the first half of QUESTION-204. The four examples behind this entry’s 0.40 to 2.92 bits were not recomputed, and LIT-1038’s temporal and spatio-temporal macros are outside the bound.