Why it is a question
THEORY-222 holds the arithmetic of Hoel, Albantakis and Tononi (2013). Effective information is measured with a uniform distribution imposed over the states at each level, and under that choice a coarse-graining can gain more effectiveness than it loses in size. The choice weights the micro states differently at the two levels. The authors note that this is what lets the macro win, and present it as fairness. If the gain disappears when one distribution over micro states is used at both levels, it measures the change of distribution, not a property of the system. Until 2026-10-20 this was THEORY-222’s promote_when.
What would answer it
- A result on systems not built to display emergence, showing that the EI-optimal coarse-graining stays the same, or stays macro, when the interventions follow the system’s own stationary distribution or another distribution fixed in advance.
- An agreement between the EI-optimal level and an independent interventionist test of which level the effects track, such as realization insensitivity under changes of realizer.
What cannot settle it: more hand-built toy systems with positive causal emergence, which show only that the arithmetic works; or applications that estimate a transition matrix and report the best of many coarse-grainings with no correction for the search.
Note of 2026-10-29: the first half answered, by derivation
The first half of this question is answered, and the answer is no. THEORY-1167 (Active) shows that the test needs no system. With one distribution over micro states at both levels, each macro intervention realized by that distribution within its block, the macro EI is I(f(X); f(Y)) ≤ I(X; Y), the micro EI, by data processing. A macro can at most tie, on any system and under any distribution fixed in advance, the stationary one included. Hoel’s positive causal emergence is the micro mechanism’s gain from changing the distribution, less a non-negative loss. On Eberhardt and Lee’s matrices (11), (13) and (14) the decomposition is 0.456 = 0 + 0.456, 0.194 = −0.150 + 0.344 and 0.0001 = −0.074 + 0.074 bits. Comolatti and Hoel, LIT-1843 (NOTE-1416), concede the stationary case (“This was known”). Eberhardt and Lee, LIT-1839 (NOTE-1413), name the maximum-entropy distribution as the source of the artefact, but do not themselves compute EI under a common one.
The second half, whether the EI-optimal level agrees with an independent interventionist test, is open. Eberhardt and Lee’s (14) is one small case where the two part. A test of realization insensitivity would keep x₁ and x₂ apart, since their effects on the macro effect variable differ (0.8/0.2 against 0.9/0.1), and EI merges them. It is a hand-built 3×3 matrix, which by this question’s own terms cannot settle agreement in general.
The bound is for EI. Measures that data processing does not bound, per-transition ones or ones normalized by log₂ n, can rise at a macro under a common distribution; they are not what this question asks about. THEORY-787 is noted the same day. The question stays Open.
Note of 2026-10-29: the second half has its test, and a partial answer
The earlier note of this date left the second half open. It now has the independent interventionist test it asks for, and part of an answer.
The test is causal feature learning: Chalupka, Perona and Eberhardt, LIT-1833 (NOTE-1420), and their overview, LIT-1834 (NOTE-1415). Its causal partition puts two micro causes in one macro state iff manipulating to either has the same effect, P(Y | do(x₁)) = P(Y | do(x₂)), and partitions effects likewise. That is realization insensitivity under changes of realizer, stated exactly, and it is independent of EI-type choices: the authors put no distribution on interventions, and the partition is defined from point manipulations alone.
The partial answer is THEORY-1169 (Active). Under one full-support distribution over micro causes at both levels, a coarse-graining keeps all of the micro EI iff it merges only causes with identical rows P(Y | do(x)) and only effects with proportional columns, whatever the distribution. So the coarse-grainings at which EI under a common distribution ties the micro are exactly the causal-feature-learning partitions, with effects grouped by normalized likelihood, and their refinements. On which coarse-grainings lose nothing, the two agree. The agreement is derived, and checked numerically on 245,394 cases, not read off a toy.
Two limits keep the question Open.
- Hoel’s EI, uniform at each level, whose optimum is the one the question names, does not agree. On Eberhardt and Lee’s (11) the two pick the same macro, since rows 1–7 are identical. On their (13) and (14), EI merges causes whose rows differ, which the causal partition keeps apart; on their note 6, EI keeps apart x₂ and x₃, whose rows are identical, because it coarsens cause and effect by one partition. These are hand-built matrices, which by this question’s terms cannot settle agreement in general, and Hoel 2013’s own examples (THEORY-222) were not recomputed.
- Under a common distribution the micro always ties, so the EI-optimal level is at best the coarsest tie. That is the causal partition: the proportionate description, not one with more causation. And the agreement needs exact equality of rows. With estimated rows only the micro ties, and causal feature learning merges by a distance threshold, a choice THEORY-1169 does not make.
The question is unchanged and stays Open.
Note of 2026-10-30: THEORY-787 rejected, and the record’s own answer
THEORY-787, the account this question was asked of, is Rejected on 2026-10-30 at the owner’s direction, by its own refutation condition: the first half of this question is answered no (THEORY-1167). The record now answers in its own voice. CLAIM-497 (Proposed, interpretive) holds that a system’s causes are found at the coarsest description whose interventions are unambiguous. ARG-041 sets out the inference from THEORY-1167 and THEORY-1169 to it. On the second half, that claim takes the common-distribution ties and causal feature learning’s partitions to be the same level, which THEORY-1169 shows. It takes Hoel’s uniform-per-level EI to fail the test, which Eberhardt and Lee’s (13) and (14) show.
The temporal and spatio-temporal macros of LIT-1038, which THEORY-1167 does not reach, are split off as QUESTION-335. What stays open here is whether a graded criterion can rank lossy coarse-grainings without bringing a distribution back. Whether that remainder leaves this question Open, or makes it Answered with that part split off, is left to the status review. The status is unchanged.