[ · · · · ] seats or rooms?erik hoel’s paper ↗

Does an explanation
need every detail?

We describe a gas using temperature, without listing every molecule.
Hoel’s paper asks when descriptions in terms of groups have useful causal structure of their own.

our test: one small mathematical example from the paper

Imagine a token moving between seats.

There are 8 seats in 5 rooms. Four rooms have one seat each. The blue room has four. At each step, the token randomly picks a seat in its current room. It never leaves that room.

Seats and rooms are our visual analogy for the paper’s states and groups, not data about people.

Start in the blue room. First ask for the next seat. Then switch to the next room.

watch the individual dotsstep 0

one sample trajectory, started at state 4

Now press “group the four.” Watch what changes, and what doesn’t.

The point is choosing what to explain.

“It stays in room e” is an exact description of the room-level behavior. It says nothing about the next seat. We kept a useful pattern and gave up detail. We did not recover the missing seat information.

Hoel’s broader theory tries to quantify what descriptions at different scales contribute causally. Our reproduction checks one worked example of that theory, not the whole theory.

What did we actually test?

We independently rebuilt the eight-state probabilities in Erik Hoel’s Quantifying emergent complexity. Counting the four shuffling seats as one room raises the selected causal-power measure from ⅔ to 1. We independently recomputed it and checked all 4,140 partitions.

separate states80.667
merge two70.758
merge three60.871
group the four51.000

262 partitions pass our exact consistency check, including the one-state partition. Five is the largest state count attaining the maximum score in this example. These counts use strong lumpability, not the paper toolkit’s tolerance-based test. Seat labels 1–8 correspond to the paper’s state labels 0–7.

What does the score mean?

This score changes the intervention distribution at each scale and divides by the logarithm of its state count. It is not “percent correct,” and it does not mean deleting data creates information.

show the definition & controls

For a transition matrix with n states: score = [entropy of the mean row − mean row entropy] / log₂(n). Each state receives equal intervention weight at the chosen scale.

Keeping the original microstate weighting instead gives 2 bits of information both before and after the useful grouping. That is our same-prior control.

A grouping is consistent here when every state inside a group has identical probabilities of moving to each destination group. The “bad grouping” fails: state 0 stays inside its group with probability 1, but state 4 does so with probability ¼.

For a single state the logarithmic normalization is undefined. We use score 0, matching the authors’ code convention: no alternative states remain to distinguish.

The moving dot is an illustrative sample from the model. The score is calculated exactly from its probabilities, not estimated from the animation. This is a worked-example reproduction, not evidence about consciousness or free will.

measured results ↓rerun in python ↓browser calculation ↓