Source
Varley (2022), LIT-953, Entropy 25(1):54, read in full in NOTE-756: §2 (local mutual information, the two frameworks), §3 (the toy and the connectome), §4 (interpretation).
What was actually shown
The identity. Local mutual information i(x; y) = log P(x,y)/(P(x)P(y)) is negative whenever P(x,y) < P(x)P(y), and the mutual information I(X; Y) is its expectation. Local excess entropy e(x) = i(x_{t−1}; x_t) is negative when P(x_t | x_{t−1}) < P(x_t). This part is exact and general.
The coarse-graining instance (§3.1). A four-node Boolean network, coarse-grained by AND over pairs, is emergent on Hoel’s measure: log(F̄(X̃)/F̄(X)) ≈ 0.202. Yet ≈52.73% of informative micro transitions map to misinformative macro transitions. They are low-probability transitions, so the average stays positive.
The ΦID instance (§3.2). The same toy’s macro system has temporal synergy {X1,X2}→{X1,X2} ≈ 0.031 bit on average, negative for 31.25% of its sixteen transitions.
The network instance (§3.1). On one Human Connectome Project matrix with Infomap macro-nodes, ≈22.47% of informative micro edges map to misinformative macro edges, and ≈80.4% of misinformative edges run between communities.
What could have come out the other way: in the toy, the misinformative share could have been negligible, and in the connectome the misinformative edges could have been spread evenly. Neither happened. The toy was built for illustration, so the first test is weak.
What this does not say
- That every emergent system flickers. The possibility is general to measures built as expectations; the occurrence is not. A deterministic, invertible dynamics with uniform state probabilities has every realised transition at e(x) = log N > 0. The paper gives no conditions under which flickering must occur.
- That the system stops being one system at a misinformative moment. The paper says the system “briefly stops being a ‘whole’”, but also that a negative value may be “merely statistical fluctuations about the mean”. Whether a negative local value marks a change in the system or a rare event under a fixed one is not settled by the statistics.
- That the two frameworks flicker together. Each was shown separately on the same toy; whether they flicker at the same transitions is not reported.
- That the frameworks are wrong. The paper disclaims critique: the point is that an average score answers a question about the ensemble.
Connections
- QUESTION-035. For the “how long does it last” clause: an
information-theoretic criterion of unity gives a verdict over an
interval, and at a moment it can give the opposite verdict. An answer
needs a window and a tolerance, which these measures leave open. No
answers:is declared, since this constrains answers and is not one. - CLAIM-157. Its duration clause, “it lasts as long as that block persists”, is stated over joint dynamics estimated over time, so it meets the same question of window. The network instance bears on its boundary clause: the macro description fails most often on edges between blocks.
- CLAIM-161. If its coherence measure is informational, “approximately conserved” has to absorb negative local values. The paper does not discuss persistence; this is the record’s application.
- THEORY-222. A complement on the same measure: THEORY-222 holds that the average effective information can be higher at a macro scale, and this holds that the macro scale can be the worse description at most of the micro transitions it summarises. Neither conflicts with the other.