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LIT-1260: Thermodynamic depth of causal states: Objective complexity via minimal representations

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StatusActive — read 2026-10-20 (NOTE-978) in full, from arXiv:cond-mat/9808147v1; worth reading for three things. It shows that the growth rate of Lloyd and Pagels' thermodynamic depth is the entropy rate of the chosen macrostate process, so lumping and splitting macrostates set it from zero up. It shows how its examples (an IID chain, a hidden fair coin, the logistic map through a non-generating partition) expose the dependence. And it proves that the causal states are the coarsest fully predictive states, with least statistical complexity and least dive (THEORY-756). Read it knowing four things. Its logistic-map "dive ≈ 0.811" is the single-symbol entropy; the entropy rate is 0.677867. Its footnote 2 makes prescience vacuous. Its dive proof covers only Markov rivals, though the result holds. Its claim that v < C_μ whenever the past and future share information fails in general. Its "objective" complexity is objective only given the measuring instrument, as its own logistic-map example shows.
Version2
Tagscomplex-systems, information-theory
Cited as “paper” byNOTE-978 — Crutchfield and Shalizi — Thermodynamic depth of causal states
Cited as “source” byTHEORY-756 — The growth rate of thermodynamic depth is the entropy rate of whichever macrostate process it is computed over, so the choice of macrostates sets it anywhere from zero up; among partitions of a stationary process's past that predict every finite stretch of the future as well as the whole past does, the causal states are the coarsest, so they have the least entropy and the least entropy rate

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Crutchfield, J. P. & Shalizi, C. R. (1999). Thermodynamic depth of causal states: Objective complexity via minimal representations. Physical Review E 59(1): 275–283. doi:10.1103/PhysRevE.59.275. Preprint: arXiv:cond-mat/9808147 (13 August 1998), under the title “Thermodynamic Depth of Causal States: When Paddling around in Occam’s Pool Shallowness Is a Virtue”.

Read 2026-10-20 in NOTE-978. The summary was checked against the reading.

Why this is here

A2 listed it as a contrastive reading: macroscopic complexity measures depend on the chosen macrostates. Bears on QUESTION-220.

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