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LIT-979: Dynamical independence: Discovering emergent macroscopic processes in complex dynamical systems

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Properties5
StatusActive — read 2026-10-14 (NOTE-791) in full; worth reading as a transformation-invariant criterion for emergent macroscopic variables: a coarse-graining is dynamically independent when the microscopic history adds nothing to its self-prediction, with a closed form and a numerical search for linear state-space models. Read it knowing that the system/environment boundary, the scale and the class of maps are inputs, that generically no exact independent variable exists beyond VAR(1), and that the optima are many. On QUESTION-065 it supplies a system-fixed target, but not the locality-based one, and it answers the uniqueness question in the negative. Held in THEORY-257.
Version2
Tagscomplex-systems, information-theory
Cited as “paper” byNOTE-791 — Barnett and Seth — Dynamical independence
Cited as “source” byTHEORY-257 — Dynamical independence, zero information flow from the microscopic history into a coarse-graining given its own history, fixes emergent macroscopic variables by the dynamics rather than by a chosen query, but for linear VAR models of order above 1 generically none is exact, and the optima are many, THEORY-733 — For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose's invariant subspaces, which exist on open sets but not generically at every scale, THEORY-764 — For a deterministic supervenient feature V of a Markov system, Rosas et al.'s practical emergence criterion Ψ equals the parts' whole-minus-sum information about V's next value minus V's one-step dynamical dependence I(V_t′; X_t | V_t), so dynamical independence raises Ψ but neither implies emergence nor is implied by it, and the paper's sufficient test for causal decoupling can pass a feature that is not dynamically independent, THEORY-820 — For a stationary discrete-valued process X and any coarse-graining Z of its past, informational closure at every horizon is the same condition as zero one-step transfer entropy from the micro past, Barnett and Seth's dynamical independence, and as causal closure, Z's ε-machine coinciding with its υ-machine; for a coarse-graining of the present state it also makes Z's causal states a coarse-graining of X's, which does not imply closure, and the closed coarse-grainings are not closed under common refinement

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Lionel Barnett and Anil K. Seth (2023), Physical Review E 108, 014304.

Key takeaways

From the reading, NOTE-791.

  • The criterion. Y = f(X) is dynamically independent of X given E iff I(Y_t : X_t⁻ | Y_t⁻, E_t⁻) ≡ 0. The micro history adds nothing to Y’s self-prediction. The same transfer entropy measures the departure from independence.
  • Properties.
    • Invariant under reparametrization.
    • Transitive across nested scales (proved in App. A).
    • Equivalent, in suitable coordinates, to a sub-graph of the causal graph with no incoming transfer. Such a variable may drive the rest of the system, but must not be driven by it.
  • Linear systems. F(X → Y) = log|LVLᵀ| from a reduced Riccati equation, or as a spectral integral. Exact independence holds iff LCA^kKMᵀ ≡ 0, or LA_kMᵀ ≡ 0 for a VAR(p). Generically it exists at every scale for VAR(1) and at none for VAR(p > 1).
  • Discovery. Minimize over the Grassmannian at each scale. The cost is multimodal, and every local optimum is kept as part of the portrait. A 9-variable VAR(7) with planted modules gave its exact optima at scales 2 and 6, nested.
  • Inputs, not outputs. The system/environment boundary is “given”, the scale is chosen, and the candidates in the linear theory are subspaces mixing units, not partitions of units.
  • Not autonomy. An independent variable need not predict itself. A white-noise coordinate can qualify.
  • Flows. For deterministic flows, the independent coarse-grainings are built from the flow’s invariants and one time-like function.

Standing in the record

Filed on 2026-10-12 for QUESTION-065 and read on 2026-10-14 in NOTE-791. Its finding is filed as THEORY-257, which answers QUESTION-065 in part on both its questions.

  • On the first, it supplies a system-fixed target, the coarse variable’s own future, so no query is chosen. That is not the locality-based target the question names. Boundary, scale and the class of maps remain the analyst’s.
  • On the second, the winner at a scale is not unique in general.

The registration’s phrase “a coarse-graining fixed by the system rather than by an analyst’s target” is right about the target and too strong about the coarse-graining.

It also bears on CLAIM-157 and CLAIM-167. Dynamical independence needs no incoming coupling but allows outgoing coupling, so a driver is emergent at its own scale, and a driven module never is. That is a different block from near-decomposability’s.

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