Source
Barnett and Seth (2023), Physical Review E 108, 014304, LIT-979, read in full in NOTE-791 (arXiv:2106.06511v2).
What was actually shown
- Definition (2.3)–(2.6).
- A coarse-graining Y = f(X) is dynamically independent of the microscopic system X, in the context of an environment E, iff T(X → Y | E) = I(Y_t : X_t⁻ | Y_t⁻, E_t⁻) ≡ 0.
- That is, the micro history adds nothing to predicting Y beyond Y’s own history.
- The same transfer entropy, the dynamical dependence, measures the departure from independence.
- Properties (proved).
- Invariance under reparametrization of source, target and conditioner (2.7).
- Transitivity across nested coarse-grainings (2.10, App. A), giving a partial order across scales.
- In suitable coordinates, independence holds iff the variable is a sub-graph of the conditional causal graph with no incoming transfer (2.12).
- Linear systems (derived).
- For an innovations-form state-space model with Σ = I and orthonormal L, F(X → Y) = log|LVLᵀ| with V = CPCᵀ + I, P from a reduced DARE (3.17)–(3.18). There is also a spectral integral form (3.21).
- Exact independence ⇔ LCA^kKMᵀ ≡ 0 for k < r (3.25). For a VAR(p) it is LA_kMᵀ ≡ 0 (3.26).
- By counting equations against the m(n − m) Grassmannian parameters, the paper expects exact solutions at every scale for VAR(1) and none for VAR(p > 1), except on a set of measure zero. THEORY-733 proves the second, and shows the first fails over the reals: exact VAR(1) solutions are the invariant subspaces of Aᵀ, which a rotation in the plane lacks on an open set.
- Search (simulation).
- Minimizing F over the Grassmannian is “in general multi-modal”, and a (1+1) evolution strategy with restarts was used.
- In a 9-variable VAR(7) built from three modules, the planted independent subspaces were found at scales 2 and 6, nested. Other runs ended in distinct local optima, some nearly orthogonal to the global one.
- What could have failed.
- The transitivity proof and the closed form could have needed further conditions.
- The search could have missed the planted subspaces or found spurious zeros. It found them at the expected scales only.
What this does not say
- It does not make the coarse-graining analyst-free.
- The system/environment boundary is “assume[d] … given”.
- The scale m is chosen, and the search is run at each m.
- The admissible maps are chosen. In the worked theory they are linear subspaces mixing all units, not partitions of the units into groups.
- What the dynamics fix is the target: the variable’s own future rather than an outside query. The authors contrast this with abstractions “driven by the agenda of the observer”.
- It does not say the optimum is unique. The authors keep every local optimum as part of an “emergence portrait”. The planted example has one exact optimum per scale because it was built that way, “in effect, … a ‘privileged’ coordinate system”.
- It does not say emergent variables are common. For generic linear models of order above 1 there is no exact one. The portrait then consists of near-independent optima, and “near” is set by a threshold or a significance test.
- It does not say an independent variable predicts itself well. A white-noise coordinate can be exactly independent (§5.2).
- It says nothing about locality. The criterion is closure against the rest of the system, not informativeness about a far environment across a buffer.
Connections
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QUESTION-065: answers in part.
- First part. Yes, a target can be supplied by the system: here it is the coarse variable’s own future. But it is not the locality-based target the question names. Boundary, scale and the class of maps remain the analyst’s.
- Second part. Under this criterion the winner at a scale is not unique in general: the cost is multimodal, and exact optima are generically absent.
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THEORY-198 and THEORY-205. Those select coarse variables by information about a far environment. This selects them by closure. On a local lattice the two could be compared directly, which is the divergence-or-match result QUESTION-065 asks for. No paper held has made that comparison.
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CLAIM-157. Near-decomposability asks for weak coupling across a block’s edge both ways. Dynamical independence asks only for none coming in. A module that drives the rest and is not driven back is exactly independent. A driven module never is.
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CLAIM-167. On this criterion a driver is a macro-variable of its own, and driver-plus-driven may form a larger nested one. That reading of one-way drive differs from treating the pair as a single system at one scale.
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CLAIM-152. It is consistent with CLAIM-152: this is still sufficiency for a target, fixed here as self-prediction. It does not refute it.
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Mori–Zwanzig memory (2026-10-20). In a linear Gaussian VAR(1), exact dynamical independence implies that the coarse variable’s Mori–Zwanzig memory vanishes, but zero memory does not imply independence (CLAIM-389). Memory is the round trip out of the coarse variable and back; dependence is the inward leg alone. Optimal prediction (NOTE-989, NOTE-975) also conditions differently. It conditions on the resolved initial value under an initial law, where this criterion conditions on the variable’s own stationary history. So the two literatures’ “closure” are not one notion.
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Weak incoming coupling (2026-10-20). For a block of coordinates, dynamical dependence is of second order in the coupling into it (CLAIM-390). That is the quantitative form of “independence forbids incoming”.
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Causal emergence (2026-10-20). THEORY-764: Rosas et al.’s Ψ subtracts exactly the one-step dynamical dependence, which bounds this account’s history-conditioned dependence from above. So Ψ’s decoupling term vanishing implies dynamical independence, but not conversely.