Source
Rosas, Geiger, Luppi, Seth, Polani, Gastpar and Mediano (2024), arXiv:2402.09090, LIT-1306, read in full in NOTE-1030. The equivalence of informational and causal closure is the paper’s Theorem 1, the implication to computational closure its Theorem 2, and the minimality lemma its Proposition 1. Dynamical independence is Barnett and Seth’s definition (LIT-979, THEORY-733). The one-step reduction, the corrected counterexample and the refinement counterexample below are the record’s.
What was actually shown
X is a stationary process with a discrete alphabet. X⃗_t is its past up to t and X⃗⁺ a future block (X_{t+1}, …, X_{t+L}). A coarse-graining is Z_t = g(X⃗_t). It is spatial if Z_t = f(X_t). Statements about conditional laws hold up to null sets.
(a) Every horizon reduces to one step. Informational closure (Definition 4) asks that I(X⃗_t; Z_{t+1}, …, Z_{t+L} | Z⃗_t) = 0 for every L. By the chain rule this equals
Σ_{ℓ=1}^{L} I(X⃗_t; Z_{t+ℓ} | Z⃗_{t+ℓ−1}).
X⃗_t is a function of X⃗_{t+ℓ−1}, so each term is at most I(X⃗_{t+ℓ−1}; Z_{t+ℓ} | Z⃗_{t+ℓ−1}). By stationarity, that is T = I(X⃗_t; Z_{t+1} | Z⃗_t). So closure at every L holds iff T = 0. T is Barnett and Seth’s transfer entropy T(X → Z), with no environment. So informational closure is exactly dynamical independence. The paper assumes this identity and does not show it.
(b) The υ-machine is the coarsest prescient partition (Prop. 1). U_t = υ(X⃗_t) groups pasts with equal laws p(z⃗⁺ | x⃗) at every L.
- U is prescient. Those laws depend only on U, so I(U_t; Z⃗⁺) = I(X⃗_t; Z⃗⁺).
- U is the coarsest. Let D = d(X⃗_t) also be prescient. Since D is a function of X⃗_t, I(X⃗_t; Z⃗⁺) = I(D; Z⃗⁺) + I(X⃗_t; Z⃗⁺ | D), so the last term is 0. Then p(z⃗⁺ | x⃗) = p(z⃗⁺ | d(x⃗)), and pasts in one D-class lie in one U-class. So U is a function of D.
- The same lemma with X replaced by Z gives the classical fact for Z’s causal states E′: they are the coarsest prescient function of Z⃗.
(c) Informational ⟺ causal closure (Thm. 1).
- Z⃗_t is a function of X⃗_t, so I(X⃗_t; Z⃗⁺ | Z⃗_t) = I(X⃗_t; Z⃗⁺) − I(Z⃗_t; Z⃗⁺) = I(U_t; Z⃗⁺) − I(E′_t; Z⃗⁺). So closure holds iff U and E′ carry equal information about Z⃗⁺ at every L.
- If they do, then E′ is a function of X⃗_t and prescient for Z⃗⁺, so by (b) U = h(E′). Then U is a function of Z⃗_t and prescient among functions of Z⃗_t. By (b) applied to Z, E′ is a function of U. Each is a function of the other, so the two partitions are equivalent: causal closure.
- Conversely, equivalent partitions carry equal information.
(d) Spatial maps: the υ-machine is a quotient of the ε-machine (Prop. 2).
- With Z_t = f(X_t), p(z⃗⁺ | x⃗) = Σ_{x⃗⁺ ∈ f⁻¹(z⃗⁺)} p(x⃗⁺ | x⃗). Pasts with equal micro future laws (one causal state E_t) therefore have equal macro future laws, so U_t = f′(E_t).
- The same sum shows the map f ↦ U_f preserves order. If f is coarser than f̃, U_f is a function of U_f̃.
(e) Causal ⟹ computational closure for spatial maps (Thm. 2). By (c), E′ ≅ U. By (d), U = f′(E). So E′_t = f*(E_t): Z’s causal states are a coarse-graining of X’s.
Neither converse holds.
- Computational without causal closure (the paper’s Counterexample 4, checked). P has rows (½, ½, 0, 0), (0, 0, ½, ½), (½, ½, 0, 0) and (0, 0, ½, ½). Z merges {1, 2} and {3, 4}. Under the uniform stationary law Z is i.i.d., so E′ is trivial and trivially a quotient of E. But X_t ∈ {1, 3} sends Z to the first block surely, so T = 1 bit.
- Spatiality is needed for (e). Take X i.i.d. and non-degenerate, and Z_t = (X_t, X_{t−1}). Then Z_{t+1} = (X_{t+1}, X_t), whose second part is in Z_t and whose first is independent of every past, so T = 0. Z’s causal state is X_t, while X’s ε-machine is trivial. So E′ is not a function of E.
- The paper’s own example for this point, Counterexample 5 (Z_t = X_{t−1}), is wrong. X⃗_t contains Z_{t+1} = X_t and Z⃗_t does not, so T = H(X_t | X_{t−1}) > 0.
Closure is not preserved by common refinement. Let X_t = (A_t, B_t, C_t) be bits, with A_{t+1} and C_{t+1} fresh fair bits and B_{t+1} = A_{t+1} ⊕ C_t.
- Every state sends probability ½ to each value of A and of B. So by THEORY-822 each of A and B is closed.
- The pair is not: A_{t+1} ⊕ B_{t+1} = C_t is known to X_t and independent of the past of (A, B).
- Computed from the exact chain: T = 0, 0 and 1 bit for A, B and (A, B), with one, two and three steps of history.
- So the closed spatial coarse-grainings do not form a sublattice of all coarse-grainings, contrary to the paper’s §III E and Theorem 2. The υ-machine map does not preserve meets either: U_A and U_B are trivial, while U_{(A,B)} is C_t.
What could have failed: informational closure could have been weaker than causal closure, since equal information need not mean equal partitions. It is not, because both partitions are the coarsest prescient ones of their kind.
What this does not say
- Not that closure marks a system. An i.i.d. noise component is closed. The paper’s own remedy is a computational quotient, not a mark of unity. Whether a closed variable is one system is QUESTION-218’s open half.
- Not interventional causation unless X is fully observed. Otherwise “causal closure” is closure in the Granger sense, as the paper says.
- Not anything approximate. Small transfer entropy is not shown to mean nearly equal machines, or the reverse.
- Not continuous or non-stationary processes. For the linear Gaussian case, THEORY-733 holds the corresponding criterion.
- Not that the closed coarse-grainings form any lattice. Closure under common coarsening is shown for strongly lumpable partitions (THEORY-822’s note), not for closure in general.
Connections
- THEORY-733. Its exact dynamical independence is this informational closure, for linear maps of a Gaussian VAR. So its criterion L A_k = B_k L is also the VAR form of causal closure. THEORY-257’s reading that the dynamics fix the variables meets the objection that closure overcounts (noise components, copies).
- THEORY-822. For Markov chains, closure is strong lumpability when the label is Markov.
- THEORY-756. Part (b) is the υ-machine version of THEORY-756’s coarsest prescient partition.
- QUESTION-218: answers in part. It confirms that the closure-side unit is a function of the whole state, or of the past. It is not a subset of members, and its closure does not extend to common refinements. It gives the question no criterion of unity.
Note of 2026-10-21: the refinement counterexample, as a change of question
A12 and A14 of the exchange of 2026-10-20, following A7, proposed a corollary: “Blackwell dominance under deterministic refinement does not imply preservation of informational closure”, from this entry’s A/B/C chain (Δ(A) = 0, Δ(A, B) = 1 bit, re-checked with one to four steps of history). For deterministic descriptions of one state, (A, B) Blackwell-dominating A is the same fact as A being a function of (A, B). So the corollary is this entry’s “not closed under common refinement” in the refinement order. Closure is not monotone the other way either: X itself is always closed and its coarse-grainings need not be.
A16 §5 and A17 §4.4 said why. For a fixed target, refining a description cannot raise its deficiency; closure’s target is the description’s own future, which refinement changes. So the three-bit chain (A and B closed, (A, B) not) is not a failure of “more information helps” but a change of question. Both halves are filed as CLAIM-409, which rests on CLAIM-407. A16 §2 also reconciled a governance-response failure with this entry by the entry’s own “Not interventional causation”; that is CLAIM-407 and CASE-125.
A13 writes the common refinement A ∨ B, where this entry calls it a meet. TERM-068 fixes the convention as this entry’s.