The claim
Fixed target: monotone. Let S carry a law, I be drawn by ρ independently of S, and Y^(I) be the outcome, as in CLAIM-407, with Γ(T) = I(S; Y^(I) | T(S), I). If T₂ = f(S) and T₁ = g(T₂), then by the chain rule
I(S; Y^(I) | T₁, I) = I(T₂; Y^(I) | T₁, I) + I(S; Y^(I) | T₂, I),
since T₁ and T₂ are functions of S and T₁ of T₂. The first term is non-negative, so Γ(T₂) ≤ Γ(T₁), with equality exactly when Y^(I) is independent of T₂ given T₁ and I. A16 §5 gives the same argument in entropies: H(Y^(I) | T₂, I) ≤ H(Y^(I) | T₁, I) “because conditioning on (T_2) supplies at least the information in (T_1), while (H(Y^{(I)}\mid S,I)) remains unchanged.” Nothing in it is special to governance: it is any fixed-target sufficiency, as in CLAIM-152.
Closure: monotone in neither direction. Informational closure (TERM-068, sense 1) of Z_t = g(X_t) is Δ_X(Z) = I(X⃗_t; Z_{t+1} | Z⃗_t) = 0 (THEORY-820). Take THEORY-820’s chain, X_t = (A_t, B_t, C_t) bits, with A_{t+1} and C_{t+1} fresh fair bits and B_{t+1} = A_{t+1} ⊕ C_t. Computed from the exact chain with one, two and three steps of history:
| Z | A | B | C | (A, C) | (B, C) | (A, B) | A⊕B | X | constant |
|---|---|---|---|---|---|---|---|---|---|
| Δ_X(Z), bits | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
- Up the refinement order. A ⪯ (A, B) ⪯ X, with Δ = 0, 1, 0. Refining a closed description can break closure, and refining further can restore it. The full state is always closed.
- Down the order. A⊕B ⪯ (A, B), with Δ = 1 and 1; constant ⪯ A⊕B, with Δ = 0 and 1. Coarsening a non-closed description can leave it non-closed or make it closed.
So the closed coarse-grainings are neither upward- nor downward-closed in the refinement order. THEORY-820 has the first half (not closed under common refinement). THEORY-822’s closure under common coarsening holds for strongly lumpable partitions, and is not a general coarsening property: the identity is strongly lumpable and its coarsening (A, B) is not closed.
Why. Δ_X(Z) conditions on Z’s own past to predict Z’s own future. Refining Z enlarges the conditioning set and also changes the target Z_{t+1}. The data-processing inequality covers the first change alone. A7: “There is no general data-processing inequality ordering these quantities, because both the predicted variable and the conditioning information change.” In the chain, by the chain rule, Δ(A, B) = I(X⃗; A_{t+1} | A⃗, B⃗) + I(X⃗; B_{t+1} | A⃗, B⃗, A_{t+1}) = 0 + 1 bit: the bit is carried by the pair’s future jointly, A_{t+1} ⊕ B_{t+1} = C_t. A16 §6 makes the same point for marginal against joint targets: with Y₁ = U and Y₂ = U ⊕ K for fair independent U and K, I(K; Y₁) = I(K; Y₂) = 0 and I(K; Y₁, Y₂) = 1 bit. A16: “What fails here is the assumption that sufficiency for each marginal target entails sufficiency for the joint target.” And A16 §5’s moral: “Adding information cannot hurt exact sufficiency for an unchanged question, but changing what the description is expected to explain can expose new dependencies.”
Blackwell. If Z₁ = h(Z₂), Z₁ is a garbling of Z₂ as an experiment about X_t, with h as the garbling kernel. By THEORY-717, Z₂ is then at least as good as Z₁ in every decision problem. So A ⪯ (A, B) in Blackwell’s order, and closure goes from 0 to 1 bit. A7’s corollary: “Decision informativeness and dynamical closure are not order-compatible.” The order that guarantees an ideal decision-maker can do no worse (CLAIM-225) does not order closure, because closure is not a fixed-target sufficiency.
Checked on 2026-10-21: every entry of the table by exact enumeration of the chain’s path law at history lengths 1, 2 and 3; the XOR values; and the monotone inequality on 3,000 random finite systems (up to seven states, three outcomes and three interventions, with random refinement pairs), with no violation.
What it does not say
- Not that richer descriptions are less autonomous. A7’s title, “When More Information Reveals Less Autonomy”, and its “a description can become strictly more informative while becoming less autonomous”, are true as an existence claim and wrong as a trend: the full state is always closed, and (A, C) refines A and stays closed.
- Not that a finer description is better all told. A17: refinement “does not worsen ideal sufficiency for an unchanged response target, although it may increase computational cost.”
- Not that Γ orders descriptions not related by refinement. The inequality holds along refinement chains only. That optimal partitions at two budgets need not nest (CLAIM-387, CLAIM-392) is about optima, not about one description’s deficiency.
- Nothing approximate. Small Δ is not shown to behave better.
- Not the causal-abstraction papers’ transitivity. Rubenstein et al.’s Lemma 5 and Beckers and Halpern’s Theorem 3.9 compose abstractions of abstractions; nothing in those papers or in Beckers, Eberhardt and Halpern orders a deficiency along refinement (NOTE-1037, NOTE-1032). The nearest analogue is the latter’s composition bounds for approximate abstraction (THEORY-843).
- Not that closure marks a system, or its failure a non-system. Noise is closed (THEORY-820), and that closure gives no part–whole order on candidate units is the open half of QUESTION-218.