The claim
Take n ≥ 2 units with X_t = A X_{t−1} + ε_t, ε_t i.i.d. N(0, Σ), Σ ≻ 0, and uniform coupling A = αI + β(11ᵀ − I), stable (|α + (n−1)β| < 1 and |α − β| < 1). Let Y = 1ᵀX, the sum.
Closure. 1ᵀA = (α + (n−1)β)1ᵀ, so L = 1ᵀ meets THEORY-733’s condition L A = B L with B = α + (n−1)β. By its sufficiency step, Y_t − B Y_{t−1} = 1ᵀε_t, which is independent of the whole past of X. So the transfer entropy I(Y_t : X⁻ | Y⁻) is zero: Y is exactly dynamically independent, for every β, including |β| ≥ |α|, and for any Σ.
No block. Every off-diagonal entry of A is β. For any partition of the units into two or more groups, each cross-group coupling equals each within-group coupling. So no partition makes the coupling across its edges small against the coupling inside, which is what near-decomposability asks (THEORY-661), and no proper group of units is a block for CLAIM-157’s test.
So closure of a collective variable does not imply that it is the aggregate of a weakly coupled block. The converse failure, a block whose aggregate is not exactly closed, is generic by THEORY-733 and is not the interesting direction; whether near-decomposability bounds the dependence is QUESTION-221.
The check
CASE-120, n = 6, Σ = I unless stated. The dependence was computed as ½ log det Var(Y_t | Y_{t−1..t−60}) / det Var(Y_t | X_{t−1}) from the exact stationary covariances. For the sum it was zero to machine precision (|T| ≤ 2.3 × 10⁻¹⁶) at (α, β) = (0.2, 0.01), (0.2, 0.05), (0.2, 0.1), (0.2, 0.13), (0.05, 0.15) and (0.3, −0.1), and with a random positive definite Σ. For a single unit it was 2.7 × 10⁻⁴ to 8.1 × 10⁻², and for the sum of three units 4.7 × 10⁻⁴ to 8.5 × 10⁻², rising with |β|.
What it does not say
- Not that the sum is one system. Closure is not CLAIM-157’s notion of unity, and neither is an agent or a self. Which of them “one system” should track is QUESTION-218.
- Not that near-decomposability is coordinate-free. In the eigenbasis of A (the sum and its orthogonal complement) the system is exactly decomposable. The claim is about partitions of the units, which is what CLAIM-157 groups. The gap that makes the mode definite is the one CLAIM-164 names, between A’s eigenvalues α + (n−1)β and α − β. That is a gap between modes, not between groups of members.
- Not anything beyond the model. Linear maps, Gaussian noise and VAR(1). For VAR(p ≥ 2), lags that are all of the uniform form keep the condition; generic lags do not (THEORY-733).
Note of 2026-10-20: every partition aggregates, none is a block
Shpak, Stadler, Wagner and Hermisson (LIT-1242, read in NOTE-979) give the exact condition for a partition’s block sums to close jointly: column-equitability (THEORY-755). Under uniform coupling every partition meets it. So in this system grouping the members does find closed objects, namely the vector of block sums for any grouping, even though no group is a weakly coupled block and no single group’s sum closes on its own. Re-checked: with six units split three and three, the transfer entropy into the pair of block sums was 0 to machine precision, and into one block sum 5.0 × 10⁻² at (α, β) = (0.2, 0.1). The claim, which is about blocks and about the sum of all units, stands. What changes is QUESTION-218: its choice between “a subset of members” and “a function of the whole state” leaves out the equitable partition.
The same holds for Markov chains. With uniform weights every partition is strongly lumpable, so every block label is closed (THEORY-822), and Rosas et al.’s “modular” random walk closes at every ratio of between- to within-module rates, by symmetry rather than modularity (NOTE-1030).
Note of 2026-11-01: closure is tested by the interventions the constitution debate allows
Baumgartner and Gebharter (NOTE-1493, p. 20) note an asymmetry between levels. A whole’s parts can be changed without the whole changing, since several micro states realize one macro state. The whole cannot be changed without some part changing (THEORY-1288). This claim’s closure is a statement about the first kind of change. The sum is exactly dynamically independent because any redistribution of activity among the units that leaves the sum fixed leaves the sum’s future unchanged. Those redistributions are interventions inside a fibre of the map from units to sum. In the debate’s terms they are surgical bottom-up interventions, which can exist. So closure, unlike constitution, is in principle testable by intervention. The test is to perturb within the fibre and watch the aggregate. CLAIM-526 uses this.
No status, title or warrant changes.