The case
A5 gave the continuous-time case of two units, ẋ = Ax with A = [[−a, b], [b, −a]], a > b > 0. The coordinate z = x₁ + x₂ obeys ż = (b − a)z exactly, while for b large the two units are strongly coupled rather than separate blocks. A5: “Exact aggregation ⇏ near-decomposability.”
Here it is in the record’s setting for dynamical independence, a linear Gaussian VAR(1) (THEORY-733): six units, X_t = A X_{t−1} + ε_t with A = αI + β(11ᵀ − I) and ε_t ~ N(0, Σ). Held fixed: the coupling between every pair is the same β. Varied: β from 0.01 to 0.15 with α = 0.2 or 0.05, one inhibitory case (α, β) = (0.3, −0.1), and Σ = I or a random positive definite matrix. Measured: the transfer entropy from X to three coarse variables, the sum of all six, one unit, and the sum of three.
Result (CLAIM-388 gives the numbers): the sum of all six is exactly independent in every setting; the other two are not, and their dependence grows with |β|.
What it can show
That a test for a unit by closure and a test by weak coupling across a group’s edge can disagree when the coupling is symmetric, not only when it is one-way, as CLAIM-157 already records. It is the closure-without-block cell of A5’s table:
| strong closure | weak closure | |
|---|---|---|
| strong block separation | both agree | block selected, dynamics not closed |
| weak block separation | this case | neither selected |
It cannot show that the sum is one system in any further sense. Nothing here compensates, entrains or maintains itself, and the units are not agents. Nor does it bear on systems whose coupling is not uniform: a generic VAR(p ≥ 2) has no exactly closed linear variable at all (THEORY-733).