The claim
Let X = (Y, R) with X_t = AX_{t−1} + ε_t stable, ε_t ~ N(0, Σ), Σ ≻ 0, and write Y_t = A_YY Y_{t−1} + A_YR R_{t−1} + ε_{Y,t}. Dynamical dependence is T(X → Y) = ½ log det Var(Y_t | Y⁻) / det Var(Y_t | X⁻) (THEORY-257, THEORY-733), and Var(Y_t | X⁻) = Σ_YY.
The predictor A_YY Y_{t−1} + A_YR E[R_{t−1} | Y⁻] uses only Y’s past. Its error is A_YR(R_{t−1} − E[R_{t−1} | Y⁻]) + ε_{Y,t}, and the two terms are independent. So
Var(Y_t | Y⁻) ⪯ Σ_YY + A_YR Cov(R_{t−1} | Y⁻) A_YRᵀ ⪯ Σ_YY + A_YR C A_YRᵀ,
where C = Cov(R_{t−1} | Y_{t−1}) under the stationary law. The second step holds because conditioning on less cannot shrink a Gaussian conditional covariance. Since log det is monotone on positive definite matrices,
T(X → Y) ≤ ½ log det(I + Σ_YY⁻¹ A_YR C A_YRᵀ) ≤ ½ tr(Σ_YY⁻¹ A_YR C A_YRᵀ).
C ⪯ Π_RR, so for A_YR = εB with the rest fixed, T = O(ε²). The coupling out of the block, A_RY, enters only through C and the stationary covariance.
The check. Random five-variable systems with a two-coordinate block, twenty at each ε ∈ {0.2, 0.1, 0.05, 0.025}, with random noise covariances, and the dependence computed from exact stationary covariances. The bound held in all 80 cases. The ratio T/ε² stayed between 8.5 and 10.5 across the four values of ε, which is second order.
What it does not say
- Not about the block’s aggregate. The bound is for the block’s full state. A sum over the block can be dependent even with no coupling into the block, if the block’s internal dynamics do not close on the sum. That part is CLAIM-386.
- Not a reason to ask for weak coupling both ways. Only the coupling into the block enters the bound. Near-decomposability’s weak coupling in both directions is sufficient for a small bound, not necessary.
- Not beyond linear Gaussian VAR(1). For VAR(p) the same argument applies lag by lag and was not written out.