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QUESTION-221: Does near-decomposability bound dynamical dependence: when the coupling into a block from the rest of the system is at most ε, is the dynamical dependence of the block's aggregate small to a stated order in ε?

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Properties7
StatusOpen
Version3
Tagscomplex-systems, information-theory, mathematics
Linedistributed-agency
RefinesQUESTION-065 — Can the target that a sufficiency-based boundary is relative to be supplied by the system's own locality, as informativeness about its far environment across a buffer, so that the coarse variables kept no longer depend on the analyst, and is the partition that wins for a given target unique?
Refined byQUESTION-242 — Does exact dynamical closure of an aggregate require a nearly decomposable block, or can a collective variable be exactly dynamically independent when no partition of the units is weakly coupled?
BearingTHEORY-771 — In a singularly perturbed system, fast states x̃ with dx̃/dt = f̃(x̃, y) + ε·δf and slow states y with dy/dt = ε·ḡ(y) + ε·δg, if f̃ is contracting in x̃ at rate β_f with x̃ = 0 an equilibrium for every y, ḡ is contracting at rate β_g, and |δf| ≤ d_f + α_f,x|x̃| + α_f,y|y|, |δg| ≤ d_g + α_g,x|x̃|, then for every ε below ε_c = β_f / [χ_f(α_f,x + χ_g α_f,y α_g,x / β_g)] the fast states stay within O(ε) of zero and, when d_g = 0, the slow states within O(ε) of the reduced trajectory, uniformly in time after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε), CLAIM-390 — For a block of coordinates in a stable linear Gaussian VAR(1), dynamical dependence is at most ½ log det(I + Σ_YY⁻¹ A_YR C A_YRᵀ), with A_YR the coupling into the block and C the covariance of the rest given the block's present state, so it is of second order in the coupling into the block, whatever the coupling out of it, CLAIM-386 — For x(t+1) = (A* + εC)x with A* block-diagonal, the Simon–Ando aggregate  = QAV has one-step closure defect exactly QA − ÂQ = QA(I − VQ); the part from the blocks' own dynamics vanishes iff every block has equal column sums, and then the whole defect is ε·QC(I − VQ)

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Why it is a question

The record has two criteria for a unit. CLAIM-157 names a block whose coupling across its edge is weak (THEORY-661, Simon). THEORY-257 names a coarse variable into which no information flows from the rest given its own history (Barnett and Seth). CLAIM-157 already records that they pick different blocks when coupling is one-way. What is missing is a bound in the symmetric case. THEORY-661 states the link only qualitatively: “Near decomposability gives approximate closure of block aggregates over the long run, by coupling strength.”

What the record holds

  • THEORY-733: for a stable linear Gaussian VAR, Y = LX is exactly independent iff L A_k = B_k L for every lag. A5 proposed the closure defect min_B E‖LAX_t − BLX_t‖² as the approximate version, and noted that it differs from total prediction error, since a closed macroprocess can still have large innovation noise.
  • THEORY-700 and THEORY-257: in the Gaussian case the dependence is a ratio of generalized variances. A5’s ½ log det R / det Q, with R and Q the innovation covariances of Y given its own past and given the micro past, is that form.
  • CLAIM-388: the converse fails. Under uniform coupling the sum of all units is exactly closed while no partition is nearly decomposable.
  • The forward half, answered in part on 2026-10-20:
    • For a block’s full state in a linear Gaussian VAR(1), dynamical dependence is at most ½ log det(I + Σ_YY⁻¹ A_YR C A_YRᵀ), of second order in the coupling into the block, whatever the coupling out of it (CLAIM-390).
    • For a block’s aggregate, the Simon–Ando one-step closure defect is exactly QA(I − VQ). Its part from the blocks’ own dynamics vanishes iff each block has equal column sums (CLAIM-386). Numerically, the relative defect after the transient is then second order in ε, and first order otherwise.
    • For a partition of a Markov chain, exact closure of the label is strong lumpability (THEORY-822). Rosas et al. name quasi-lumpability methods but give no bound. In one jittered random walk the transfer entropy into the module label rose from 3 × 10⁻⁸ to 4 × 10⁻⁶ as the between-module rate rose from 0.001 to 1, roughly in step with the spread of block row sums (NOTE-1030). One example, not a rate. The open target: a bound on transfer entropy from that spread and the smallest probability of leaving a block.
    • Still open: the aggregate’s dynamical dependence itself, rather than the deterministic defect, and any bound over a horizon longer than one step. The open statement of Simon–Ando (NOTE-977) gives no rate.
    • A rate for trajectory error, not dependence: under contraction of the fast modules and of the reduced slow system, and a small-gain separation ε < ε_c, fast states stay O(ε) from quasi-steady values and slow states O(ε) from the reduced trajectory, uniformly in time after a transient (Bousquet and Slotine, THEORY-771). It is deterministic. With that estimate the block sums’ closure defect ε·QCx̃ is O(ε²), matching CLAIM-386’s measured second order. One numerical check in the reading, not a proof, put the transfer entropy into block sums at second order in ε.

What would count as an answer

For a stated class (linear Gaussian VAR(1) first), a bound on the dynamical dependence of a block’s aggregate in terms of the size of the off-block coupling, the spectral gap and the noise covariance, over a stated horizon; or a family of nearly decomposable systems in which it stays bounded away from zero as ε → 0. A5 was explicit that the useful results “would be conditional bounds, with the distributions and norms explicit.”

Split, 2026-10-21 (ADR-049). The converse is QUESTION-242, Answered by CLAIM-388. This question keeps the forward bound for the block’s aggregate. CLAIM-390 answers it for the block’s full state and CLAIM-386 gives the one-step deterministic defect, but neither bounds the aggregate’s dynamical dependence.

Note of 2026-10-21: a non-implication claimed as established

A2 of the exchange of 2026-10-20 listed “an approximate reduced dynamics can be accurate without establishing informational independence” among non-implications “already established”, on THEORY-771 alone. A3 called the same case “a promising conceptual separation”. Neither cited this question, CLAIM-390 or CLAIM-386. The record bounds the dependence (second order in the coupling into a block) but holds no construction of the non-implication as stated. It is one of the cells QUESTION-252 asks to be filled.

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