The claim
Let A = A* + εC with A* = diag(A₁, …, A_m). Let Q be the block-sum map (Q_Ii = 1 for i in block I). Let V be the n × m matrix whose I-th column is block I’s leading eigenvector v_I, normalized so that 1ᵀv_I = 1, so QV = I. The Simon–Ando aggregate, in the form NOTE-977 reads from Shpak et al.’s (3.13b), is  = QAV.
- The identity. D := QA − ÂQ = QA − QAVQ = QA(I − VQ). Since A = A* + εC, D = QA*(I − VQ) + ε QC(I − VQ).
- The blocks’ own part. Block I’s row of QA*(I − VQ) is 1ᵀA_I(I − v_I1ᵀ) on block I’s coordinates. If every column of A_I sums to λ, then 1ᵀA_I = λ1ᵀ, and λ1ᵀ(I − v_I1ᵀ) = λ(1ᵀ − 1ᵀ) = 0. Conversely, if 1ᵀA_I(I − v_I1ᵀ) = 0, then 1ᵀA_I = (1ᵀA_I v_I)1ᵀ, a constant row: the column sums are equal. Equal column sums is THEORY-755’s column-equitability for the trivial partition of each block, and includes stochastic blocks.
So with equal column sums the one-step defect is exactly εQC(I − VQ). Without them it has an O(1) part from the blocks themselves, which Simon and Ando’s argument relies on the fast transient to remove.
The checks. Three blocks of sizes 3, 4 and 3, with a random C. The identity held to 2 × 10⁻¹⁶. The blocks’ own part was 2.6 × 10⁻¹⁶ with column-stochastic blocks, and 0.51 with unequal column sums. Measured on states after 200 steps, the relative defect ‖Dx‖/‖QAx‖ was 5.6 × 10⁻⁵, 4.2 × 10⁻⁷ and 4.0 × 10⁻⁹ at ε = 10⁻², 10⁻³ and 10⁻⁴ with equal column sums (second order), and 2.3 × 10⁻³, 1.7 × 10⁻⁴ and 1.6 × 10⁻⁵ without (first order). Those orders are numerical. They rest on the standard estimate that post-transient states lie within O(ε) of the slow subspace, which was not proved here.
What it does not say
- No bound on dynamical dependence. The defect is deterministic and one-step. For the information-theoretic version, for a block’s full state, see CLAIM-390.
- Nothing about the length of the window. Simon and Ando’s aggregate is meant to hold between the end of the fast transient and the onset of global relaxation. This identity holds at every step, but the smallness of D on actual states needs the transient to have passed.
- Not that Shpak et al. state it. They give  and no error term.
Note of 2026-10-20: the estimate the orders rested on
The post-transient orders above rested on “the standard estimate that post-transient states lie within O(ε) of the slow subspace”. Bousquet and Slotine’s Lemma 3 (THEORY-771) is that estimate in continuous time, under contraction of the fast modules and of the reduced slow system: the fast part x̃ stays O(ε) uniformly after a transient, so the coupling defect ε·QCx̃ is O(ε²). In continuous time block sums are slow coordinates iff QA* = 0, the counterpart of equal column sums. The claim, discrete-time and one-step, is unchanged.
Note of 2026-11-01: the closure defect as a response to within-block redistribution
The defect QA(I − VQ) is the aggregate’s one-step response to the part of the state that the aggregation map Q discards. VQ keeps each block’s sum and spreads it along the block’s leading eigenvector, and since Q(I − VQ) = 0, (I − VQ)x is a redistribution within blocks that leaves every block sum unchanged. In the terms of the constitution debate read today (NOTE-1493, p. 20; THEORY-1288), these are changes to the parts that leave the whole fixed, the one kind of intervention across levels that can be surgical. So the defect is measurable by intervention, by perturbing within blocks and reading the aggregate’s next value. The relation between an aggregate and its blocks cannot be measured that way, since every intervention on an aggregate also changes some block. Same structure, not shown to be the debate’s part–whole relation: Q is a linear map, not a mechanism.
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