Source
Simon (1962), LIT-905, “The Architecture of Complexity”, Proceedings of the American Philosophical Society 106(6):467–482, read in full in NOTE-704. The theorem is Simon & Ando (1961), which the lecture cites and the record could not obtain (LIT-1127, Unavailable). The nearest open statement of it is Shpak, Stadler, Wagner and Altenberg (2004), LIT-1241, read in NOTE-977: Theorems 2.1–2.2 there give the eigenprojector convergence and the four time windows as existence statements, with no rate in ε and no error term on the aggregate.
What was actually shown
The propositions. “(a) in a nearly decomposable system, the short-run behavior of each of the component subsystems is approximately independent of the short-run behavior of the other components; (b) in the long run, the behavior of any one of the components depends in only an aggregate way on the behavior of the other components.” Formally, the matrix of interaction coefficients is block-diagonal with off-diagonal entries below a small ε.
The worked example. A building of rooms divided into cubicles. Within hours each room equalises its temperature; over days the building does. “After the intraroom short-run equilibria have been reached, a single thermometer in each room will be adequate”.
Hierarchy by interaction. Simon defines a hierarchy as a system of interrelated subsystems, each hierarchic in turn down to some elementary level, and includes systems “in which there is no relation of subordination among subsystems”. Spatial and social hierarchies are reconciled “by defining hierarchy in terms of intensity of interaction”: nerve cells and telephone wires “permit very specific strong interactions at great distances”.
Description. Near decomposability is one reason hierarchic systems can be described briefly: “most things are only weakly connected with most other things”.
What could have come out otherwise. Weak coupling between blocks could have left no separation of time scales. The theorem says it leaves one for ε small enough. The open statement the record holds gives no rate in ε (NOTE-977), so this entry does not claim one. A rate is available in a different frame: Bousquet and Slotine (THEORY-771) give a first-order bound in ε on trajectory error, uniform in time after a transient, for nonlinear continuous-time systems whose fast modules and reduced slow system are contracting and whose slow coordinates are given. Del Vecchio and Slotine (THEORY-773) give the same kind of rate for the quasi-steady-state reduction, with explicit constants. Neither is Simon and Ando’s theorem, and neither bounds dependence.
What this does not say
- That complex systems are in fact nearly decomposable. That is Simon’s empirical generalisation across examples, with no count. He allows that complex systems that are not hierarchic “may to a considerable extent escape our observation”, and leaves open which way that dependence runs.
- That social systems are. “Near decomposability is generally very prominent” in social systems is asserted from formal organisations and channels of communication.
- That higher levels of an organisation are slower. “It is probably true” that higher-frequency dynamics belong to subsystems; that executives’ horizons lengthen with rank is “generally believed”. No data.
- How fast hierarchies evolve. The watchmaker parable (Hora against Tempus, about 4,000 times slower) shows that the time to assemble a complex form “depends critically on the numbers and distribution of potential intermediate stable forms”. Simon calls its assumptions “not particularly realistic” and says the number cannot be taken seriously.
- Where a block’s boundary is. The theorem takes the partition as given. Choosing it, and how small ε must be, are separate questions.
Connections
- The record’s own argument. CLAIM-157 holds that a combined form is one system while its joint dynamics are nearly decomposable with the form as a block whose gap beats the Occam factor; this is the result it relies on for “nearly decomposable”. CLAIM-331 holds that higher-level variables are characteristically slower than their constituents; this supplies the fast-inside, slow-aggregate-outside half, and does not supply Simon’s unsupported claim about organisations. CLAIM-154 prices the block.
- Dynamical independence (THEORY-257) asks when a coarse variable is closed given its own history. Near decomposability gives approximate closure of block aggregates over the long run, by coupling strength. The record has no bound of the one by the other (QUESTION-221), and the converse fails: a collective variable can be exactly closed while no grouping of the units is a block (CLAIM-388). Shpak et al. (LIT-1242, unread) separate aggregation from decomposition in Simon and Ando’s setting.
- Modular synchronisation (THEORY-265) is a model of the same shape: coupling strong inside modules makes each module respond as a unit.
- Coupling is not constitution (CLAIM-173, CLAIM-194). A block can be strongly coupled because one side drives the other, or because a boundary was imposed on it. The theorem is silent on both.
- Bounds the record has derived (2026-10-20). For a block’s full state in a linear Gaussian VAR(1), dynamical dependence is of second order in the coupling into the block (CLAIM-390). For a block’s aggregate, the one-step closure defect splits exactly into a first-order part from coupling between blocks and a part from the blocks’ own dynamics, which vanishes iff each block has equal column sums (CLAIM-386).
- Nearly decomposable is a spectral condition. NOTE-977’s flat landscape has entrywise weak coupling between blocks at every mutation rate and no fast–slow separation at any. Read entrywise, the hypothesis is not sufficient.
- Scale locality is a different pattern (THEORY-754). In a turbulent cascade, coupling between scales falls off as a power of their ratio with no gap, and the whole flux crosses every scale. It is banded, not block-diagonal, so this theorem does not apply to it.