Measuring Frozen Accidents: Path-Dependent Emergence in Complex Adaptive Systems

Abstract

Emergent order in technologies, living systems, and now learned machines is often a frozen accident: a system reliably organizes, but which organization forms is selected by history, not fixed by fundamentals. This is path dependence: an early lead under increasing returns locks in, as with QWERTY or VHS. Yet our measures of emergent structure are synchronic, so they cannot separate an inevitable order from a history-selected one, a structural limit, not a measurement problem. We supply the missing diachronic measure: identifying it with non-ergodicity, we define a path-dependence degree μpath ∈ [0,1], the across-history variance of the emergent order, exact in the reinforced urn. We measure it in a complex adaptive system whose history is cheap to replay: a transformer language model retrained from a new seed. Across nine seed replicates μpath is small but nonzero and tracks capability fragility. At a second size it recovers the two runs an independent stability analysis flags as anomalous. Emergence thus gains a second axis: not just what order forms, but how much of it was accidental. Because such an accident is not fixed by the recipe, controllability becomes a measured quantity rather than an assumption.