An information-theoretic foreshadowing of mathematicians' sudden insights.

The "eureka" insights that drive progress in science and mathematics remain shrouded in mystery. Sudden, unexpected, appearing like "flashes of lightning", these insights have the hallmarks of critical transitions in complex systems. Here, zooming in on mathematicians working on proofs in their own departments, we show that sudden insights are anticipated by a system-agnostic, information-theoretic early warning signal. Using dense behavioral recordings of mathematicians' moment-to-moment activity, we find that their blackboard interactions (e.g., writing, gesturing; [Formula: see text]) became increasingly unpredictable before an insight, analogous to the critical fluctuations that anticipate transitions in physical and ecological systems. We explore analytically when this early warning signal applies to varied systems with discrete, symbolic dynamics. While bibliometric analyses offer a zoomed-out perspective on innovation, publications are a coarse-grained record of individuals' insights. Explaining the sudden insights of innovators, from scientists to sculptors, requires attending to the local, distributed systems of their intellectual activity.