The role of edge states for early-warning of tipping points

The paper proposes, with simulations and heuristic arguments, that observables aligned with the edge state/instanton direction exhibit the strongest early-warning signals, and documents alignment of precritical fluctuations with that direction in multiple models, including a realistic ocean model. However, the paper explicitly states that a general theoretical underpinning and a definitive criterion are lacking. By contrast, the model’s solution provides a standard OU/linear-response derivation that variance diverges like 1/(-Re α1) and lag-Δ autocorrelation tends to e^{α1Δ} along the leading eigenvector, plus a center-manifold/instanton alignment argument in the gradient case—precisely the missing theoretical backbone. The paper’s key claims (edge/instanton alignment and directional EWS) are supported empirically but not proved, and the authors themselves call for theory. Hence: paper incomplete, model correct. See the paper’s abstract and discussion of instanton-directed fluctuations and edge-state-guided observables, as well as the explicit note that more theory is needed for a general criterion .