Selecting the Right Early-Warning Signal: Structural Failure of Critical-Slowing-Down Indicators at Scale-Mixture Transitions in Nonlinear Stochastic Dynamics

The selection of the monitoring signal is the decisive design choice in any early-warning system, yet it has received little theoretical attention compared with the indicators developed for fold bifurcations by Scheffer and collaborators. We study this choice for scale-mixture transitions in nonlinear stochastic dynamics, in which the driving noise is conditionally Gaussian with a slowly ramping amplitude σ(t), so that the increment law is locally Gaussian within any observation window but aggregately heavy-tailed across the ramp as a single amplitude parameter crosses a critical value σc. We compare five candidate indicators on a three-dimensional Itô reserve diffusion: (C2) an amplitude estimate from the quadratic variation of observation increments; (B2) sliding-window kurtosis; (A3) coupling-consistency residuals; and the two classical Scheffer indicators (Variance, AR(1)). Our central result is analytical, not merely empirical: we prove that the two critical-slowing-down (CSD) indicators cannot carry an early signal in this regime. Sliding-window kurtosis has a population value that is independent of the control parameter (Proposition 1); lag-1 autocorrelation approaches a bounded limit fixed by the relaxation rate rather than diverging, and its only dependence on the control parameter is an inverted one mediated by observation noise (Proposition 2). A Monte Carlo study over nine (Tramp, σobs) configurations confirms both propositions: B2 never crosses threshold and AR(1) fires only at the highest observation noise, and then late. The C2 indicator, by contrast, is an unbiased amplitude estimator and delivers the earliest warning in the low-observation-noise regime, with a closed-form lead-time scaling law L = κ(σobs) Tramp − W dt/2 that matches the simulations to within a few percent. The dominance is regime-dependent: sliding-window variance, although a biased and lagging amplitude proxy that C2 strictly improves upon at low observation noise, remains competitive and overtakes C2 once σobs becomes comparable to the dynamic range of the ramp signal. The contribution is a sharp, proof-backed delimitation of when classical EWS indicators fail by design and of the regime in which a noise-amplitude estimator is the correct alternative.