Generalized Stochastic Resilience for Early Warning Signals Based on Koopman Operator

The paper’s core decomposition of the stochastic KMD residual into a Koopman-truncation term plus a stochastic covariance term is correct (their Eq. (29) via Corollary III.1) and matches the model’s bias–variance identity. However, the jump to Theorem III.3 (“residual → ∞ as β → β*”) silently imports extra assumptions that are not stated: (i) that the L2 measure μ is the stationary law of the Markov chain (so time-stationary AR(1)/Lyapunov variance formulas apply), and (ii) that the observable’s U(1) image has nonzero component outside the chosen finite subspace. The paper’s derivation of the covariance blow-up (their Eq. (32)) relies on a stationarity/independence argument for φλn(xt+1)=λnφλn(xt)+ηn,t and implicitly conflates the stationary time-variance with the one-step, x-integrated variance that defines their covariance operator, unless μ is the stationary law μβ; this is not made explicit. As a result, Theorem III.3 is false in general: for a simple linear system x_{t+1}=ρx_t+ω_t with g(x)=x, a fixed bounded μ, and m chosen so Φm resolves g, the residual is constant in β and does not blow up as ρ→1. The candidate solution identifies these missing hypotheses, gives the counterexample that falsifies the universal claim, and proves a corrected theorem under natural conditions (μβ and a nontrivial unresolved component), using the discrete Lyapunov series to show the critical-mode variance scales like (1−|λc|^2)^{-1} and feeds into the unresolved bias term. This reconciles the paper’s decomposition with a mathematically valid divergence statement. See Eq. (29), Eq. (32), and Theorem III.3 in the paper for the points discussed.