Action Functional as an Early Warning Indicator in the Space of Probability Measures via Schrödinger Bridge

The paper defines the early-warning indicator I(ρ*_t) as 1/t times the minimal Benamou–Brenier kinetic action to connect (ρ0, ρ*_t) (its Eq. (25)), and motivates using Schrödinger bridges to obtain the path marginals ρ*_t, but it does not prove the time-rescaling identity that yields I(ρ*_t)=W2^2(ρ0,ρ*_t)/t^2, nor does it supply theoretical conditions ensuring existence/regularity of the SB interpolation or quantitative tipping guarantees beyond experiments. The candidate solution supplies these missing ingredients: a correct time-rescaling proof for the Benamou–Brenier action on [0,t], standard hypotheses for SB existence, and an explicit Gaussian/Brownian example that produces a discontinuous jump in I. Minor notational issues in the paper (e.g., writing W2 instead of W2^2 in the dynamic formulation) reinforce the need for a rigorous reconciliation. Taken together, the paper’s claims are conceptually sound but theoretically incomplete, while the model’s solution is correct and fills the gaps (indicator definition: ; SB–BB background: ; tipping definition: ).