Non-Markovian Reduced Models to Unravel Transitions in Non-equilibrium Systems

The paper’s Theorem IV.2 explicitly states the conditional-expectation identity E_{ρ_{θ_t ω}}[G|Π_c](X) = A_c X + Π_c G_2(X,X) + 2 Π_c G_2(X, Φ^*(θ_t ω,X)) + ⟨G_2(Y,Y)⟩_{ρ^X_{θ_t ω}} and its corollary that, under assumption (H) and when ⟨G_2(Y,Y)⟩_{ρ^X_{θ_t ω}} = 0, the non-Markovian optimal reduced model is dX = E_{ρ_{θ_t ω}}[G|Π_c](X) dt; it does not claim pathwise exactness of the resolved dynamics, only optimal closure via conditional expectation . The candidate’s derivation of the identity (part 1) matches the paper and correctly uses the A-invariant spectral splitting implicit in the setup, consistent with the paper’s eigenmode decomposition and assumption (H) that noise acts only on unresolved modes . However, in part (2) the candidate asserts an “exact closed reduced model,” conflating conditional-expectation closure with the actual pathwise drift of X; exactness would require stronger measurability (e.g., Y determined by X), which the paper does not assume. Thus the paper is correct; the model over-claims exactness.