Variational Characterization and Identification of Reaction Coordinates in Stochastic Systems

The paper’s Theorem 5.1 states the variance perturbation bound and Lemma 5.2 gives Var_μ(f_L∘ξ)=Var_{\bar μ}(f_L) (see the statement and discussion around Theorem 5.1 and Lemma 5.2) . In Appendix B, the core L^1 bound Lemma B.1 is derived by expanding the z- and level-set integrals and using coarea, yielding a factor called π̄(z), followed by the ε-lumpability assumption to get a linear-in-ε estimate . However, in the subsequent variance estimate the paper invokes “as μ is finite, L^2_μ⊂L^1_μ, hence there exists C>0 such that ||·||_2≤C||·||_1”, to pass from L^2 to L^1 norms (cf. the line leading to the term C^2||·||_{L^1}||·||_{L^1})—this direction is not generally valid on finite measure spaces without additional boundedness assumptions; the embedding provides ||·||_1 ≤ K||·||_2, not the reverse . Aside from this gap, the remaining steps produce the linear term and an O(ε^2) remainder consistent with the theorem’s statement . The proof of Var_μ(f_L∘ξ)=Var_{\bar μ}(f_L) via coarea is clear and correct . The candidate solution fixes the main gap by using the symmetric two-sample identity for variance and a purely L^1-based bound on mixed terms, and it also clarifies a notational conflation in the paper between effective densities along ξ versus along ϑ when extracting a level-set mass factor (the paper writes π̄(z) where the integral is over Σ_ϑ(z)) . Overall, the theorem is correct, but the paper’s proof as written is incomplete due to the L^2→L^1 step; the model’s proof is correct and self-contained.