Fast reaction limits via Γ-convergence of the Flux Rate Functional

The paper proves Γ-convergence of the flux-level large-deviation functionals Ĩ^ε_0 + J̃^ε to Ĩ^0_0 + J̃^0 on the topology Θ (Theorem 1.2), identifying limit constraints (3.11) and the slow/damped/fast-cycle decomposition, including the quadratic moderate-deviation limit on fast cycles and the measure-valued damped part . Compactness and the fast-cycle equilibration/divergence-free condition are derived via the mild continuity equation (Lemma 3.10) and the derivation of (3.11) . The Γ-liminf for the fast-cycle part is obtained by the dual, small-parameter expansion p = √ε ζ (as the paper shows in detail) leading to ½∫(j̃^2/A)dt, while slow and damped parts follow by convex duality and measure-level entropy (1.12) . A key step for Γ-limsup is the recovery construction: the paper regularizes to enforce positivity, adds small fluxes along chains to V1, and uses fast-cycle adjustments to satisfy the ε-level continuity equations exactly (Proposition 4.5 and Lemma 4.9) . The model’s liminf analysis and identification of the limit constraints are largely consistent with the paper. However, the recovery sequence is flawed: (i) it incorrectly claims one can “add a divergence-free correction on R_damp within each time slice … at no cost,” which is false for the strictly convex integrand s(·|·); the paper instead transports small amounts along connecting chains to V1, with vanishing cost as ε→0 ; (ii) it does not include the positivity/regularization step that the paper uses to control the fast-cycle quadratic term and ensure nonnegativity of fluxes, cf. Lemma 4.7 and the subsequent construction ; and (iii) its fast-cycle correction cannot fix residuals at V0_slow or V1 nodes, whereas the paper enforces the ε-level continuity equations globally via a coordinated construction (again Lemma 4.9) . Because these gaps affect the Γ-limsup (recovery) direction in an essential way, the candidate solution is incorrect, while the paper’s argument is complete and correct under its stated assumptions.