Contraction Dynamics in Heterogeneous Spatial Environments

The paper’s Theorem 1 establishes contraction for the ODE–PDE interconnection with θ-diffusion under the small-gain condition λ1λ2 > (−/(2√(m1,∞m2,∞)))^2 and yields the sharp 2×2-eigenvalue rate; its full derivation is supplied in the SI and follows a standard quadratic energy argument with a mean-free coupling term G^∘ and the θ-diffusion spectral gap . The candidate solution copies the paper’s structure but (i) loses a factor of 1/2 in the coupling constant and (ii) consequently imposes the stricter condition λ1λ2 > κ^2 (instead of λ1λ2 > (κ/2)^2) and states a weaker rate; therefore it does not prove the claimed theorem under the paper’s stated hypotheses .