Pattern alternations induced by nonlocal interactions

The paper derives the standard three-mode cubic amplitude equations for a 2π/3 triad near a Turing point (its eqs. (34)-(36)) and proves Theorem 6, which classifies homogeneous, stripe, hexagon (H0/Hπ), and mixed branches and gives the same existence and stability thresholds μ1, μ2, μ3, μ4 as the model solution; see the amplitude system and phase reduction in (34)-(35) and Theorem 6’s branch-by-branch formulas and eigenvalues . The model’s steps (normal-form reduction, steady states, explicit Jacobian/eigenvalues, and stability conditions, including the always-saddle mixed branch) match the paper’s proof and results. Minor editorial tension appears in one concluding sentence about stripes, but elsewhere the paper explicitly states stripe stability thresholds and bistable regions consistent with Theorem 6 and with the model solution . No substantive discrepancies affecting the main results were found.