Geometric perspective of linear stability in finite networks of nonlinear oscillators

Both the paper and the candidate solution correctly derive the finite‑N linear stability test for q‑twisted states on circulant networks, namely λ_{m,q} = (γ_{q+m}+γ_{q−m})/2 − γ_q (paper: Eq. (2)/(5) and Appendix Eq. (23) ). However, for the selection claim (“which q has the largest basin”), the paper only provides a heuristic/empirical argument referencing prior work on basins and stable eigenvalues (no self‑contained proof here) , while the model’s attempt to prove that the argmax of γ_q maximizes a certified local basin margin relies on an unsupported inequality. Thus, linear stability is handled correctly; basin selection is not rigorously established by either, leaving both incomplete.