Phase transition in a kinetic mean-field game model of inertial self-propelled agents

The paper’s stability proof hinges on claiming that the non-Hamiltonian linearization N and a symmetrized Hamiltonian matrix Ns have the same characteristic equation and therefore are similar, i.e., Ns = P^{-1} N P. Having identical spectra does not, by itself, imply similarity; additional conditions (e.g., diagonalizability with matched Jordan structure) are missing. This gap affects the subsequent change of variables using P in Lemma 1. The rest of the argument (CARE, invariant graph, decay on the stable subspace, and uniqueness under invertibility of P11+P12X+) is otherwise standard and well-aligned with Riccati/invariant subspace theory. The candidate solution treats the similarity as an explicit hypothesis and then completes a correct, standard proof via the invariant graph subspace; thus, under that assumption, the model solution is correct. Key elements such as the BVP setup and the role of the stabilizing Riccati solution match the paper’s framework (e.g., the BVP and block structure of N, Theorem 1, and the uniqueness condition) .