Multi-Population Phase Oscillator Networks with Higher-Order Interactions

The paper’s Theorem 3.13 rigorously proves Lyapunov stability of the all-synchronized set SM under the assumptions gσ ∈ C1 and aσ := ∂γgσ(0,0) > 0, with stability defined via Wasserstein-1 neighborhoods (Definition 3.6). The candidate solution establishes the same conclusion by a different route: it derives a uniform one-sided Lipschitz bound for the phase flow inside small support tubes and proves forward-invariant tubes with exponential diameter decay. This yields a correct special-case proof (for support-concentrated neighborhoods), and the candidate remarks how to adapt to open neighborhoods, though those technical estimates are only sketched (and handled in detail in the paper via inside/outside mass decompositions and perturbation bounds). Hence, the conclusions agree; the proofs differ in technique and generality.