Synchronization of mean-field models on the circle

The paper’s main criterion (Theorem 2.1) precisely states that if τ∫|f‴|+ ≤ 4(1+τ/(2π))f′(0) and f′(x)<0 off [−τ,τ], then every stationary point of (S1) is either synchronized or locally unstable, and it proves this via cut-stability and an integration-by-parts argument culminating in Lemma 3.2 and Corollary 3.3 (with M=2π), which is rigorous and internally consistent . By contrast, the candidate solution hinges on the false inequality α(J) ≥ λmax((J+Jᵀ)/2) and treats positivity of vᵀSv as sufficient for instability; in general α(J) ≤ λmax(S), so the direction is reversed. It also invokes an unproven “two-scale averaging” step to obtain the 1+τ/(2π) factor and uses a stationarity identity that implicitly assumes extra symmetry not present in Theorem 2.1. Hence the paper’s argument is correct while the model’s proof is flawed/incomplete.