Sufficiently dense Kuramoto networks are globally synchronizing

The paper proves that for the homogeneous Kuramoto model on any connected graph with modified connectivity μ̃ > 3/4, the only stable equilibrium is the all‑in‑phase state, yielding μc ≤ 0.75 for loopless graphs. The proof uses moment inequalities for ρ1 and ρ2, a self‑loop/twinning reduction, and a final bound (Theorem 5) that rules out any nontrivial stable equilibria . By contrast, the candidate’s argument hinges on asserting that any equilibrium’s phases can be placed in a single interval of length < π (so that all maximum–neighbor phase differences have nonpositive sine). That assertion is false: a configuration can have all pairwise geodesic distances < π yet fail to lie inside any semicircle, invalidating the key sign argument and the ensuing two‑case dichotomy. Consequently, the model’s proof is flawed, even though parts of it (e.g., that any two‑cluster 0/π equilibrium is a saddle) are correct.