EMERGENT BEHAVIORS OF HIGH-DIMENSIONAL KURAMOTO MODELS ON STIEFEL MANIFOLDS

The paper’s Theorem 1.3 asserts that for the homogeneous second-order Stiefel model (Ξi ≡ 0, m,γ,κ>0), every global solution satisfies limt→∞ ||Ṡi(t)||F = 0 and, moreover, limt→∞ G(t) = 0 (complete consensus) without further restrictions on initial data. The energy dissipation part is correct (via Proposition 4.1, yielding dE/dt = -(2γ/N)∑||Ṡi||^2 when Ξ≡0), hence velocities indeed vanish . However, the universal consensus claim contradicts explicit stationary “balanced” equilibria: for N=2, take S2(0) = -S1(0) and Ṡ1(0)=Ṡ2(0)=0. Then the right-hand side of (1.4) vanishes identically and Si(t) ≡ Si(0), hence G(t) stays constant and strictly positive (here G = 2p) rather than approaching 0, contradicting Theorem 1.3’s second conclusion . The paper’s key inequality mG̈ + γĠ + 2κξG ≤ … (Lemma 4.2) would force G→0 when the forcing tends to 0, but for the stationary non-consensus state one has G̈=Ġ=0 and thus 2κξG ≤ 0, which is false for G>0, revealing a gap in Lemma 4.2 or in its applicability . The candidate solution correctly proves velocities go to zero and supplies the balanced counterexample, so the universal consensus claim is false as stated, absent an additional non-degeneracy assumption (e.g., M(0)≠0).