ASYMPTOTIC CONVERGENCE OF HETEROGENEOUS FIRST-ORDER AGGREGATION MODELS: FROM THE SPHERE TO THE UNITARY GROUP

The paper’s Theorem 5.1 asserts convergence of the Lohe matrix model on U(d) under κ > D(a), a scalar-frequency ansatz Hj = −aj Id, and a smallness condition on D(U0), introducing the barrier β solving 1 − s = 2d sin(s/2). Its proof reduces to Kuramoto phases θj (5.3), moves to the rotating frame Vj = e−iθj Uj (5.4), and uses the diameter inequality d/dt D(V) ≤ −κ(1 − 2 sin(D(Θ)/2))D(V) + κD(V)2 (5.6) to conclude Vj → V∞ and then Uj → eiθ∞j V∞, i.e., U∞i U∞,†j = ei(θ∞i−θ∞j)Id (Theorem 5.1) . However, the proof invokes Proposition 5.1, which requires the additional initial-phase smallness D(Θ0) < δ (5.7), but Theorem 5.1’s hypotheses (5.9) do not assume D(Θ0) < δ. The step “Since (5.9) holds, initial conditions for D(V0) and D(Θ0) in (5.7) are fulfilled” is therefore not justified as written . Moreover, the appeal to Kuramoto phase convergence relies on Proposition 2.1(ii), which assumes κ > 1.6 D(a)/R0 and R0 > 0; this is also not assumed in Theorem 5.1, so existence of θ∞ is not guaranteed from the stated hypotheses alone . The candidate solution mirrors the paper’s strategy (phase reduction, rotating frame, diameter/Riccati estimate, barrier β), but it also omits the needed D(Θ0) < δ condition and asserts global Kuramoto convergence under κ > D(a) without the extra initial-condition hypotheses actually used in the paper. Hence both the paper and the model solution are incomplete.