ASYMPTOTIC INTERPLAY OF STATES AND ADAPTED COUPLING GAINS IN THE LOHE HERMITIAN SPHERE MODEL

The paper proves that, under the anti-Hebbian law with Stuart–Landau coupling pair and small initial Lyapunov functional, complete aggregation occurs and all adaptive couplings vanish (Theorem 3.1). Its core ingredients are exactly the dissipation identity for L_{ij} (Lemma 4.2), a barrier/invariance argument ensuring Rij(t)>0 uniformly (Lemma 4.3), and then a Barbalat-based case analysis to conclude Dij→0 and κij→0 (proof of Theorem 3.1). The candidate solution establishes the same dissipation identity and barrier, then closes with a clean LaSalle-invariance argument on a compact positively invariant set. Thus, the analytical skeleton is aligned on the key identity and barrier, while the final convergence mechanism differs (Barbalat vs. LaSalle). No contradictions were found; the model’s proof is sound under the same hypotheses as the paper’s anti-Hebbian/SL setting. Citations: the model and L_{ij} are set up in Section 4 (anti-Hebbian law and L_{ij}) and Lemma 4.2 gives L̇_{ij}≤0 in the acute-angle regime; Lemma 4.3 delivers the barrier; boundedness of κ follows from κ̇≤−γκ+4μ; and Theorem 3.1 states the aggregation and vanishing of κ (paper: anti-Hebbian/SL case) .