EMERGENCE OF SYNCHRONIZATION IN KURAMOTO MODEL WITH GENERAL DIGRAPH

The paper’s Theorem 1.1, the Qk-inequalities, and the hierarchical propagation across the node decomposition are coherent and mutually consistent; the proof controls convexified path functionals Qk and yields finite-time entrance of D(θ(t)) into any prescribed D∞ with the explicit threshold κ > (1 + ((d+1)α)/(α − D(θ(0))))(4c)^d c̃/(β^{d+1} D∞) as stated (Theorem 1.1 and Lemmas 3.3–4.3) . By contrast, the candidate solution relies on a global scalar inequality for D(t) obtained by (i) dropping a strictly positive term from the right-hand side of an upper-bound differential inequality (invalid under ≤), and (ii) asserting sinγ/γ · c0 = 1 for c0 = S_{N−1}(η)γ/sinγ, which is false (the product equals S_{N−1}(η) ≠ 1). It also uses the bound |sin x| ≤ sin γ for |x| ≤ γ, which need not hold when γ > π/2, whereas the paper carefully uses sin x ≥ (sinγ/γ)x and tailored convexifications instead (Appendix A) .