SYNCHRONIZATION IN THE COMPLEXIFIED KURAMOTO MODEL

The paper’s Theorem 3.5 proves, under the strong-coupling/initial-separation hypotheses (λ > λ_c/sin δ and x(0) ∈ [0, π−δ]^N with ∑ω_n = 0), that phase synchronization occurs if and only if all natural frequencies are identical; the proof leverages Lemma 3.1 (real-part full phase-locking), Lemma 3.2 (imaginary-part contraction), Theorem 3.3 (full phase-locking), and Theorem 3.4 (frequency synchronization), then passes to the limit in the real-part equation to conclude phase synchronization and necessity of identical ω’s . The model gives a different direct proof: for ω_1=…=ω_N (hence λ_c=0), it shows diameter contraction of the real parts and exponential decay of the imaginary-part variance (after the real-part diameter enters a cone), and for necessity it uses a Cesàro-average argument on the pairwise difference equation. The two arguments are logically consistent with the same statement; the model’s route via a gradient-flow potential and a clean Ḋ_x-inequality is distinct from the paper’s energy-function and lemma chain. Minor technical gaps in the model (handling non-unique maximizers of x_i(t) when differentiating D_x) are easily fixed with Dini-derivatives; overall, both are correct.