On synchronization in Kuramoto models on spheres

The paper states and uses that, on the Poisson-kernel invariant family for the real model, the centroid and the image of zero are colinear with c(t) = μ_{d−1}(|a|) a and μ_{d−1} given by a Gauss hypergeometric expression; substituting f = K c into the closed ODE for a then yields the scalar radial law ṗ = (K/2) μ_{d−1}(p) p (1 − p^2) (Propositions 4–5; equations (5)–(9)) . For the complex model, the centroid equals the Poisson–Szegő parameter along the invariant family (Propositions 7–8), which implies c satisfies the same closed ODE and, for p = Kc, ṙ = K(r − r^3) (Proposition 10) . The candidate solution reproduces these results with a compatible hypergeometric form for μ_{d−1} and the same symmetry reduction and radialization. The only flaw is an aside incorrectly asserting that the complex m = 1 case has vanishing coupling; with the paper’s Hermitian inner-product convention, the coupling does not cancel and Proposition 10 still applies .