Asymptotic exactness of the mesoscopic SDE near synchronization transitions (conjecture)

Prove that the Langevin equation for the complex amplitude A(t) (Eq. (5), derived via the Dean–Kawasaki equation and center-manifold reduction) is asymptotically exact near synchronization transition points for large but finite population size N. Specifically, demonstrate that the empirical density dynamics stays close to the deterministic center manifold and that the reduced stochastic differential equation accurately captures finite-size fluctuations in the small-noise regime.

Background

The central technical advance of the paper is a mesoscopic SDE for the order-parameter amplitude derived from the Dean–Kawasaki equation using a center-manifold expansion. This yields closed-form predictions for steady-state distributions and moments that match simulations near criticality.

The authors expect this reduction to be asymptotically exact near transitions, citing supporting rigorous results in related mean-field systems, but they do not provide a formal proof here, presenting it as a conjecture.