Derivation of the half-wave maps equation from Calogero–Moser spin systems

We prove that the energy-critical half-wave maps equation \[ ∂_t S =S × |∇| S, (t,x) ∈ R × T \] arises as an effective equation in the continuum limit of completely integrable Calogero-Moser classical spin systems with inverse square 1/r² interactions on the circle. We study both the convergence to global-in-time weak solutions in the energy class as well as short-time strong solutions of higher regularity. The proofs are based on Fourier methods and suitable discrete analogues of fractional Leibniz rules and Kato-Ponce-Vega commutator estimates. In a companion paper, we further extend our arguments to study the real line case and more general spin interactions.