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We prove that the energy-critical half-wave maps equation \[ \partial_t\mathbf{S} =\mathbf{S} \times |\nabla| \mathbf{S}, \quad (t,x) \in \mathbb{R}\times \mathbb{T} \] arises as an effective equation in the continuum limit ofcompletely integrable Calogero-Moser classical spin systems with inverse square$1/r^2$ interactions on the circle. We study both the convergence toglobal-in-time weak solutions in the energy class as well as short-time strongsolutions of higher regularity. The proofs are based on Fourier methods andsuitable discrete analogues of fractional Leibniz rules and Kato-Ponce-Vegacommutator estimates.  In a companion paper, we further extend our arguments to study the real linecase and more general spin interactions.

Derivation of the Half-Wave Maps Equation from Calogero--Moser Spin Systems