  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 of
completely integrable Calogero-Moser classical spin systems with inverse square
$1/r^2$ 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.
