Energy minimization of repelling particles on a toric grid

We explore the minimum energy configurations of repelling particles distributed over $n$ possible locations forming a toric grid. We conjecture that the most energy-efficient way to distribute $n/2$ particles over this space is to place them in a checkerboard pattern. Numerical experiments validate this conjecture for reasonable choices of the repelling force. In the present paper, we prove this conjecture in a large number of special cases---most notably, when the sizes of the torus are either two or multiples of four in all dimensions and the repelling force is a completely monotonic function of the Lee distance between the particles.