  We use specialized Monte Carlo simulation methods and moment free energy
calculations to provide conclusive evidence that dense polydisperse spheres at
equilibrium demix into coexisting fcc phases, with more phases appearing as the
spread of diameters increases. We manage to track up to four coexisting phases.
Each of these is fractionated: it contains a narrower distribution of particle
sizes than is present in the system overall. We also demonstrate that,
surprisingly, demixing transitions can be nearly continuous, accompanied by
fluctuations in local particle size correlated over many lattice spacings.
