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The magnetism of nanoparticles and thin films of wide-bandgap oxides that include no magnetic cations is an unsolved puzzle. Progress has been hampered both by the irreproducibility of much of the experimental data, and the lack of any generally-accepted theoretical explanation. The characteristic signature is a virtually anhysteretic, temperature-independent magnetization curve which saturates in an applied field that is several orders of magnitude greater than the magnetization. It appears as if a tiny volume fraction, < 0.1%, of the samples is magnetic and that the energy scale of the problem is unusually high for spin magnetism. Here we investigate the effect of dispersing 4 nm CeO2 nanoparticles with powders of gamma-Al2O3, sugar or latex microspheres. The saturation magnetization, Ms ~ 60 A/m for compact samples, is maximized by 1 wt% lanthanum doping. Dispersing the CeO2 nanopowder reduces its magnetic moment by up to an order of magnitude. There is a characteristic length scale of order 100 nm for the magnetism to appear in CeO2 nanoparticle clusters. The phenomenon is explained in terms of a giant orbital paramagnetism that appears in coherent mesoscopic domains due to resonant interaction with zero-point fluctuations of the vacuum electromagnetic field. The theory explains the observed temperature-independent magnetization curve and its doping and dispersion dependence, based on a length scale of 300 nm that corresponds to the wavelength of a maximum in the UV absorption spectrum of the magnetic CeO2 nanoparticles. The coherent domains occupy roughly ten percent of the sample volume.