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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 by both 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 that saturates in an applied field that is several orders of magnitude greater than the magnetization. It would seem as if a tiny volume fraction, ≲0.1%, of the samples is magnetic and that the energy scale is unusually high for spin magnetism. Here we investigate the effect of dispersing 4 nm CeO2nanoparticles with powders of γAl2O3, sugar or latex microspheres. The saturation magnetization, Ms≍ 60 A m−1for compact samples, is maximized by 1 wt% lanthanum doping. Dispersing the CeO2nanopowder reduces its magnetic moment by up to an order of magnitude, and there is a characteristic length scale of order 100 nm for the magnetism to appear in CeO2nanoparticle 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 ultraviolet absorption spectrum of the magnetic CeO2nanoparticles. The coherent domains occupy roughly 10% of the sample volume.