Using a sum-rule approach, we develop an exact theoretical framework for polarizability and asymptotic van der Waals correlation energy functionals of small isolated objects. The functionals require only monomer ground-state properties as input. Functional evaluation proceeds via solution of a single position-space differential equation, without the usual summations over excited states or frequency integrations. Explicit functional forms are reported for reference physical systems, including atomic hydrogen and single electrons subject to harmonic confinement, and immersed in a spherical-well potential. A direct comparison to the popular Vydrov-van Voorhis density functional shows that the best performance is obtained when density decay occurs at atomic scales. The adopted sum-rule approach implies general validity of our theory, enabling exact benchmarking of van der Waals density functionals and direct inspection of the subtle long-range correlation effects that constitute a major challenge for approximate (semi)local density functionals.