Before revision:  Surprisingly, we show that when the number of point sources becomes large, there is a phase transition where the sample complexity goes from polynomial to exponential , and we pin down its location to within a universal constant that is independent of the number of point sources. Thus it is rigorously possible both to break the diffraction limit and yet to prove strong impossibility results depending on the setup. This is the first non-asymptotic statistical foundation for resolution in a model arising from first principles in physics, and helps clarify many omnipresent debates in the optics literature .
After revision:  Surprisingly, we show that when the number of point sources becomes large, there is a phase transition where the sample complexity goes from polynomial to exponential . Surprisingly, we show that this does not occur at the Abbe limit, which has long been presumed to be the true diffraction limit .