Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers, we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.