This paper is the first part of a trilogy dedicated to a proof of global
well-posedness and scattering of the (4+1)-dimensional mass-less
Maxwell-Klein-Gordon equation (MKG) for any finite energy initial data. The
main result of the present paper is a large energy local well-posedness theorem
for MKG in the global Coulomb gauge, where the lifespan is bounded from below
by the energy concentration scale of the data. Hence the proof of global
well-posedness is reduced to establishing non-concentration of energy. To deal
with non-local features of MKG we develop initial data excision and gluing
techniques at critical regularity, which might be of independent interest.Comment: 59 page
