  We uncover the full null structure of the Maxwell-Dirac system in Lorenz
gauge. This structure, which cannot be seen in the individual component
equations, but only when considering the system as a whole, is expressed in
terms of tri- and quadrilinear integral forms with cancellations measured by
the angles between spatial frequencies. In the 3D case, we prove
frequency-localized $L^2$ space-time estimates for these integral forms at the
scale invariant regularity up to a logarithmic loss, hence we obtain almost
optimal local well-posedness of the system by iteration.
