  In this article, we compute the gluon fusion Higgs boson cross-section at
N3LO through the second term in the threshold expansion. This calculation
constitutes a major milestone towards the full N3LO cross section. Our result
has the best formal accuracy in the threshold expansion currently available,
and includes contributions from collinear regions besides subleading
corrections from soft and hard regions, as well as certain logarithmically
enhanced contributions for general kinematics. We use our results to perform a
critical appraisal of the validity of the threshold approximation at N3LO in
perturbative QCD.
