  The single-logarithmic enhancement of the physical kernel for Higgs
production by gluon-gluon fusion in the heavy top-quark limit is employed to
derive the leading so far unknown contributions, ln^{5,4,3}(1-z), to the N^3LO
coefficient function in the threshold expansion. Also using knowledge from
Higgs-exchange DIS to estimate the remaining terms not vanishing for z =
m_H^2/s^hat -> 1, these results are combined with the recently completed soft +
virtual contributions to provide an uncertainty band for the complete N^3LO
correction. For the 2008 MSTW parton distributions these N^3LO contributions
increase the cross section at 14 TeV by (10 +- 2)% and (3 +- 2.5)% for the
standard choices mu_R = m_H and mu_R = m_H/2 of the renormalization scale. The
remaining uncertainty arising from the hard-scattering cross sections can be
quantified as no more than 5%, which is smaller than that due to the strong
coupling and the parton distributions.
