  The production of a Higgs boson in association with at least two jets
receives contributions both from the fusion of weak vector bosons (VBF) and
from QCD processes, especially gluon fusion (GF). The former process is
important for measuring the coupling of the Higgs boson to weak bosons, whereas
the latter process plays an important role in determining any $CP$-admixtures
in the Higgs sector. In this paper we go beyond the current state-of-the-art
for fixed order calculations of the GF process (i.e. one loop $H+2j$ including
full quark mass effects) by including the all-order effects in leading
$\log(\hat s/p_t^2)$, together with full quark mass and loop-propagator
kinematic effects. We calculate the mass-dependent components and implement the
resummation within the framework of High Energy Jets.
  The high energy effects suppress the prediction compared to fixed order at
large $\Delta y_{12}$ and $m_{jj}$ (and therefore within the usual VBF cuts of
widely separated jets), just as found in the limit of $m_t\to \infty$. The mass
dependence is more significant than at fixed order, because the systematic
inclusion of the leading logarithms in $\hat s/p_t^2$ results in a hardening of
the transverse momentum of the Higgs boson, which in turn probes in more detail
the loop-structure of the coupling. In particular, the full mass dependence
reduces the cross section within VBF cuts by 11\% compared to a calculation
based just on the infinite top mass limit, but the impact of the bottom quark
remains small. This all implies that the gluon-fusion contribution within
VBF-cuts is less severe than current estimates suggest.
