  We bootstrap the three-point form factor of the chiral part of the
stress-tensor supermultiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory,
obtaining new results at three, four, and five loops. Our construction employs
known conditions on the first, second, and final entries of the symbol,
combined with new multiple-final-entry conditions, ``extended-Steinmann-like''
conditions, and near-collinear data from the recently-developed form factor
operator product expansion. Our results are expected to give the maximally
transcendental parts of the $gg\to Hg$ and $H\to ggg$ amplitudes in the
heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of
functions we describe contains all transcendental functions required for
four-point amplitudes with one massive and three massless external legs, and
all massless internal lines, including processes such as $gg\to Hg$ and
$\gamma^*\to q\bar{q}g$. We expect the extended-Steinmann-like space to contain
these amplitudes at higher loops as well, although not to arbitrarily high loop
order. We present evidence that the planar $\mathcal{N}=4$ three-point form
factor can be placed in an even smaller space of functions, with no independent
$\zeta$ values at weights two and three.
