  We present fully differential predictions for the production cross section of
a Higgs boson via the gluon fusion mechanism at next-to-next-to-next-to leading
order (N$^3$LO) in QCD perturbation theory. To perform our calculation we apply
the Projection-to-Born method for the first time to the calculation of the
non-factorising production of a colorless final state at the LHC at N$^3$LO. We
predict differential distributions for the two photon final state produced by
the decay of the Higgs boson and apply fiducial cuts on the photon rapidities
and momenta. The N$^3$LO corrections to these differential distributions have
complex features and are in part larger than the inclusive N$^3$LO corrections
to the production cross section. Overall, we observe that the inclusion of the
N$^3$LO QCD corrections significantly reduces the perturbative uncertainties
and leads to a stabilisation of the perturbative expansion.
