  We present the coefficient functions for deep-inelastic scattering (DIS) via
the exchange of a scalar phi directly coupling only to gluons, such as the
Higgs boson in the limit of a very heavy top quark and n_f effectively massless
light flavours, to the third order in perturbative QCD. The two-loop results
are employed to construct the next-to-next-to-leading order physical evolution
kernels for the system (F_2 F_phi) of flavour-singlet structure functions. The
practical relevance of these kernels as an alternative to MSbar factorization
is bedevilled by artificial double logarithms at small values of the scaling
variable x, where the large top-mass limit ceases to be appropriate. However,
they show an only single-logarithmic enhancement at large x. Conjecturing that
this feature persists to the next order also in the present singlet case, the
three-loop coefficient functions facilitate exact predictions (backed up by
their particular colour structure) of the double-logarithmic contributions to
the fourth-order singlet splitting functions, i.e., of the terms (1-x)^a
ln^k(1-x) with k = 4, 5, 6 and k = 3, 4, 5, respectively, for the off-diagonal
and diagonal quantities to all powers a in 1-x.
