  We have computed the even-$N$ moments $N\leq 20$ of the pure-singlet quark
splitting function $P_{\,\rm ps}$ at the fourth order of perturbative QCD via
the anomalous dimensions of off-shell flavour-singlet operator matrix elements.
Our results, derived analytically for a general gauge group, agree with all
results obtained for this function so far, in particular with the lowest six
even moments obtained via physical cross sections. Using these results and all
available endpoint constraints, we construct approximations for $P_{\rm ps}$ at
four loops that should be sufficient for most collider-physics applications.
Together with the known results for the non-singlet splitting function $P_{\rm
ns}^{\,+}$ at this order, this effectively completes the quark-quark
contribution for the evolution of parton distribution at N$^{\:\!3}$LO
accuracy. Our new results thus provide a major step towards fully consistent
N$^{\:\!3}$LO calculations at the LHC and the reduction of the residual
uncertainty in the parton evolution to the percent level.
