The full, explicit, scale dependence of the inclusive N 3 LO cross section for single Higgs hadroproduction is obtained by calculating the convolutions of collinear splitting kernels with lower-order partonic cross sections. We provide results for all convolutions of splitting kernels and lower-order partonic cross sections to the order in ǫ needed for the full N 3 LO computation, as well as their expansions around the soft limit. We also discuss the size of the total scale uncertainty at N 3 LO that can be anticipated with existing information.