RVV$\times$V: Interference contributions to inclusive Higgs boson and Drell-Yan production at N$^4$LO in QCD

We present partonic contributions to the inclusive gluon-fusion Higgs boson and Drell-Yan production cross sections at Hadron colliders at next-to-next-to-next-to-next-to leading order (N$^4$LO). Specifically, we compute contributions due to the interference of one-loop amplitudes with two-loop amplitudes with three QCD partons and the Higgs boson or a virtual photon. Our result is in the form of a Laurent expansion in the dimensional regulator $\epsilon$, the coefficients of which are analytic functions in the ratio of the Higgs boson or virtual photon mass to the partonic center of mass energy. Furthermore, we introduce and deploy a new package implemented in Mathematica, CIFAR ("Color Invariant Feynman Amplitude Reducer"), to express color factors in terms of general Casimir invariants of simple compact Lie algebras.