N$^3$LO gravitational spin-orbit coupling at order $G^4$

In this paper we derive for the first time the N$^3$LO gravitational spin-orbit coupling at order $G^4$ in the post-Newtonian (PN) approximation within the effective field theory (EFT) of gravitating spinning objects. This represents the first computation in a spinning sector involving three-loop integration. We provide a comprehensive account of the topologies in the worldline picture for the computation at order $G^4$. Our computation makes use of the publicly-available \texttt{EFTofPNG} code, which is extended using loop-integration techniques from particle amplitudes. We provide the results for each of the Feynman diagrams in this sector. The three-loop graphs in the worldline picture give rise to new features in the spinning sector, including divergent terms and logarithms from dimensional regularization, as well as transcendental numbers, all of which survive in the final result of the topologies at this order. This result enters at the 4.5PN order for maximally-rotating compact objects, and together with previous work in this line, paves the way for the completion of this PN accuracy.