The $g^6$ pressure of hot Yang-Mills theory: Canonical form of the integrand

We present major progress towards the determination of the last missing piece for the pressure of a Yang-Mills plasma at high temperatures at order $g^6$ in the strong coupling constant. This order is of key importance due to its role in resolving the long-standing infrared problem of finite-temperature field theory within a dimensionally reduced effective field theory setup. By systematically applying linear transformations of integration variables, or momentum shifts, we resolve equivalences between different representations of Feynman sum-integrals. on the integrand level, transforming those into a canonical form. At the order $g^6$, this results in reducing a sum of O(100000) distinct sum-integrals which are produced from all four-loop vacuum diagrams down to merely 21. Furthermore, we succeed to map 11 of those onto known lower-loop structures. This leaves only 10 genuine 4-loop sum-integrals to be evaluated, thereby bringing the finalization of three decades of theoretical efforts within reach.