The study of the interaction of a massive black hole binary with its gaseous environment is crucial in order to be able to predict merger rates and possible electromagnetic counterparts of gravitational-wave signals. The evolution of the binary semimajor axis resulting from this interaction has been recently debated, and a clear consensus is still missing because of several numerical limitations, i.e., fixed orbit binaries or lack of resolution inside the cavity carved by the binary in its circumbinary disk. Using on-the-fly particle splitting in the 3D meshless code gizmo, we achieve hyper-Lagrangian resolution, which allows us to properly resolve the dynamics inside the cavity—in particular, for the first time, the disks that form around the two components of a live binary surrounded by a locally isothermal gaseous circumbinary disk. We show that the binary orbit decays with time for very cold and very warm disks and that the result of the interaction in the intermediate regime is strongly influenced by the disk viscosity, as this essentially regulates the fraction of mass contained in the disks around the binary components, as well as the fraction that is accreted by the binary. We find the balance between these two quantities to determine whether the binary semimajor axis decreases with time.