Not yet reviewed by Pith; the record is open.

Circumbinary disks are found in a variety of astrophysical scenarios, spanning binary star formation to accreting supermassive black hole binaries. The interaction with a circumbinary disk can yield opposite effects on the binary orbit leading to circularization, or exciting the eccentricity, widening the orbit or shrinking it and facilitating mergers. We present a new formalism for the long-term evolution of the disk-binary interaction based on the results of recent suites of hydrodynamic simulations, which resolve the complex geometry of the gas in the vicinity of the binary and fully account for the gravitational and accretion forces. We release a python package, \texttt{spindler}, that implements our model. We show that, unless the mass reservoir feeding the disk is comparable to the mass of the binary, accretion onto the binary depletes the disk mass before inducing a significant change in orbital separation or mass ratio. This finding implies that, in most scenarios, interaction with a circumbinary disk is not an efficient mechanism to shrink the orbit of the binary. However, as long as the mass of the disk is at least a few percent of the mass of the binary, the interaction can excite the eccentricity up to an equilibrium value, and induce a statistical correlation between mass ratio and eccentricity. We consider the applicability of our model to a variety of astrophysical scenarios: during star formation, in evolved stellar binaries, triples and in supermassive black hole binaries. We discuss the theoretical and observational implications of our predictions.