The paper investigates the long-term orbital evolution of binary systems influenced by circumbinary disks, which are common in various astrophysical contexts, from binary star formation to supermassive black hole binaries. The authors present a new formalism based on hydrodynamic simulations to predict the outcome of disk-binary interactions, considering both gravitational and accretion forces. They release a Python package called *spindler* to implement their model. Key findings include: - Under the thin disk model with fixed thickness and viscosity, accretion onto the binary depletes the disk mass before significantly altering the orbital separation or mass ratio, unless the disk mass is comparable to the binary's mass. - The interaction can excite the eccentricity up to an equilibrium value and induce a statistical correlation between the mass ratio and eccentricity, provided the disk mass is at least a few percent of the binary's mass. - The model is applied to various astrophysical scenarios, including star formation, evolved stellar binaries, triples, and supermassive black hole binaries, with implications for observational signatures and theoretical understanding. The study highlights the complex interplay between the binary potential, disk morphology, and resulting forces, emphasizing the need for extensive parameter exploration in hydrodynamic simulations. The authors also compare their results with those from other models, noting differences in behavior at small eccentricities and orbital decay. The implications for binary star formation and the detection of disk signatures in young stellar objects are discussed, suggesting that the presence of a stable thin circumbinary disk can be discerned through the eccentricity and mass ratio distributions of young binaries.The paper investigates the long-term orbital evolution of binary systems influenced by circumbinary disks, which are common in various astrophysical contexts, from binary star formation to supermassive black hole binaries. The authors present a new formalism based on hydrodynamic simulations to predict the outcome of disk-binary interactions, considering both gravitational and accretion forces. They release a Python package called *spindler* to implement their model. Key findings include: - Under the thin disk model with fixed thickness and viscosity, accretion onto the binary depletes the disk mass before significantly altering the orbital separation or mass ratio, unless the disk mass is comparable to the binary's mass. - The interaction can excite the eccentricity up to an equilibrium value and induce a statistical correlation between the mass ratio and eccentricity, provided the disk mass is at least a few percent of the binary's mass. - The model is applied to various astrophysical scenarios, including star formation, evolved stellar binaries, triples, and supermassive black hole binaries, with implications for observational signatures and theoretical understanding. The study highlights the complex interplay between the binary potential, disk morphology, and resulting forces, emphasizing the need for extensive parameter exploration in hydrodynamic simulations. The authors also compare their results with those from other models, noting differences in behavior at small eccentricities and orbital decay. The implications for binary star formation and the detection of disk signatures in young stellar objects are discussed, suggesting that the presence of a stable thin circumbinary disk can be discerned through the eccentricity and mass ratio distributions of young binaries.