Investigating the residuals in the

We investigate the scaling relation between black hole (BH) and stellar mass (M• − M*), diagnosing the residual Δlog(M•/M⊙) (Δ) in this relation to understand the coevolution of galaxies and BHs in the cosmological hydrodynamic simulation S<ns0:sc>IMBA</ns0:sc>. We show that S<ns0:sc>IMBA</ns0:sc> reproduces the observed M• − M* relation well, with little difference between central and satellite galaxies. By using the median value to determine the residuals, we find that the residual correlates with galaxy cold gas content, star formation rate, colour, and BH accretion properties. Both torque and Bondi models implemented in S<ns0:sc>IMBA</ns0:sc> contribute to this residual, with torque accretion playing a major role in high-redshift and low-mass galaxies, while Bondi (including BH mergers) dominates at low redshift and massive galaxies. By dividing the sample into two populations (Δ > 0 and Δ < 0), we compare their evolutionary paths by following the main progenitors. From this evolutionary tracking, we propose a simple picture for BH-galaxy coevolution: early-formed galaxies seed BHs earlier, with stellar mass increasing rapidly to reach the point of triggering ‘jet mode’ feedback. This process reduces the cold gas content and halts the growth of M*, effectively quenching galaxies. Meanwhile, during the initial phase of torque accretion growth, the BH mass is comparable between galaxies formed early and those formed later. However, galaxies that formed earlier appear to attain a marginally greater BH mass when transitioning to Bondi accretion, aligning with the galaxy transition time. As the early-formed galaxies reach this point earlier – leaving a longer period for Bondi accretion and mergers – their residuals become positive, i.e. having more massive BHs at z = 0 compared to these late-formed galaxies at the same M*. This picture is further supported by the strong positive correlation between the residuals and the galaxy age, which we propose as a validation with observation data of the scenario suggested by S<ns0:sc>IMBA</ns0:sc>.