Specific star formation rates in the $M_{\rm bh}$-$M_{\rm *,sph}$ diagram and the evolutionary pathways of galaxies across the sSFR-$M_{\rm *}$ diagram

It has been suggested that the bulge-to-total stellar mass ratio or feedback from black holes (BHs), traced by the BH-to-(total stellar) mass ratio, might establish a galaxy's specific star formation rate (sSFR). We reveal that a galaxy's morphology -- reflecting its formation history, particularly accretions and mergers -- is a far better determinant of the sSFR. Consequently, we suggest that galaxy formation models which regulate the sSFR primarily through BH feedback prescriptions or bulge-regulated disc fragmentation consider acquisitions and mergers which establish the galaxy morphology. We additionally make several new observations regarding current ($z\sim0$) star-formation rates. (i) Galaxies with little to no star formation have bulges with an extensive range of stellar masses; bulge mass does not dictate presence/absence on the `star-forming main sequence'. (ii) The (wet merger)-built, dust-rich S0 galaxies are the `green valley' bridging population between elliptical galaxies on the `red sequence' and spiral galaxies on the blue star-forming main sequence. (iii) The dust-poor S0 galaxies are not on the star-forming main sequence nor in the `green valley'. Instead, they wait in the field for gas accretion and/or minor mergers to transform them into spiral galaxies. Mid-infrared sample selection can miss these (primordial) low dust-content and low stellar-luminosity S0 galaxies. Finally, the appearance of the quasi-triangular-shaped galaxy-assembly sequence, previously dubbed the Triangal, which tracks the morphological evolution of galaxies, is revealed in the sSFR-(stellar mass) diagram.