Clash of the Trident and Tuning Fork: insights from bar and spiral strength in the (massive black hole)-stellar mass diagrams, and the `Triangal' galaxy evolution schema

The `Triangal' galaxy evolution schema is used to assess whether the Tuning Fork (bar strength) or the van~den~Bergh Trident and ATLAS$^{3D}$ Comb (spiral strength) offer greater evolutionary insight. A new catalogue of quantitative bar strengths (measured by the bar-to-total luminosity ratio, $P$), refined galaxy morphologies, and dust bin classifications is presented. It contains 137 galaxies with spheroid stellar masses, obtained from multi-component decompositions, and directly measured black hole masses, $M_{\rm bh}$. By placing these galaxies within the $M_{\rm bh}$-($M_{\rm\star,sph}$, $M_{\rm\star,gal}$) parameter space, an evolutionary reference frame reflecting integrated growth is established. Galaxies with varying bar strengths, and double bars, are observed to not occupy preferred locations, highlighting that bars are products of secular evolution-and can be transient or recurrent phenomena-and that they track neither hierarchical mass assembly nor galaxy speciation. In contrast, three physically distinct formation channels for S0/a galaxies are identified: (primeval S0)-to-S transitions; faded spiral galaxies; and, most commonly, wet-major-merger-built dust-rich S0 galaxies (on the `green mountain'). Galaxies with particularly strong spirals appear on the right-hand side of the spiral galaxy distribution. Furthermore, a `Dust Attrition/Retention' sequence places S0 (and compact massive ES,b) galaxies with dusty nuclear discs between the dust-poor and dust-rich S0 galaxies, and a `Disc Down-sizing' sequence is revealed, in which E galaxies with dusty nuclear discs-potentially formed through `damp' mergers-bridge the ES,e (ellicular) galaxies with intermediate-scale stellar discs and the dust-poor pure E galaxies. Extensive historical context is provided, and, finally, suspected biases in precision cosmology stemming from neglected precision galaxy morphology are discussed.