Using the latest sample of 48 spiral galaxies having a directly measured supermassive black hole mass, M BH , we determine how the maximum disk rotational velocity, v max (and the implied dark matter halo mass, M DM ), correlates with the (i) black hole mass, (ii) central velocity dispersion, σ 0 , and (iii) spiral-arm pitch angle, φ. We find that M BH ∝ v 10.62±1.37 max ∝ M 4.35±0.66 DM , significantly steeper than previously reported, and with a total root mean square scatter (0.58 dex) similar to that about the M BH -σ 0 relation for spiral galaxies-in stark disagreement with claims that M BH does not correlate with disks. Moreover, this M BH -v max relation is consistent with the unification of the Tully-Fisher relation (involving the total stellar mass, M * ,tot ) and the steep M BH ∝ M 3.05±0.53 * ,tot relation observed in spiral galaxies. We also find that
DM, consistent with past studies connecting stellar bulges (with σ 0 100 km s −1 ), dark matter halos, and a nonconstant v max /σ 0 ratio. Finally, we report that tan |φ| ∝ (−1.18 ± 0.19) log v max ∝ (−0.48 ± 0.09) log M DM , providing a novel formulation between the geometry (i.e., the logarithmic spiral-arm pitch angle) and kinematics of spiral galaxy disks. While the v max -φ relation may facilitate distance estimations to face-on spiral galaxies through the Tully-Fisher relation and using φ as a proxy for v max , the M DM -φ relation provides a path for determining dark matter halo masses from imaging data alone. Furthermore, based on a spiral galaxy sample size that is double the size used previously, the self-consistent relations presented here provide dramatically revised constraints for theory and simulations.4 If we calculate the Tully-Fisher relation from our dataset, we find that M * ,tot ∝ v 3.94±1.01 rot , which is consistent with Equation (1) at the level of 0.05 σ. 5 The dynamics of newly assembled massive objects (DY-NAMO) project (Green et al. 2010(Green et al. , 2014Bassett et al. 2014) demonstrated that their more local (z ∼ 0.1) sample of galaxies, selected to have high star-formation rates, are analogs of turbulent z 2 disk galaxies, with similar vrot/σ 0 ratios. 46 The quantity vmax should not be confused with the circular rotation velocity of the disk of our Galaxy at the location of the Sun and equal to (238 ± 15) km s −1 (Bland-Hawthorn & Gerhard 2016) or (229.0 ± 0.2) km s −1 (Eilers et al. 2019).