Evolution of the spin parameter a as a function of R_qso for the sEdd_0 (solid blue), sEdd_y (solid orange), sEdd_+z (solid red), sEdd_0.05_0 (dotted blue), sEdd_0.05_y (dotted orange), and sEdd_+z_LM (dashed dotted red) simulations, from the moment super-Eddington was allowed. Equation (13) is shown in green. The markers show the starting a–R_qso for each simulation. Each step represents a mass increase of ΔM = 8 × 10⁴ M_⊙. Only BHs in corotation (a > 0) during the entirety of the simulation are shown here. The sEdd_y and sEdd_+z BHs have self-regulated, therefore their trajectories on the a–R_qso plane are very short compared to the rest of the simulations shown.