Comparison of values for the critical mass ratio for stable mass transfer. The four systems correspond to initial periods of approximately P_i ≃ 1.8, 7.1, 10.5, and 72.6 days (for a representative q = 0.25). Panel a shows values for q_crit obtained using the ζ method (q_crit, ζ; sphere hashing) and using MESA calculations (q_crit, calc; diagonal hashing). For β = 1, the two methods coincide by construction, since we calibrate ζ_d from the conservative borderline and then solve ζ_L(q,β) = ζ_d to obtain q_crit,ζ. Panel b shows the ratio of these critical mass ratios q_crit,calc/q_crit,ζ. In both panels, the shaded widths are chosen for readability. The vertical extent indicates the uncertainty owing to the finite bisection resolution. The colours correspond to different mass-transfer efficiencies, as shown in the legend.