Limits of stability for Pop III SMSs accreting angular momentum at a fraction f of the Keplerian momentum. The solid red lines show the hylotropic limits for the indicated f (Eq. (81)), with T_c = 1.8 × 10⁸ K and μ = 0.6, relevant for Pop III SMSs. The dotted red lines indicate the limit of stability for constant ratios 0.001–0.01–0.1 of rotational to gravitational energy (Eq. (90)). The coloured areas reflect the relative contribution by gas and rotation in the determination of the stability limit (81): From yellow to red, the rotation term represents < 0.1, 0.1−1, 1−10, and > 10 times the gas term , respectively. The Pop III GENEC tracks of Haemmerlé et al. (2018b, 2019) are shown for the indicated accretion rates. The mass fraction of the convective core is indicated by the grey diagonals.