Key parameters at the onset of core collapse. Panel (a): final compactness parameter ξ_2.5. Panel (b): central specific entropy s_c. Panel (c): iron core mass M_Fe. Panel (d): binding energy above the iron core E_bind. Panel (e): C-free core mass M_C-free, all plotted as functions of the initial helium star mass $M_{\mathrm{He}}^{\mathrm{i}}$. The top axis shows the CO core masses of He stars at the end of core He burning. The light gray shading indicates radiative core C burning, while the darker gray shading represents radiative core Ne burning in the panel (a). The red dotted line at ξ_2.5 = 0.35 separates models likely to explode (below the line) from those likely to implode (above the line). Black crosses indicate models predicted to explode according to both the explodability criteria of Ertl et al. (2016) and Müller et al. (2016). White crosses mark models predicted to explode by the Ertl et al. (2016) criterion but not the Müller et al. (2016) criterion. Black circles indicate models predicted to fail to explode and collapse into BHs according to both criteria. The black dashed vertical line marks the $M_{\mathrm{He}}^{\mathrm{i}}=21M_{\odot }$ model (M_CO = 9.86 M_⊙), serving as a proxy for the ∼10 M_⊙ threshold for the upper limit of successful explosions suggested by Patton & Sukhbold (2020).