Seed-mass bound from Eddington-limited growth ($M_{0,\min }=M_f{e}^{-\Delta t/t_{\mathrm{Sal}}}$)

For fixed radiative efficiency $\eta$, Eddington-limited accretion without mergers gives $M(t)=M_0{e}^{t/t_{\mathrm{Sal}}}$ with $t_{\mathrm{Sal}}=\eta {\sigma }_{T}c/[(1-\eta )4\pi G{m}_p]$. A specified final mass and growth interval therefore require $M_0\ge M_f{e}^{-\Delta t/t_{\mathrm{Sal}}}$. The retained factor $1-\eta$ distinguishes actual hole growth from rest-mass supply.