Efficiency upper bound from Eddington-limited growth ($\eta \le \Delta t/[\Delta t+t_E\log (M_f/M_i)]$)

For fixed efficiency, growth from $M_i$ to $M_f$ within $\Delta t$ at no more than the Eddington luminosity requires $[\eta /(1-\eta )]t_E\log (M_f/M_i)\le \Delta t$. Solving gives an upper bound, because greater radiative efficiency reduces retained mass growth at a fixed luminosity. A duty cycle below unity tightens the requirement; mergers and super-Eddington episodes invalidate its stated premises.