Exponential closed-box gas consumption ($M_g=M_0{e}^{-(1-F)kt}$)

With gross star formation rate $\psi =k{M}_g$, constant returned mass fraction $F$, and the instantaneous recycling approximation, the net locked mass rate is $(1-F)\psi$. Mass conservation gives ${\dot{M}}_g=-(1-F)k{M}_g$. For constant stellar yield $Y$ per unit locked mass and initially pristine gas, metal conservation gives $M_g\dot{Z}=Y(1-F)\psi$, so gas-phase metallicity grows as $Z=Y(1-F)kt$. Gross star formation and permanent locking have different normalizations.

A fixed-efficiency closed box with permanent locking fraction $\alpha$ predicts exponential decay of gas and gross star formation. Backward extrapolation therefore grows exponentially with lookback time. It requires a stated present age and assumes the same gas reservoir and efficiency over the entire interval. Cosmological inflow, outflow, mergers and changing gas conditions generally invalidate such an extrapolation over many depletion times.