= Exponential closed-box gas consumption
{title2=$M_g=M_0e^{-(1-F)kt}$}
With gross <star formation rate> $\psi=kM_g$, constant returned mass fraction $F$, and the <instantaneous recycling approximation>, the net locked mass rate is $(1-F)\psi$. <Mass conservation> gives $\dot M_g=-(1-F)kM_g$. 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.
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