= Steady fold of a thermosolutal Lorenz model
{title2=$r_{\rm min}=(\sqrt{r_s\tau}+\sqrt{1-\tau^2})^2$}
A nonzero steady <Lorenz model of thermosolutal convection> branch obeys $r(x)=(1+x)+r_s\tau(1+x)/(\tau^2+x)$, with $x=a^2$. For $0<\tau<1$, it has an interior <saddle-node bifurcation> precisely when $r_s>\tau^3/(1-\tau^2)$. Differentiating gives $x_*=-\tau^2+\sqrt{r_s\tau(1-\tau^2)}$ and the stated minimum. The weaker-versus-stronger threshold distinction matters: a sufficient bound need not be the sharp subcriticality condition. Steady-branch existence alone does not decide oscillatory stability.
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