Most unstable softened disk wavenumber ($s_{\ast }=e^{-\delta s_{\ast }}(1-\delta s_{\ast }),s_{\ast }<(1+\delta {)}^{-1}$)

With $s=Q{c}_{s}k/\kappa$ and $\delta =\kappa ϵ/(Q{c}_s)$, maximize $F(s)=2s{e}^{-\delta s}-s^2$. Its unique positive critical point satisfies the displayed equation. For positive gravitational softening, $e^{-\delta s_{\ast }}<1$ and $1-\delta s_{\ast }>0$, giving the strict bound. Unsoftened gravity has $s_{\ast }=1$.