= Empty strict decay domain of the symmetric two-derivative formula
{title2=$\mathcal D_{\rm decay}=\varnothing$}
On the <Dahlquist test equation>, write the <amplification polynomial> as $D\xi^2-16z\xi-A$, where $D=z^2-7z+15$ and $A=z^2+7z+15$. The identity $|A|^2-|D|^2=28\operatorname{Re}z(|z|^2+15)$ shows that both <roots of a polynomial> can lie inside the open <unit disk> only if $\operatorname{Re}z<0$. But the <complex quadratic Schur criterion> would then require $32|\operatorname{Re}z|(|z|^2+15)<28|\operatorname{Re}z|(|z|^2+15)$, which is impossible. Thus the <strict linear stability domain> is empty. This concerns robust damping of all modes, rather than convergence on fixed time intervals or the <bounded-root stability set of the symmetric two-derivative formula>.
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