A tiny CARE residual is inconclusive near the stability boundary

The scalar CARE is $2\epsilon X-X^2+{\epsilon }^2=0$, with positive root $X_{\ast }=(1+\sqrt{2})\epsilon$ and stable closed loop $A_c(X_{\ast })=-\sqrt{2}\epsilon$. At the candidate, ${\widehat{A}}_c=-0.1\epsilon <0$, the exact residual is $r=\mathrm{\mid }{F}_c(\widehat{X})\mathrm{\mid }=1.99{\epsilon }^2=1.99\cdot {10}^{-16}$, and $g=1$.

The scalar Lyapunov derivative is $\mathcal{L}(E)=2{\widehat{A}}_{c}E=-0.2\epsilon E$, so its inverse norm is $\kappa =1\mathrm{/}(0.2\epsilon )=5\cdot {10}^8$. For every $\rho >0$, $r+{\rho }^2\ge 2\sqrt{r}\rho$, so $\kappa (r+{\rho }^2)\ge 2\kappa \sqrt{r}\rho >\rho$ because $2\kappa \sqrt{r}>14$. Thus no radius satisfies even the self-map condition, despite the absolute residual being below $2\cdot {10}^{-16}$ and the exact stabilizing root existing. The residual alone is inconclusive because the inverse Lyapunov factor is large near the imaginary axis.