\begin{tabular}{@{}L{0.3\linewidth}L{0.2\linewidth}L{0.42\linewidth}@{}}
\toprule
Family & boundary type & evidence \\
\midrule
$B(h)$ window edges $h_\pm$ & escape & $h_\pm$ recovered exactly as roots of Cain's equality for a Schur complement; $\ks\sim\delta^{-1}$ (Table~\ref{tab:exponents}) \\
coNP gadgets (Pell cores with one attachment) & escape (heuristic) & exact face certificates at 40 of 40 certified births; $\ks$ grows like $\delta^{-1}$ (Table~\ref{tab:exponents}) \\
$B_3$ (Example~\ref{ex:b3}) & tangential & exact: unique contact, of rank one; $\ks=3$ \\
$D_0A_\varepsilon$, $\varepsilon=1/1000$ (Example~\ref{ex:tangential}) & tangential contact & exact contact of rank one; isolated (Jacobian rank 6) \\
community $S(J-\varepsilon I)$ at $\varepsilon_H$ (Section~\ref{sec:w1}) & tangential & $\ks\to5$ as $\varepsilon\uparrow\varepsilon_H$ \\
circulant $J-\varepsilon I$ (normal) & -- & D-stable iff Hurwitz; $P=I$ certifies for $\varepsilon>\varepsilon_H$ \\
\bottomrule
\end{tabular}