Beta-stable cardinal property ($\beta$-stable)

A formula $\Phi (x,\kappa )$ defines a $\beta$-stable property of $\kappa$ when it is absolute between the universe and every transitive $M\supseteq V_{\kappa +\beta }$.

If a $\beta$-strong embedding has critical point $\kappa$ and the $\beta$-stable property $\Phi (\kappa )$ holds, then $\{\mu <\kappa :\Phi (\mu )\}$ is unbounded in $\kappa$. For each $\gamma <\kappa$, the target sees $\kappa$ as a witness between $\gamma$ and $j(\kappa )$; elementarity reflects a witness between $\gamma$ and $\kappa$.

A property $P$ reflects below a cardinal $\kappa$ when $\{\mu <\kappa :P(\mu )\}$ is unbounded in $\kappa$. For an elementary embedding with critical point $\kappa$, one often proves this by using $\kappa$ as a witness below $j(\kappa )$ in the target and then invoking elementarity.