User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20180927133742/@comment-11227630-20180927155954

Just now I found something. See definition 3.6.

Ordinal \(\alpha\) is \(\beta\)-\(\Pi_n\)-reflecting if for every \(\Pi_n\) formula \(\phi\), \(L_{\alpha+\beta}\models\phi\rightarrow\exists\alpha_0<\alpha\exists\beta_0<\alpha(L_{\alpha_0+\beta_0}\models\phi)\). Similar to this: \(\alpha\) is \(\beta\)-\(\Pi_n^1\)-shrewd if for every \(\Pi_n^1\) formula \(\phi\), \(V_{\alpha+\beta}\models\phi\rightarrow\exists\alpha_0<\alpha\exists\beta_0<\alpha(V_{\alpha_0+\beta_0}\models\phi)\).

Such things exist, but the document does not show the relation between \(\beta\)-stable ordinals and \(\beta\)-\(\Pi_0\)-reflecting ordinals.