User blog comment:Alemagno12/Defining stage n cardinals in terms of stationary sets/@comment-1605058-20180612183310

Okay, I was talking with Deedlit and we think we've reached an exact conclusion. S(T,a) enumerates 1-hyper-Mahlo cardinals, which are cardinals in which hyper-Mahlos are stationary.

First, note that every hyper-Mahlo cardinals H satisfies H=M(H,0), so, since hyper-Mahlos are stationary in any 1-hyper-Mahlo, so are cardinals of the form M(a,0). This in particular disproves your claim that those cardinals are precisely weakly compact ones.

Conversely, suppose we have a cardinal \(\kappa\) which is not 1-hyper-Mahlo. We may as well assume it is hyper-Mahlo (if it's not, M(a,0) surely aren't stationary in it). Since hyper-Mahlos are not stationary in it, there is a club \(C\subseteq\kappa\) which doesn't contain any hyper-Mahlos. Let \(D\) be the closure of the set \(\{M(\beta,1),\beta<\kappa\}\), also a club in \(\kappa\) since it is hyper-Mahlo. We note that the only elements of \(D\) of the form \(M(\alpha,0)\) are hyper-Mahlos (if \(M(\alpha,0)\) is a limit of \(M(\beta,1),\beta<\alpha\), then \(M(\alpha,0)\) has cofinality at most \(\alpha\), which is possible only if \(M(\alpha,0)=\alpha\)). But then it follows that \(C\cap D\) is a club which contains no cardinal of the form \(M(\alpha,0)\), which is exactly what we wanted.

(thanks to Deedlit for noting the first part of the direction, and for noting an argument I've figured out works for any non-1-hyper-Mahlo)