User blog comment:Alemagno12/Defining stage n cardinals in terms of stationary sets/@comment-5529393-20180611195103

S(1+y,x) won't enumerate the y-Mahlos when y is a limit ordinal. If x is the first y-Mahlo for y limit, then S(1+y,z) will be x for all z <= x.

I see the difference between S(T,x) and the normal definition of hyper-Mahlo, but what makes you think that S(T,x) enumerates the weakly compacts? This seems highly unlikely to me.

We need an actual definition of "stage x" cardinals before we can say that any particular function enumerates them.