User blog comment:Scorcher007/Hierarchy of ordinals Z 2. Help me to understand/@comment-35470197-20180825090441

> 5• M ~ recursively Mahlo ~ weakly Mahlo = (zoooo 2.5) ~ OCF→PTO KPM

Do you have a precise reference of it?

Madore refers to [Rathjen 1990], but the paper did not deal with PTO of KPM at all. What Rathjen proved in another related paper is just the inequality \(\textrm{PTO}(\textrm{PTO}) \leq \psi_0(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\).

The equation can be found in several wikis, but there is no precise reference. If you know a precise reference or a reason why it holds, please tell me it. (I note that Deedlit has stated the equality for his OCF, but his OCF is not Rathjen's OCF.)