This function is almost like in the definition of middle-growing hierarchy https://googology.fandom.com/wiki/Middle-growing_hierarchy https://googology.fandom.c

This function is almost like in the definition of middle-growing hierarchy https://googology.fandom.com/wiki/Middle-growing_hierarchy https://googology.fandom.com/wiki/Middle-growing_hierarchy But this hierarchy can be defined for any ordinal with a system of notations. I'm wondering if there is some deeper sense in this Patcail's predecessor function? Are there some follow up research on that?