User blog comment:Vel!/FGH Gripe/@comment-30754445-20170220134751/@comment-30754445-20170220164134

That condition might be "desirable", but it is an impossible requirement.

As an example, consider this function:

Given fω(k) ≤ n < fω(k+1), we define g(n)=fω(k+1)

What FGH ordinal would you attach to g(n)?

And what ordinal would you attach to gn(n)?

Any reasonable definition of "an X-level function on the FGH" would rate both g(n) and gn(n) at the ω-level. There's no way to avoid this kind of thing.