There are two kinds of ambiguities on BEAF:
- Undefinedness of arrays past tetrational arrays (formalization by Deedlit11 and Ikosarakt1 doesn't satisfy condition about number of entries in array, but there was not another formalization: seems to be partially resolved until X^^^X by me using climbing method and formal strings + operations on them like "delete this part and append something to end of it")
- The legion arrays are expectedly broken: the smallest legion array in Bowers' description has power of only LVO. That is the reason why I won't never define legion arrays unless Bowers himself fixes the definition. There are two ways of defining them: one-layer (legion arrays = LVO) and multi-layer (legion arrays = psi(Omega_omega using Madore's psi function). IF we choose multi-layer way of definition and IF estimation by Hyp cos is correct, then I agree that the power of four-entry legiattic arrays is some expression in Hyp cos' OCF under weakly Mahlo cardinals. (But not in Rathjen's psi function using M cardinals - despite of some sources by Hyp cos' OCF using chi function in addition to psi and this seems like Rathjen's function, this is supposedly different OCF!) It is possible that in analysis of multi-layer arrays not using ill-defined catching hierarchy we need some even higher OCF's: using psi function on weakly compact cardinals, П4-reflections, stable cardinals and even beyond. But even Rathjen's psi function using M cardinals is highly complicated. I have some knowledge about similar (but different) OCF with examples of FSs (but incomplete) and the examples of expressions in it are ψ(ψ_χ(M^M)(0)), ψ(ψ_χ(M_3)(0)), ψ(ψ_χ(M_M)(Ω^Ω)), ψ(ψ_χ(M_M,1)(0)) and ψ(ψ_χ(M(1,0))(0)) where M(1,0) is the least fixed point of Mahlo cardinals, that is, M_M_M_M... .
UPD: I warn all to not using comparison of BEAF with any ordinal notation because BEAF past {L,X}n,n becomes very ambiguous and there is no unique way to work with them.