User blog comment:Ubersketch/Lexicographic ordinal notation up to the Bachmann-Howard ordinal/@comment-35470197-20190913052145/@comment-35392788-20190913110116

I'll get to the point somewhat more directly : you should not be working with ordinals when building your notation. Only strings. Why the hell not, you might ask ?

Well, it's quite simple actually. If you're working with actual ordinals, you have \(\omega^\omega=(1+\omega)^{1+\omega}=\text{TREE}(3)+(\text{Croutonillion}+\omega+0)^{G_64+\omega}\). When you have all of these equivalent forms, there isn't a single way to map them to strings.