User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 3/@comment-5529393-20130530121945/@comment-5529393-20130530141332

Um, ordinal collapsing functions are nothing like your "k-brackets is k-1-brackets nested n times" rule. Each cardinal level collapses functions down to the next lower cardinal level in a very powerful manner. Nesting n times is nowhere near as powerful. I suggest you study ordinal collapsing functions in greater depth, they really are a very elegant and powerful construction.

The Buchholz Hydra actually uses a subtle rule involving going down to the highest ancestor that has a lower index than the leaf node, and this case it really does correspond very closely to ordinal notation. Buchholz proved his hydras were fast-growing by showing the exact isomorphism between his hydras and the corresponding ordinal notation. You haven't done so, and it certainly appears to me that your notation is nothing like ordinal collapsing functions.