User blog comment:Mh314159/FOX notation updated/@comment-39585023-20191213105844

"For example, since [n](x) is defined only for the case n=m, the recursion [2](0) = [1]<2>(2) does not make sense."

Where did you get this impression? Is it because of repeating n in "[n]‹n›(x) = is recursed using the same rule as f‹n›(x), treating [n] as f." I will change to "[u]‹n›(x) = is recursed using the same rule as f‹n›(x), treating [u] as f.". The definition of [u](x) is parallel to the definition of f(x), therefore [u](x) =[u]‹m-1›[u]‹m›(x-1)(x). The recursion is defined all the way down to [u]<0>(x) which then recurses to [u-1](x) with S being a long string. I don't think I'm ever going to formalize my work to the extent that you would choose to, you being trained in formal math.