User blog comment:Billicusp/Uggh/@comment-12.144.5.30-20160112070930/@comment-27173506-20160116182823

I belive you have a mistake there: X (9 u(9 u 9)]u) Y should be X (9 u[[(9 u 9)u) Y

I just realized that you have a weak definition of multiple u's. Under your definition, n uuu...uu n=n u n u n u n... n u n~n^^^n. This analysis is based on the mistake that n uuu...uu n = n uuu...u n uuu...u n... n uuu...u n. The growth rate of n uuu...u n with m u's in your system is f_4m(n), so n un n~f_5(n). Here's the analysis:

n u n~n^^n~f_4(n)>

n uu n = n u n u n u n... n u n~n^^n^^n^^n...n^^n=n^^^n~f_5(n)

n un n~n^nn~f_n+2(n)~f_w(n)

n u(n u[(n u n)]u) n~n^n^ n^. . n. . n n n~f_w+1(n)

n u(n u(n u n)) n=n u(n u[(n u[(n u[(...(n u n)...)] n)] n) n~f_w+1(f_w+1(f_w+1...f_w+1(n)...))=f_w+2(n)

n u(n u{n u n}) n=n u(n u[..[[n u n...]]] n) n~f_w2(n)=f_w+n(n)

n u(n u) n=n u(n u{n u{n u{...{n u n}...} n} n} n)n~f_w2+1(n)

This is the weakness of your notation. As you have it, each additional curly bracket does n nestings of the previous thing, so it reaches f_w2+n(n)=f_w3(n) (or using the current rules f_w2(n)). In HAN this is about n![1,3] (or using the current rules n![1,2]). If you did it so that [...[[...]]] with any amount of curly brackets and "n" square brackets is equal to [...[]...] with n u n nestings and "n-1" square brackets in each pair (or in a more readable form: n[m{n u nm}n]=n-1[m{n un-1[m{n u ...m{n u nm... nm}n-1] nm}n-1]), and each time the curly bracket is the outside thing, decompose it into n square brackets, it would reach f_w^2(n)=f_wn(n).

This is the strength of n->n->n->...n->n with n arrows in Chained arrow notation, {n,n,n,n} in Bowers Exploding Array Function, n![1,1,2]=n![1,n] in Hyperfactorial array notation, and more importantly, at least to the limit of everything in your other pages (if you changed the base rule in your notation to using Bower's dimensional arrays, but that isn't really your work, so I'm not counting it), which just shows that smart recursion is more important than the length of the explanation.