User blog comment:Primussupremus/More ideas for my notation./@comment-28606698-20170426201145

1) You did not explain how it works (a,b,c#[x]|{y}|{k}[n])= f_w+k+4(n), how does your notation obtain such properties. As for now it seems that without mediation of FGH your notation is powerless.

2) First three entries do not work after you have passed limit of up-arrow notation

3) You still did not define set of rules.

Let me show how Bowers's array works for case of four entries, may be it will be helpful for you (I will talk about 4-entries example becose it is simplier, than linear array with arbitrary amount of entries, and that is closer to level, where you are working):

1) a,b=a^b 2) a,b,c,1=a,b,c 3) a,1,c,d=a 4) a,b,c,d=a,(a,b-1,c,d),c-1,d 5) a,b,1,d=a,a,(a,b-1,1,d),d-1

The main idea is the following: if third or fourth entry increases by one then previous entry becomes the original array with the second element decreased by 1. Those rules give next properties

$$a,b+1,c+1,d+1=f_{\omega d+c}^b(a)$$

and the limit $$\omega^2 $$.