User blog comment:89Pepsi Polka/the hyper-square hierarchy/@comment-30118230-20180218131850

I can help you make this well-defined. Just remember that if the notation isn't well-defined,interpreting it is difficult to people who want to help you.

This is a well-defined ruleset:

Let $$C$$ denote any valid string in this notation. $$C$$ can also be empty.

$$\square(n)=n+1$$

$$C_1C_2(n)=C_1(C_2(n))$$

$$\boxed{C_1\square}_{C_2}(n)=\boxed{C_1}^n_{C_2}(n)$$ where $$C^1=C\land C^{n+1}=CC^n$$

$$\square_{C\square}(n)=\square_{C\square}[n](n)$$ where $$\square_{C\square}[1]=\square_C\land \square_{C\square}[k+1]=\boxed{\square_{C\square}[k]}_C$$

I hope this is what you intended,and yes,with a proper fundamental sequence,this notation reaches $$f_{\zeta_0}(n)$$.