The Diagonalizable @ Notation is an attempt to make a much stronger version of X-Sequence Hyper-Exponential Notation. Like the original, it includes inputs a(#)b, and a pseudo-ordinal, which is @ (pronounced at) in this notation instead of X.
a(b)*c=a(b)a(b)...(b)a(b)a with c a's
a[(b)^c]d=a(b)(b)...(b)(b)d with c (b)'s
Inputs inside the parentheses are called inners, other outputs are called outers.
All outers must be positive integers. All inners must be in some form involving 0, a positive integer, @, or @ having hyperoperations applied to it in a way which uses positive integers or another @
# (pronounced hash) is any series of inners
All parenthesis must be ordered greatest to least, assuming @ is some number larger than all other entries. For example, a(4)(5)b is invalid.
The notation a(b)*c\(d)e can be used for a(b)a(b)...(b)a(b)a(d)e with c a's. It is exclusively used in naming numbers.
All expressions are solved right to left.
a(0)b=a^b
a#(0)b=a#*b (In this case, the # MUST contain some valid string)
a#(b)c=a#[(b-1)^c]a
For a non-natural number input, a(#)b=a(#[b])a
@[b]=b
a(#+c)b=a(#+c[b])a=a(#+c-1)ba only if c>0
(#+b)[n]=#+b[n] if b is some limit ordinal
#*(b+1)[n]=#*b+#[n]
(#*b)[n]=#*b[n] if b is some limit ordinal
#b+1[n]=#b*#[n]
#b[n]=#b[n] is b is some limit ordinal
(#{b}(c+1))[n]=(#{b}c){b-1}#
(#{b}c)[n]=#{b}c[n] if c is some limit ordinal
(#{b}c)[n]=#{b[n]}c if b is some limit ordinal
(#(0)b)[n]=#^b
Define (v) as # from B@N
(#(v)(0)b)[n]=#(v)#...#(v)# (with b #'s)
(#(v)(b)c)[n]=#(v)(b-1)c#
(#(0)b)[n]=#(0)b[n] if b is some limit ordinal
(#(v)(0)b)[n]=#(v)(0)b[n] if b is some limit ordinal
(#(v)(b)c)[n]=#(v)(b)c[n] if c is some limit ordinal
(#(v)(b)c)[n]=#(v)(b[n])c if b is some limit ordinal
Prioritize: in order, in a(b)c, b, in a{b}c, b, in a(b)c and a{b}c, c, otherwise right to left