This is my version of SpongeTechX's copy notation. It relies on repeating digits and has a relation with hypermathematics.
In case you don't know what hypermathematics is, it's basically normal mathematics but slightly funnier.
a*b in hypermathematics = a[b] using copy notation.
E.g. 2[2] = 22, 10[10] = 10101010101010101010.
Now here's where the juicy stuff comes in. Let me define some formal rules regarding this, where a,b ε Ζ+. Regarding the superscripted brackets, this just means how many layers of brackets there are. E.g. [3 = [[[. I call this basic copy notation (BCN).
Base rule, a[b] = aaaaa...aaa with b digits
Prime rule, a[n1]n = a
Recursion rule, a[nb]n = a[n-1a[n-1a[n-1... ...a[n-1a]n-1... ...]n-1]n-1]n-1 for a total of b a's
Now we can diagonalize this further by representing the nested brackets as arrays. I call this arrayed copy notation (ACN). Here. all letters (a,b,c,d,...,n) shown here ε Ζ+.
From here the array notation will be based on "dummy recursion", which is recursion without a rule similar to BEAF's catastrophic rule and is the recursion seen throughout Saibian's Extensible-E system and my Graham Array Notation.
Base rules, a[b] = aaaaa...aaa with b digits, and a[b,c] = a[cb]c.
Tailing rule, a[b,c,d,...,n,1] = a[b,c,d,...,n]
Prime rule, a[1,b,c,d,...,n] = a
Recursion rule: a[b,c,...,x,y,z] = a[b,c,...,x,a[b,c,...,x,y,z-1]]
You might have noticed that the recursion rule is significantly different to and much weaker than Tech's original notation. Some examples to illustrate this better:
2[2] = 22
2[3,2] = 2[[3]] = 2[2[2]] = 2,222,222,222,222,222,222,222 = dutrimevalka
2[3,3,2] = 2[3,2[3,3,1]] = 2[3,2[3,3]] = 2[3,2[[[3]]]] = 2[3,2[[2[[2]]]]] = 2[3,2[[22]]] = 2[[[...[[[3]]]...]]] (many nests of brackets)
We can diagonalize the linear arrays further until we reach a point where we have to move on. This extension, hyperiated copy notation (HCN), introduces hyperia (#), like the original copy notation. This works similarly to Hyper-E Notation. Here @ is the rest of the array and % d entries between hyperia.
Let any valid array be a[b X c X d X e... X n] where X is a separator.
Base rules, a[b] = aaaaa...aaa with b digits, a[b,c] = a[cb]c, and a[x#y] = a[x,x,...,x] for y x's.
Tailing rule, a[@n,1] = a[@n]
Prime rule, a[1#n] = a
Keep in mind that there can be no commas between the hyperia or before the first hyperion.
I don't have a recursion rule just yet, so I will be doing this on a case by case basis.
a[b#c] = a[b,b,...,b] for c b's.
a[@m,n] = a[@a[@m,n-1]]
a[%b#c] = a[%b,b,...,b] for c b's
Examples:
10[10#10] = 10[10,10,10,10,10,10,10,10,10,10]
10[10#10,10] = 10[10#10[10#10,9]]
Next extension, I define a[@b#nc] = a[@b#n-1b#n-1b#n-1... ...b#n-1b] with c b's. @ is the rest of the array or can be empty.
# is shorthand for #1
a comma is shorthand for #0
I'll give you an example. E.g., 3[3#33] = 3[3#23#23] = 3[3#23#3#3] = 3[3#23#3,3,3] = 3[3#23#3,3[3#23#3,3,2]] = 3[3#23#3,3[3#23#3,3[3#23#3,3]]], I think you know what to do from here.
Analysis time, using base 9.
9[9] is 1 smaller than 10^9 = 10^9 -1
9[9[9]] < 10^10^9 > 9^^3
9[[9]] > 9^^9
9[[[9]]] > 9^^^9
9[a,b] > 9{9}9, has growth rate w
9[a,b,c] > 9{{1}}9, has growth rate w+1
9[a,b,c,d] > 9{{2}}9, has growth rate w+2
9[a#b] > 9{{9}}9, growth rate w2
9[a#b,c] > 9{{{1}}}9, growth rate w2+1
9[a#b#c] > {9,9,9,3}, growth rate w3
9[a#2b] = 9[a#a#a#...#a] with b a's > {9,9,9,9}, growth rate w^2
9[a#2b,c] has growth rate w^2 +1
9[a#2b#c] has growth rate w^2 +w
9[a#2b#c#d] has growth rate w^2 +w2
9[a#2b#2c] has growth rate (w^2)2
9[a#3b] has growth rate w^3
9[a#[1]b] = 9[a#ba] has growth rate w^w
MORE COMING SOON
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