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I have decided to change the format of my notation to a[X{a,b,c}]b. The previous form is not completely incorrect, but I prefer this.

Basic Separators

First, let's make a[X{x[0]y}]b = a[X{x,x,x...x}]b with y entries. That's the very first and basic separator. Now, before we think about the next separator, we have to think how multiple [0] separators would react. For example, what would this solve to: 9[X{3[0]2[0]3}3 = 9[X{3[0]2,2,2}]3. We have a few options.

  1. Weak option. 9[X{3[0]2,2,2}]3 = 9[X{3,3,2,2}]3. I think it's pretty obvious why this is lame.
  2. Medium option. 9[X{3[0]2,2,2}]3 = 9[X{3[0]9[X{2,2,2}]3}]3. This is way stronger but still rather weak.
  3. Strong option. This is a bit more complex, and the option I chose. 9[X{3[0]2,2,2}]3 = 9[X{3[0]2,9[X{3[0]2,9[X..{3[0]2,2,1}..]}]}] with 9[X{3[0]2,2,1}]3 nestings. In other words, by treating the series of entries as a sort of array in itself, we reach much higher strength.

Firstly though, I need to explain a few things. The nesting that works like normal arrays works only up to when there is only one entry left: the one to the right of the separator. From here, it's pretty clear how to continue with separators: a[X{x[y+1]z}]b = a[X{x[y]x[y]x[y]x...x[y]x}]b. Furthermore, separators are always solved right to left. That is, this: 7[X{2[4]3[1]2}]2 would decompose into this: 7[X{2[4]3[0]3}]2 = 7[X{2[4]3,3,3}]2 and not into this: 7[X{2[3]2[3]2[1]2}]2.

Here's a small example: 3[X{3[1]4[0]0,0}]2 = 3[X{3[1]4[0]3}]2 = 3[X{3[1]4,4,4}]2 = 3[X{3[1]4,3[X{3[1]4,3[X..{3[1]4,4,3}..]}]}]2 with 3[X{3[1]4,4,3}]2 nestings. This will eventually decompose into 3[X{3[1]x}]2 with a crazily large x, and that's equal to 3[X{3[0]3[0]3[0]...3[0]3}] with x 3's, and we've seen how big the numbers get with merely one [0] at the end!

Now that we've defined basic separators, surely we can go beyond. Well, yes, we can.

Nested Separators

We have defined separators, and we've defined arrays. Why not let the separators become arrays, like this: a[X{x[1,0]y}]b = a[X{x[0,a[X{x[0,a[X..[0,0]..]y}]y}]b with a[X{x[0,a[X{x[0,0]y}]b}]b nestings. With that in mind, we can make the separator an array in itself, even containing a separator! For example: 6[X{3[4[7]9,10[2[4]3]8]4}]5. This brings me to the limit of my notation so far (but not for long): the G(n) function.

G(1) = 1
G(n) = n[X{n[n[n[...n[n]n...]n]n]n}]n with G(n-1) nestings. This function is at least f_w^w2(n), and I think much much more. Let's check the first values of this function:

G(1) = 1 (by definition)
G(2) = 2[X{2}]2 = 4 (trivial case)
G(3) = 3[X{3[3[3[3]3]3]3}]3 = 3[X{3[3[3[2]3[2]3]3]3}]3 = 3[X{3[3[3[2]3[1]3[1]3]3]3}]3 = 3[X{3[3[3[2]3[1]3[0]3[0]3]3]3}]3 = 3[X{3[3[3[2]3[1]3[0]3,3,3]3]3}]3 = 3[X{3[3[3[2]3[1]3[0]3,3[X{3[3[3[2]3[1]3[0]3,3[X{3[3[3[2]3[1]3[0]3,3[X..3[X{3[3[3[2]3[1]3[0]3,3,2}]3..]3]3]3}]3]3]3}]3]3]3}]3 with 3[X{3[3[3[2]3[1]3[0]3,3,2]3]3}]3 nestings.

That exploded pretty fast! I think my notation now deserves one or two numbers. G(10100) is called Googox (pronounced GOO-gox), and G10100(10100) is Supergox (pronounced SU-per-gox).

Rules

Definitions

An entry is a single non-negative integer or 2 non-negative integers separated by a separator, which is an entry inside square brackets.

"#" means a linear array, an array composed only of non-negative integers separated by commas. If there is a separator somewhere in the array, then the linear array counts only from the number to the right of the separator.

"N" means a non-zero number.

"&" means a single number or array composed of entries inside curly brackets separated by commas.

// means any amount of brackets and entries inside a separator on the left side (including zero, that is no nesting).

\\ means any amount of brackets and entries inside a separator on the left side (including zero).

Y means a string of zeroes of length zero or more in a linear array.

@ means rest of expression on the same level of nesting.

+++ means rest of expression.

Actual Rules

  1. a[X1]b = a^ba
  2. a[X&]1 = a^a
  3. a[Xx+1]b+1 = a[Xx](a[Xx+1]b)
  4. a[X{0}]b = a[Xa]b
  5. a[X{n+1}]b = a[Xa[Xa[X..{n}..]b]b]b with a[X{n}]b nestings
  6. a[X{+++,//Y,0,0\\}]b = a[X{+++,//Y,a\\}]b
  7. a[X{+++,//#,n+1,0\\} = a[X{+++,//#,n,a[X{+++,#,n,a[X..{+++,//#,n,0\\}..]b\\}]b\\}]b with a[X{+++,//#,n,a[X{+++,//#,n,0\\}]b\\}]b nestings
  8. a[X{+++,//#,m,n+1\\}]b = a[X{+++,//#,a[X+++,//#,a[X...{+++,//#,m,n\\}..]b\\}]b\\}]b with a[X{+++,//#,m,n\\}]b nestings.
  9. a[X{+++,//@x[0]y\\}]b = a[X{+++,//@x,x,x...x,x\\]b with y x's.
  10. a[X{+++,//@x[n+1]y\\}]b = a[X{+++,//@x[n]x[n]x[n]x...x[n]x\\}b with y x's.

The strength of the notation reaches up to about f_w^{w2+1}(n). Analysis coming soon!

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