This will be a cite source for the page update. Please do not post errors here, rather post in my last post about 7th place.

**Fundamental laws**

All reworks/old versions of GAN follow the fundamental laws here

* An # means the rest of an array or just an array.

* An & means any kind of symbol, let it be , ^ / \ A and so on.

1. Regarding 1's. [#,1,#] = [#]

2. Regarding 0's. [#,0,#] = [#,#]

3. Symbols. [a&1] = [a]

4. If the symbol is decomposable. [a&1] = [a&[a&0]]. The way the symbol decomposites can change regarding which symbol is used. It can be [a&1] = [a&[a&[.....[a&0]...]]] with a &'s or another definition.

5. If the symbol is decomposable. [a&0] = [a,a]

6. Any single argument. [a] = a

7. Any emtpy slot. [] = the nearest number, righ to left ([5.[].6] = [5,6,6]). Just an empty slot ([]) is 10.

**Basic Rules**

Rules can be changed/followed (by) the fundamental laws.

[a,b] = a^{b}

[a,b,c] = a↑↑...↑↑b with c up-arrows

[a,b,c,d] = a{a{a...a{c}b...b}b}b with d {}'s using BEAF. \(a\{\{1\}\}b\) = [a,a,a,b-1] *(May contain errors)*

**Arrays**

with 5 or more entries, GAN, like CCAN or BEAF, we define degenerate arrays

The last entry in the array is called the "Helper" The second-to-last entry in the array is called the "Leader"

This only applies to 5+ entrie arrays. 4 or lower entry arrays should follow the "Basic Rules" section.

e.g. [a,b,c,d,e] -> d is the "leader" the leader is the second-to last entry in an array

[a,b,c,L,d] = [a,b,c,L-1[a,b,c,L-1,d]]

(L means the "Leader")

The process stops when L reaches 1 e.g. [3,3,3,3,3] = [3,3,3,2,[3,3,3,2,3]] = [3,3,3,2,[3,3,3,1,[3,3,3,1,3]]] = [3,3,3,2,[3,3,3,[3,3,3,3]]]

Continues :

[3,3,3,3,3,3] = [3,3,3,3,2,[3,3,3,3,2,3]] [5,6,4,2,8,9] = [5,6,4,2,[5,6,4,2,8,8],8]

So we can move on to the rules

The leader decreases by one and the last entry is turned into the whole array itself, with the leader reduced by 1.

**Rules**

[a^b] = [a,a,a,...,a] with b a's

[a^^b] = [a^a^...^a] with b a's

[a^^^b] = [a^^a^^...^^a] with b a's

[a(^)b] = [a^^...^^a] with b ^'s

[a(^^)b] = [a(^)a(^)a(^)...(^)a] with b a's

[a(^^^)b] = [a(^^)a(^^)a(^^)...(^^)a]

[a((^))b] = [a(^^...^^)a] with b ^'s

[a((^^))b] = [a((^))a((^))a((^))...((^))a]

[a((^^^))b] = [a((^^))a((^^))a((^^))...((^^))a]

[a/b] = [a((...((^^...^^))...))a] with b ()'s and b ^'s [a//b] = [a/a/a/.../a] with b a's

[a///b] = [a//a//a//...//a] with b a's

[a(/)b] = [a///...///a] with b /'s

[a(//)b] = [a(/)a(/)a(/)a....a(/)a] with b a's

[a((/))b] = [a(///...///)a] with b /'s

[a((//))b] = [a((/))a((/))a((/))a.....a((/))a] with b a's

**Dimensional arrays**

Dimensional arrays are arrays that need 2 or more spaces to represent.

they are done by [a**c], think of it as a ^{b}&c in BEAF **

**[a<2>b] is a 2-row 2-dimensional array of a,b = **

a,b a,b

**Which is [[a,b]^[a,b]]. All spaces are connected by the ^ **

**n-dimensinal arrays are n-dimensional, n-row arrays with n-1 dimensional arrays connected by the ^ **

**Rules (Continued)**

We define a powerfull function, [a\b]

[a\b] = [[[a**a<b>a.....a<b>a]((...((///...///))...))[a<b>a<b>a.....a<b>a]((...((///...///))...))[a<b>a<b>a.....a<b>a]......[a<b>a<b>a.....a<b>a]((...((///...///))...))[a<b>a<b>a.....a<b>a]\b-1]
**

**<b>with b a's in [a<b>a<b>a<b>a...<b>a] and a [a<b>a<b>a<b>a...<b>a]'s overall. there are b ()'s and a /'s.
**

**<b>[a\\b] = [a\a\a...a] with b a's **

**And so on (You already know what [a\\\b] means, do you :3) **

**[A,a] = [a\\\...\\\a] with a \'s **

**[A,A,a] = [A,[A,a]] **

**[A,a,a] = [A,[a,a]] **

**[A2,a] = [A,A,...,A,a] with a A's **

**[An,a] = [An-1,An-1,...,An-1,a] with a An-1's**