Katakana!'s extended arrow notation

Katakana!'s extended arrow notation is an extension for arrow notation created by Katakana!Katakana! Ultimate Number List Segment 1 @ 22:42 YouTube..
==Definition==
Since there is no formulation of the notation by the creator, the definition of this notation is a verbal description. The original text of the description is given below.
You can diagonalize over any amount of up arrows by using \([\uparrow]\). The number after the \([\uparrow]\) denotes how many arrows you should put. Ex: \(10[\uparrow]5 = 10\uparrow\uparrow\uparrow\uparrow\uparrow10\)
You can also put arrows after the \([\uparrow]\), so you can have things like \([\uparrow]\uparrow\) and \([\uparrow]\uparrow\uparrow\). These behave similarly to before: \([\uparrow]\uparrow\) is repeated \([\uparrow]\), just like \(\uparrow\uparrow\uparrow\) is repeated \(\uparrow\uparrow\).
And after \([\uparrow]\uparrow\uparrow\uparrow\uparrow\cdots\)? You can just add another \([\uparrow]\) onto the end, leaving you with \([\uparrow][\uparrow]\). You can continue doing things like \([\uparrow][\uparrow]\uparrow\), \([\uparrow][\uparrow][\uparrow]\), \([\uparrow][\uparrow][\uparrow][\uparrow]\)... etc.
After all of this, we reach \([\uparrow\uparrow]\), diagonalizing over how many \([\uparrow]\)'s you want to put, like \([\uparrow]\) diagonalizing over the amount of \(\uparrow\)'s. Then, we can do \([\uparrow\uparrow]\uparrow\), \([\uparrow\uparrow][\uparrow]\), \([\uparrow\uparrow][\uparrow\uparrow]\), etc using all we learned before.
Notice a pattern here? That's right, if we do \([\uparrow\uparrow\uparrow]\), it diagonalizes over the amount of \([\uparrow\uparrow]\)! We can continue doing things like \([\uparrow\uparrow\uparrow\uparrow]\), \([\uparrow\uparrow\uparrow\uparrow\uparrow]\), and so on as well.
Now, the same thing we did to arrows before, we can do again now. Since \([\uparrow]\) diagonalizes over \(\uparrow\uparrow\cdots\uparrow\uparrow\), you can put this all inside brackets:
\(\uparrow\) diagonalizes over \([\uparrow\uparrow\cdots\uparrow\uparrow]\). So: \(10[ [ \uparrow ] ] 5 = 10[\uparrow\uparrow\uparrow\uparrow\uparrow]10\)...
It looks like we can keep up putting brackets inside of one another to make the notation stronger. So, we can have \([ [ [\uparrow] ] ]\) diagonalize over \([ [ \uparrow\uparrow\cdots\uparrow\uparrow ] ]\), and then keep going until we reach \([ [ \cdots [ [\uparrow] ] \cdots ] ]\), with arbitrarily many brackets.
==Analysis==
* Analysis by Sirootooo
==Sources==
Category:Notations
Category:Arrow