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I don't know if any of you people remember, but way back like 2 years ago, I created something called UltraF notation, which was the most powerful notation I have ever made.
I could've continued a bit further quite easily by going the Saibian route and shoving my notation inside BEAF as far as it'll go, but I decided against it. Instead, I envisioned a fabulously powerful extension which could defeat anything that approach had to offer. It was fiendishly complicated and messy, but had sooooo much potential. I called it DistortionF. Pastme figured out most of it, but hit a critical snag that halted progress. Then, for a really really long time, I kinda lost interest in googology. However, a few days ago, I started thinking about Distortioâ€¦
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I made most of a PRBN extension a while back, but then I lost it and now I can't be bothered to figure it all out again. I've also been working on a horrifyingly overcomplicated mess I call "columnar mathematics".
Both of those went badly, so I went back to the drawing board and made this instead, which turned out to be a lot easier to make work.
UltraF notation is based on using the entity F to substitute subexpressions into themselves, in such a way that the presence of each additional F amplifies the power of the others. F doesn't stand for anything, I just needed some symbol to use.
I used more plain language here than normal  The rules are a lot easier to work with than they are to formalise.
The entity F only makes sense within a subeâ€¦
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This is a new operator I've come up with, which not only grows faster than any conventional hyperoperator (e.g. pentation), but also grows faster than any extended hyperoperator (e.g. megotion). It doesn't fit precisely into the framework thanks to BEAF's weirdness so I can't really express it as the limit of a sequence of other established functions, but it's not too far away and I'm sure I could make a modified version which could fit in with a little fiddling.
It will use the notation:
\[A\circledcirc B\]
It is defined as follows, using PRBN:
\[A\circledcirc B = \circledast[A]_B[A](A)\]
The growth rate of
\[A\circledcirc A\]
in FGH should be something close to
\[f_{\omega^\omega}(n)\]
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I find the higher hyperoperators to be a bit confusing, so for a while now I've been trying to develop a system which will allow me to define them more easily.
My first attempt was "arrayspace functions"  functions with of a base, an index in the form of an array, and an "arrayspace" consisting of a hierarchy of indices, with each index being used to define another function with different indices in a lower "arrayspace". This is an interesting concept I might return to, but my implementation ended up extremely weak.
Eventually I came up with this. It's nothing too new or interesting, and it doesn't grow particularly fast, but it's defined to make things convenient.
\[\circledast [\&Z](X)\]
The most basic rule is:
\[\circledast [Z](X) = ZX\]
Thâ€¦
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This is my first serious attempt at googology. I hope I didn't screw up too badly ^.^
The Wfunction takes the following form:
\[W(X_n,X_{n1},...,X_2,X_1)\]
Rule 1. If there is only one entry:
\[W(X) = 2 ^ X\]
Rule 2. If there are two or more entries, with no leading zeroes:
\[W(X_n,X_{n1},...,X_2,X_1) = W(X_n1,\underbrace{W(X_n1,X_{n1},...,X_1,X_0),...,W(X_n1,X_{n1},...,X_2,X_1)}_{n1})\]
This is less confusing when looking at the 2entry instance:
\[W(X_2,X_1) = W(X_21,W(X_21,X_1))\]
Rule 3. Leading zeroes are deleted if they exist:
\[W(0, X_n,X_{n1},...,X_1) = W(X_n,X_{n1},...,X_1)\]
Let
\[WW(X) = W(\underbrace{X,X,X...,X,X,X}_X)\]
\[WW^n(X) = \underbrace{WW(WW(WW...(WW(X))...))}_n\]
Let
\[WW[1]Y(X) = WW^{WW(X)}(X)\]
\[WW[n]Y(X) = WW[n1]Y^{â€¦
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