BEAF

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BEAF, or Bowers Exploding Array function, is an extremely fast-growing function.

Defination

The base, or b, is the first entry in the array.

The prime, or p, is the second entry in the array.

The pilot is the first non-1 entry after p.

The copilot is the entry before the pilot. It is sometimes p.

A previous entry is entries before the pilot, but is on the same row as the pilot; a previous row is rows before the pilot, but is on the same plane as the pilot; and so on.

1. Base rule: If any entry that exist after p is 1, then the value of the array (called \(v(A)\)) is \(b^p\).

2. Prime rule: If p is 1, then \(v(A)=1\).

3. Catastrophic rule: If Rules 1-2 don't apply, then pilot decreases by 1, copilot becomes the original array but with p decreased by 1, and every previous structure \(X^n\) is filled with a \(\underbrace{b \times b \cdots \times b}_n\) hypercube.

Examples

\begin{eqnarray*} \{3,3,3,3\} &=& \{3,\{3,2,3,3\},2,3\} \\ &=& \{3,\{3,\{3,1,3,3\},2,3\},2,3\} \\ &=& \{3,\{3,3,2,3\},2,3\} \\ &=& \{3,\{3,\{3,2,2,3\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,\{3,1,2,3\},1,3\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,1,3\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,2,1,3\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,3,\{3,1,1,3\},2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,3,3,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,1,2\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,2,1,2\}\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,3,\{3,1,1,2\}\}\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,3,3\}\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,3\uparrow^{3\uparrow\uparrow\uparrow3}3,1,2\},2,2\},2\},1,3\},2,3\} \end{eqnarray*}


\begin{eqnarray*} \{3,3(1)3,3\} &=& \{3,3,3(1)2,3\} \\ &=& \{3,\{3,2,3(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,3,2(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,2,2(1)2,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,3(1)2,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,3,3(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,2,3(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,3,2(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,2,2(1)1,3\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3(1)1,3\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,2(1)1,3\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,3(1)3,2\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,3,3(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,2,3(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,3,2(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,2,2(1)2\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,3(1)2\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,3,3\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,3\uparrow\uparrow\uparrow3(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{\underbrace{3,\cdots,3}_{3\uparrow\uparrow\uparrow3}\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \end{eqnarray*}

Hyperdimentional arrays

\(\{3,3(0,1)2\}=???\)

What does (0,1) even mean? It is a \(X^X\) structure, or \(X^p\). So we can replace (0,1) with (p), or (3). The process is something like diagonalization, but it isn't.

\begin{eqnarray*} \{3,3(0,1)2\} &=& \{3,3(3)2\} \\ &=& \{3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3\} \end{eqnarray*}

You get the idea how big this number is.