FANDOM


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.

\[\circledast [\&Z](X)\]

One-entry Level

The most basic rule is:

\[\circledast [Z](X) = ZX\]

This is the only rule necessary for a one-entry Z-array.

A useful feature of PRBN as opposed to W-notation is that things will always work out so that X and functions thereof can be placed in the index and its meaning will still be easy to write down in terms of X.

\[\circledast [1](X) = X\]

\[\circledast [2](X) = 2X\]

\[\circledast [4](X) = 4X\]

\[\circledast [a](X) = aX\]

\[\circledast [X](X) = X^2\]

\[\circledast [X+5](X) = X^2+5X\]

\[\circledast [2X](X) = 2X^2\]

\[\circledast [X^2](X) = X^3\]

\[\circledast [X^3](X) = X^4\]

Obviously, we're hitting a limit pretty fast, so we need to introduce the second entry.

Two-entry Level

It is worth noting that: The rules for two-entry arrays are:

\[\circledast [\&Z,1](X) = X\]

\[\circledast [Z_2,Z_1](X) = \circledast [Z_2-1,\circledast [Z_2,Z_1-1](X)](X)\]

\[\circledast [0,\&Z](X) = \circledast [\&Z](X)\]

Therefore:

\[\circledast [1,1](X) = X\]

\[\circledast [1,2](X) = X^2\]

\[\circledast [1,3](X) = X^3\]

\[\circledast [1,a](X) = X^a\]

\[\circledast [1,X](X) = {}^2X\]

\[\circledast [2,1](X) = X\]

\[\circledast [2,2](X) = {}^2X\]

\[\circledast [2,3](X) = {}^3X\]

\[\circledast [2,a](X) = {}^aX\]

\[\circledast [2,X](X) = X\uparrow\uparrow\uparrow2\]

\[\circledast [3,2](X) = X\uparrow\uparrow\uparrow2\]

\[\circledast [3,3](X) = X\uparrow\uparrow\uparrow3\]

\[\circledast [3,X](X) = X\uparrow\uparrow\uparrow\uparrow2\]

\[\circledast [a,2](X) = X\{a\}2\]

\[\circledast [a,b](X) = X\{a\}b\]

\[\circledast [a,X](X) = X\{a\}X\]

\[\circledast [X,a](X) = X\{X\}a\]

\[\circledast [X,X](X) = X\{X\}X\]

\[\circledast [X\{X\}X,X](X) = X\{X\{X\}X\}X\]

\[\circledast [X\{X\{X\}X\}X,X](X) = X\{X\{X\{X\}X\}X\}X\]

At this point, we are approaching another limit.

Multiple-entry level

\[\circledast [\&Z_2,1,\&Z_1](X) = X\]

\[\circledast [Z_n,Z_{n-1},Z_{n-2},...,Z_2,Z_1](X) = \circledast [Z_n-1,[\circledast [Z_n,Z_{n-1}-1,Z_{n-2},...,Z_2,Z_1](X),Z_{n-2},...,Z_2,Z_1](X)\]

Summary of rules:

1. Ending Rule: If there's a single entry, multiply it by X.

2. Termination Rule: If there's a 0 at the beginning of the array, delete it.

3. Default Rule: If there's a 1 in the array anywhere except the beginning, the result is just X.

4. Iterating Rule: If there are multiple entries, the first entry is greater than 0 and all others are greater than 1, then reduce the first entry by 1 and replace the second entry with the entire function with the second entry reduced by 1.

Therefore:

\[\circledast [1,1,1](X) = X\]

\[\circledast [1,1,2](X) = X\]

\[\circledast [1,2,2](X) = X\{X\}2\]

\[\circledast [1,2,a](X) = X\{X\}a\]

\[\circledast [1,2,X](X) = X\{X\}X\]

\[\circledast [1,3,X](X) = X\{X\{X\}X\}X\]

\[\circledast [1,a,X](X) = X\{\{1\}\}a\]

\[\circledast [1,X,X](X) = X\{\{1\}\}X\]

\[\circledast [2,2,X](X) = X\{\{1\}\}X\]

\[\circledast [2,3,X](X) = X\{\{1\}\}X\{\{1\}\}X\]

\[\circledast [2,X,X](X) = X\{\{2\}\}X\]

\[\circledast [3,X,X](X) = X\{\{3\}\}X\]

\[\circledast [a,X,X](X) = X\{\{a\}\}X\]

\[\circledast [X,X,X](X) = X\{\{X\}\}X\]

\[\circledast [1,2,X,X](X) = X\{\{X\}\}X\]

\[\circledast [1,3,X,X](X) = X\{\{X\{\{X\}\}X\}\}X\]

\[\circledast [1,4,X,X](X) = X\{\{X\{\{X\{\{X\}\}X\}\}X\}\}X\]

\[\circledast [1,a,X,X](X) = X\{\{\{1\}\}\}a\]

\[\circledast [1,X,X,X](X) = X\{\{\{1\}\}\}X\]

\[\circledast [2,3,X,X](X) = X\{\{\{1\}\}\}X\{\{\{1\}\}\}X\]

\[\circledast [2,X,X,X](X) = X\{\{\{2\}\}\}X\]

\[\circledast [3,X,X,X](X) = X\{\{\{3\}\}\}X\]

\[\circledast [a,X,X,X](X) = X\{\{\{a\}\}\}X\]

\[\circledast [X,X,X,X](X) = X\{\{\{X\}\}\}X\]

The limit growth of this I conjecture to be:

\[f_{\omega^2}(X)\]

Extension 1: Subblocks

A subblock is a shorthand entity indicating a string of Xs.

\[\circledast [1,[3]](X) = X\{\{\{1\}\}\}X\]

\[\circledast [a,[3]](X) = X\{\{\{a\}\}\}X\]

\[\circledast [[4]](X) = X\{\{\{X\}\}\}X\]

\[\circledast [1,[a]](X) = X\underbrace{\{\{...\{\{}_a1\}\}...\}\}X\]

\[\circledast [1,[X]](X) = X\underbrace{\{\{...\{\{}_X1\}\}...\}\}X\]

\[\circledast [X,[X]](X) = X\underbrace{\{\{...\{\{}_XX\}\}...\}\}X\]

\[\circledast [X,[\circledast [X,[X]](X)]](X) = X\underbrace{\{\{...\{\{}_{\circledast [X,[X]](X)}X\}\}...\}\}X\]

Extension 2: Array-Subblocks

Similar rules to the main array also apply to subblocks:

5. Breakout Rule: If there's a 1 in a subblock anywhere except the beginning, the answer to the entire expression is X.

6. Terminating Subblock Rule: If there's a 0 at the beginning of a subblock, it is deleted.

7. Ending Subblock Rule: If a subblock contains a single entry, replace it with that many entries all equal to X in the main array.

8. Iterating Subblock Rule: Otherwise, reduce the subblock's first entry by 1 and replace the second with the entire expression with the subblock's second entry reduced by 1.

\[\circledast [X,[1,1]](X) = X\]

\[\circledast [X,[1,2]](X) = X\underbrace{\{\{...\{\{}_{\circledast [X,[X]](X)}X\}\}...\}\}X\]

\[\circledast [X,[1,3]](X) = X{^\{_\{}1{^\}_\}}3\]

\[\circledast [X,[1,a]](X) = X{^\{_\{}1{^\}_\}}a\]

\[\circledast [X,[1,X]](X) = X{^\{_\{}1{^\}_\}}X\]

And we've finally reached megotion!

\[\circledast [X,[2,2]](X) = X{^\{_\{}1{^\}_\}}X\]

\[\circledast [X,[2,3]](X) = X{^\{_\{}1{^\}_\}}X{^\{_\{}1{^\}_\}}X\]

\[\circledast [X,[2,X]](X) = X{^\{_\{}2{^\}_\}}X\]

\[\circledast [X,[3,X]](X) = X{^\{_\{}3{^\}_\}}X\]

\[\circledast [X,[a,X]](X) = X{^\{_\{}a{^\}_\}}X\]

\[\circledast [X,[X,X]](X) = X{^\{_\{}X{^\}_\}}X\]

\[\circledast [X,[1,2,X]](X) = X^\{_\{X^\}_\}X = \{X,X,X,1,2\}\]

\[\circledast [X,[1,3,X]](X) = X^\{_\{X^\{_\{X^\}_\}X^\}_\}X\]

\[\circledast [X,[1,X,X]](X) = X{^\{_\{}{^\{_\{}1{^\}_\}}{^\}_\}}X = \{X,X,1,2,2\}\]

\[\circledast [X,[2,2,X]](X) = X{^\{_\{}{^\{_\{}1{^\}_\}}{^\}_\}}X\]

\[\circledast [X,[2,3,X]](X) = X{^\{_\{}{^\{_\{}1{^\}_\}}{^\}_\}}X{^\{_\{}{^\{_\{}1{^\}_\}}{^\}_\}}X\]

\[\circledast [X,[2,X,X]](X) = X{^\{_\{}{^\{_\{}2{^\}_\}}{^\}_\}}X = \{X,X,2,2,2\}\]

\[\circledast [X,[3,X,X]](X) = X{^\{_\{}{^\{_\{}3{^\}_\}}{^\}_\}}X = \{X,X,3,2,2\}\]

\[\circledast [X,[a,X,X]](X) = X{^\{_\{}{^\{_\{}a{^\}_\}}{^\}_\}}X = \{X,X,a,2,2\}\]

\[\circledast [X,[X,X,X]](X) = X{^\{_\{}{^\{_\{}X{^\}_\}}{^\}_\}}X = \{X,X,X,2,2\}\]

\[\circledast [X,[X,X,X,X]](X) = X{^\{_\{}{^\{_\{}X{^\}_\}}{^\}_\}}X = \{X,X,X,3,2\}\]

\[\circledast [X,[X,[X]]](X) = \{X,X,X,X,2\}\]

\[\circledast [X,[X,[X,[X]]]](X) = \{X,X,X,X,3\}\]

\[\circledast [X,[X,[X,[X,[X]]]]](X) = \{X,X,X,X,4\}\]

Extension 3: Secondary Arrays

Let the secondary array determine the number of nested subblocks in the next secondary array down, in such a way that it lines up with the above.

9. Ending Secondary Array Rule I:

\[\circledast [0]_2[\&Z_1](X) = \circledast [Z_1](X)\]

10. Ending Secondary Array Rule II:

\[\circledast [Z_2]_2[\&Z_1](X) = \circledast [Z_2-1]_2[X,[Z_1]](X)\]

11. Breakout Secondary Array Rule:

\[\circledast [\&Z_{2a},1,\&Z_{2b}]_2[\&Z_1](X) = X\]

12. Iterating Secondary Array Rule: Otherwise, reduce the leading secondary array's first entry by 1 and replace the second with the entire expression with the leading secondary array's second entry reduced by 1.

\[\circledast [5]_2[X](X) = \{X,X,X,X,5\}\]

\[\circledast [6]_2[X](X) = \{X,X,X,X,6\}\]

\[\circledast [a]_2[X](X) = \{X,X,X,X,a\}\]

\[\circledast [X]_2[X](X) = \{X,X,X,X,X\}\]

\[\circledast [1,2]_2[X](X) = \{X,2,1,1,1,2\}\]

\[\circledast [1,a]_2[X](X) = \{X,2,1,1,1,2\}\]

\[\circledast [1,X]_2[X](X) = \{X,X,1,1,1,2\}\]

\[\circledast [2,2]_2[X](X) = \{X,X,1,1,1,2\}\]

\[\circledast [2,X]_2[X](X) = \{X,X,2,1,1,2\}\]

\[\circledast [3,X]_2[X](X) = \{X,X,3,1,1,2\}\]

\[\circledast [4,X]_2[X](X) = \{X,X,4,1,1,2\}\]

\[\circledast [a,X]_2[X](X) = \{X,X,a,1,1,2\}\]

\[\circledast [1,2,X]_2[X](X) = \{X,X,X,1,1,2\}\]

\[\circledast [1,X,X]_2[X](X) = \{X,X,1,2,1,2\}\]

\[\circledast [2,X,X]_2[X](X) = \{X,X,2,2,1,2\}\]

\[\circledast [a,X,X]_2[X](X) = \{X,X,a,2,1,2\}\]

\[\circledast [X,X,X]_2[X](X) = \{X,X,X,2,1,2\}\]

\[\circledast [X,X,X]_2[X](X) = \{X,X,X,2,1,2\}\]

\[\circledast [X,X,X,X]_2[X](X) = \{X,X,X,3,1,2\}\]

\[\circledast [X,X,X,X,X]_2[X](X) = \{X,X,X,4,1,2\}\]

\[\circledast [X,[X]]_2[X](X) = \{X,X,X,X,1,2\}\]

\[\circledast [X,[1,X]]_2[X](X) = \{X,X,1,1,2,2\}\]

\[\circledast [X,[2,X]]_2[X](X) = \{X,X,2,1,2,2\}\]

\[\circledast [X,[a,X]]_2[X](X) = \{X,X,a,1,2,2\}\]

\[\circledast [X,[X,X]]_2[X](X) = \{X,X,X,1,2,2\}\]

\[\circledast [X,[1,X,X]]_2[X](X) = \{X,X,1,2,2,2\}\]

\[\circledast [X,[2,X,X]]_2[X](X) = \{X,X,2,2,2,2\}\]

\[\circledast [X,[a,X,X]]_2[X](X) = \{X,X,a,2,2,2\}\]

\[\circledast [X,[X,X,X]]_2[X](X) = \{X,X,X,2,2,2\}\]

\[\circledast [X,[X,X,X]]_2[X](X) = \{X,X,X,2,2,2\}\]

\[\circledast [X,[X,X,X,X]]_2[X](X) = \{X,X,X,3,2,2\}\]

\[\circledast [X,[X,[X]]]_2[X](X) = \{X,X,X,X,2,2\}\]

\[\circledast [X,[X,[1,X]]]_2[X](X) = \{X,X,1,1,3,2\}\]

\[\circledast [X,[X,[2,X]]]_2[X](X) = \{X,X,2,1,3,2\}\]

\[\circledast [X,[X,[a,X]]]_2[X](X) = \{X,X,a,1,3,2\}\]

\[\circledast [X,[X,[X,X]]]_2[X](X) = \{X,X,X,1,3,2\}\]

\[\circledast [X,[X,[1,X,X]]]_2[X](X) = \{X,X,1,2,3,2\}\]

\[\circledast [X,[X,[X,X,X]]]_2[X](X) = \{X,X,X,2,3,2\}\]

\[\circledast [X,[X,[1,X,X,X]]]_2[X](X) = \{X,X,1,3,3,2\}\]

\[\circledast [X,[X,[X,X,X,X]]]_2[X](X) = \{X,X,X,3,3,2\}\]

\[\circledast [X,[X,[X,[X]]]_2[X](X) = \{X,X,X,X,3,2\}\]

\[\circledast [X,[X,[X,[1,X]]]_2[X](X) = \{X,X,1,1,4,2\}\]

\[\circledast [3]_2[X]_2[X](X) = \{X,X,X,X,3,2\}\]

\[\circledast [4]_2[X]_2[X](X) = \{X,X,X,X,4,2\}\]

\[\circledast [a]_2[X]_2[X](X) = \{X,X,X,X,a,2\}\]

\[\circledast [X]_2[X]_2[X](X) = \{X,X,X,X,X,2\}\]

\[\circledast [1,X]_2[X]_2[X](X) = \{X,X,1,1,1,3\}\]

\[\circledast [2,X]_2[X]_2[X](X) = \{X,X,2,1,1,3\}\]

\[\circledast [X,X]_2[X]_2[X](X) = \{X,X,X,1,1,3\}\]

\[\circledast [1,X,X]_2[X]_2[X](X) = \{X,X,1,2,1,3\}\]

\[\circledast [X,X,X]_2[X]_2[X](X) = \{X,X,X,2,1,3\}\]

\[\circledast [X,X,X,X]_2[X]_2[X](X) = \{X,X,X,3,1,3\}\]

\[\circledast [X,[X]]_2[X]_2[X](X) = \{X,X,X,X,1,3\}\]

\[\circledast [X,[1,X]]_2[X]_2[X](X) = \{X,X,1,1,2,3\}\]

\[\circledast [X,[2,X]]_2[X]_2[X](X) = \{X,X,2,1,2,3\}\]

\[\circledast [X,[X,X]]_2[X]_2[X](X) = \{X,X,X,1,2,3\}\]

\[\circledast [X,[1,X,X]]_2[X]_2[X](X) = \{X,X,1,2,2,3\}\]

\[\circledast [X,[X,X,X]]_2[X]_2[X](X) = \{X,X,X,2,2,3\}\]

\[\circledast [X,[X,X,X,X]]_2[X]_2[X](X) = \{X,X,X,3,2,3\}\]

\[\circledast [X,[X,[X]]]_2[X]_2[X](X) = \{X,X,X,X,2,3\}\]

\[\circledast [X,[X,[1,X]]]_2[X]_2[X](X) = \{X,X,1,1,3,3\}\]

\[\circledast [X,[X,[X,X]]]_2[X]_2[X](X) = \{X,X,X,1,3,3\}\]

\[\circledast [X,[X,[X,X,X]]]_2[X]_2[X](X) = \{X,X,X,2,3,3\}\]

\[\circledast [X,[X,[X,X,X,X]]]_2[X]_2[X](X) = \{X,X,X,3,3,3\}\]

\[\circledast [X,[X,[X,[X]]]]_2[X]_2[X](X) = \{X,X,X,X,3,3\}\]

\[\circledast [4]_2[X]_2[X]_2[X](X) = \{X,X,X,X,4,3\}\]

\[\circledast [5]_2[X]_2[X]_2[X](X) = \{X,X,X,X,5,3\}\]

\[\circledast [a]_2[X]_2[X]_2[X](X) = \{X,X,X,X,a,3\}\]

\[\circledast [X]_2[X]_2[X]_2[X](X) = \{X,X,X,X,X,3\}\]

\[\circledast [X]_2[X]_2[X]_2[X]_2[X](X) = \{X,X,X,X,X,4\}\]

Extension 4: Tertiary, Quaternary, and Higher-Level Arrays

To reach tetratiational-level, we need to define even higher kinds of arrays.

13. Ending Higher-Level Array Rule I:

\[\circledast [0]_Y[\&Z_1](X) = \circledast [\&Z_1](X)\]

14. Ending Higher-Level Array Rule II:

\[\circledast [Z_Y]_Y[\&Z_1](X) = \circledast [Z_Y-1]_Y[X]_{Y-1}[\&Z_1](X)\]

15. Breakout Higher-Level Array Rule:

\[\circledast [\&Z_{Ya},1,\&Z_{Yb}]_Y[\&Z_1](X) = X\]

16. Iterating Higher-Level Array Rule: Otherwise, reduce the leading array's first entry by 1 and replace the second with the entire expression with the leading array's second entry reduced by 1.

\[\circledast [4]_3[X](X) = \{X,X,X,X,X,4\}\]

\[\circledast [5]_3[X](X) = \{X,X,X,X,X,5\}\]

\[\circledast [a]_3[X](X) = \{X,X,X,X,X,a\}\]

\[\circledast [X]_3[X](X) = \{X,X,X,X,X,X\}\]

\[\circledast [1,2]_3[X](X) = \{X,2,1,1,1,1,2\}\]

\[\circledast [1,a]_3[X](X) = \{X,a,1,1,1,1,2\}\]

\[\circledast [1,X]_3[X](X) = \{X,X,1,1,1,1,2\}\]

\[\circledast [a,X]_3[X](X) = \{X,X,a,1,1,1,2\}\]

\[\circledast [X,X]_3[X](X) = \{X,X,X,1,1,1,2\}\]

\[\circledast [1,X,X]_3[X](X) = \{X,X,1,2,1,1,2\}\]

\[\circledast [X,X,X]_3[X](X) = \{X,X,X,2,1,1,2\}\]

\[\circledast [X,X,X,X]_3[X](X) = \{X,X,X,3,1,1,2\}\]

\[\circledast [X,[X]]_3[X](X) = \{X,X,X,X,1,1,2\}\]

\[\circledast [X,[1,X]]_3[X](X) = \{X,X,1,1,2,1,2\}\]

\[\circledast [X,[X,X]]_3[X](X) = \{X,X,X,1,2,1,2\}\]

\[\circledast [X,[1,X,X]]_3[X](X) = \{X,X,1,2,2,1,2\}\]

\[\circledast [X,[X,X,X]]_3[X](X) = \{X,X,X,2,2,1,2\}\]

\[\circledast [X,[X,[X]]]_3[X](X) = \{X,X,X,X,2,1,2\}\]

\[\circledast [X]_2[X]_3[X](X) = \{X,X,X,X,X,1,2\}\]

\[\circledast [1,X]_2[X]_3[X](X) = \{X,X,1,1,1,2,2\}\]

\[\circledast [X,X]_2[X]_3[X](X) = \{X,X,X,1,1,2,2\}\]

\[\circledast [X,X,X]_2[X]_3[X](X) = \{X,X,X,2,1,2,2\}\]

\[\circledast [X,[X]]_2[X]_3[X](X) = \{X,X,X,X,1,2,2\}\]

\[\circledast [X,[1,X]]_2[X]_3[X](X) = \{X,X,1,1,2,2,2\}\]

\[\circledast [X,[X,X]]_2[X]_3[X](X) = \{X,X,X,1,2,2,2\}\]

\[\circledast [X,[1,X,X]]_2[X]_3[X](X) = \{X,X,1,2,2,2,2\}\]

\[\circledast [X,[X,X,X]]_2[X]_3[X](X) = \{X,X,X,2,2,2,2\}\]

\[\circledast [X,[X,[X]]]_2[X]_3[X](X) = \{X,X,X,X,2,2,2\}\]

\[\circledast [3]_2[X]_2[X]_3[X](X) = \{X,X,X,X,3,2,2\}\]

\[\circledast [X]_2[X]_2[X]_3[X](X) = \{X,X,X,X,X,2,2\}\]

\[\circledast [3]_3[X]_3[X](X) = \{X,X,X,X,X,3,2\}\]

\[\circledast [X]_3[X]_3[X](X) = \{X,X,X,X,X,X,2\}\]

\[\circledast [1,X]_3[X]_3[X](X) = \{X,X,1,1,1,1,3\}\]

\[\circledast [2,X]_3[X]_3[X](X) = \{X,X,2,1,1,1,3\}\]

\[\circledast [X,X]_3[X]_3[X](X) = \{X,X,X,1,1,1,3\}\]

\[\circledast [1,X,X]_3[X]_3[X](X) = \{X,X,1,2,1,1,3\}\]

\[\circledast [X,X,X]_3[X]_3[X](X) = \{X,X,X,2,1,1,3\}\]

\[\circledast [X,[X]]_3[X]_3[X](X) = \{X,X,X,X,1,1,3\}\]

\[\circledast [X,[1,X]]_3[X]_3[X](X) = \{X,X,1,1,2,1,3\}\]

\[\circledast [X,[X,X]]_3[X]_3[X](X) = \{X,X,X,1,2,1,3\}\]

\[\circledast [X,[1,X,X]]_3[X]_3[X](X) = \{X,X,1,2,2,1,3\}\]

\[\circledast [X,[X,X,X]]_3[X]_3[X](X) = \{X,X,X,2,2,1,3\}\]

\[\circledast [2]_2[X]_3[X]_3[X](X) = \{X,X,X,X,2,1,3\}\]

\[\circledast [X]_2[X]_3[X]_3[X](X) = \{X,X,X,X,X,1,3\}\]

\[\circledast [1,X]_2[X]_3[X]_3[X](X) = \{X,X,1,1,1,2,3\}\]

\[\circledast [X,X]_2[X]_3[X]_3[X](X) = \{X,X,X,1,1,2,3\}\]

\[\circledast [1,X,X]_2[X]_3[X]_3[X](X) = \{X,X,1,2,1,2,3\}\]

\[\circledast [X,X,X]_2[X]_3[X]_3[X](X) = \{X,X,X,2,1,2,3\}\]

\[\circledast [X,[X]]_2[X]_3[X]_3[X](X) = \{X,X,X,X,1,2,3\}\]

\[\circledast [X,[1,X]]_2[X]_3[X]_3[X](X) = \{X,X,1,1,2,2,3\}\]

\[\circledast [X,[X,X]]_2[X]_3[X]_3[X](X) = \{X,X,X,1,2,2,3\}\]

\[\circledast [X,[X,X,X]]_2[X]_3[X]_3[X](X) = \{X,X,X,2,2,2,3\}\]

\[\circledast [2]_2[X]_2[X]_3[X]_3[X](X) = \{X,X,X,X,2,2,3\}\]

\[\circledast [X]_2[X]_2[X]_3[X]_3[X](X) = \{X,X,X,X,X,2,3\}\]

\[\circledast [3]_3[X]_3[X]_3[X](X) = \{X,X,X,X,X,3,3\}\]

\[\circledast [X]_3[X]_3[X]_3[X](X) = \{X,X,X,X,X,X,3\}\]

\[\circledast [4]_4[X](X) = \{X,X,X,X,X,X,4\}\]

\[\circledast [5]_4[X](X) = \{X,X,X,X,X,X,5\}\]

\[\circledast [X]_4[X](X) = \{X,X,X,X,X,X,X\}\]

\[\circledast [X]_4[X](X) = \{X,X,X,X,X,X,X\}\]

Extension 5: Filling In the Gap

It will be useful in future to define order-1 arrays:

17. Primary Array Rule:

\[\circledast [\&Z_2]_1[\&Z_1](X) = \circledast [\&Z_1,[\&Z_2]](X)\]

This offers an alternative notation and is necessary in order for

\[A\circledcirc 1\]

to be defined.

This extra form will become useful later.

Extension 6

Why not make the level of the array itself an array?

18. Breakout Array-Level Array Rule:

\[\circledast [\&Z_2]_{[\&Z_{3a},1,\&Z_{3b}]}[\&Z_1](X) = X\]

19. Iterating Array-Level Array Rule: Otherwise, reduce the highest-order level-array's first entry by 1 and replace the second with the entire expression with the highest-order level-array's second entry reduced by 1.

We don't need any ending rules because the single-entry case is already covered by Extensions 3 through 5.

Examples coming soon!

Extension 7: Pteriforever's Considerably Less Boring Notation

I've finally come up with an idea to truly take this to the next level! A new notation arises where entities called structural elements are used as a shorthand for the structures used in ordinary PRBN, and can then be easily extended to cover things that go far beyond it

Curly brackets:

\[\{\}\]

are now used to denote a "structural element". The entries of structural elements can be nonstructural elements, other structural elements, or single numbers (which could be considered both at the same time). Most of the rest of this section will explain how they work.

Ordinary brackets:

\[()\]

are also needed to denote nonstructural elements in order to differentiate them from structural elements. These don't represent structures in any abstract way; they /are/ exactly PRBN structures. For instance, ([1,X,X,X]) is simply the array 1,X,X,X, While {1,X,X,X} means something else entirely.

One important exception is when a single number exists: A = ([A]) = {A}. These "trivial" elements can simply be turned into single-number entries in the next-highest level of nesting. From now on, the former won't be called an element at all, while the others will be called trivial elements.

If the index, at the highest level, has only the ordinary square brackets, it is assumed to be a nonstructural element. PRBN contains only nonstructural elements, so all the notation of PRBN can remain the same. Nonstructural elements can't directly contain other elements; they can contain functions containing other elements, however.

Square brackets immediately within nonstructural elements aren't subblocks; they're the normal brackets which encase the pieces of the index in PRBN. Only square brackets within other square brackets within nonstructural elements denote subblocks. Nonstructural elements within

Structural elements with three or fewer entires can be converted directly into non-structural elements in a fairly straightforward manner:

\[\{A\} \rightarrow ([A])\]

\[\{B,A\} \rightarrow ([A,B])\]

\[\{C,B,A\} \rightarrow ([A]_C[B])\]

Therefore:

\[\circledast[\{A\}](X) = \circledast[A](X)\]

\[\circledast[\{B,A\}](X) = \circledast[A,B](X)\]

\[\circledast[\{C,B,A\}](X) = \circledast[A]_C[B](X)\]

The rules for structural elements are remarkably similar to those for PRBN, but the important distinction is that the function itself isn't placed back into the index while processing structural elements; only other structural elements are. The function itself /is/ placed back in when nonstructural elements are processed, however. For simplicity, structural elements will be converted into nonstructural elements when they reach a specific form, as this reduces the number of rules necessary.

20. Order Rule: The first element in the expression which contains no other elements is called the "active element" and is always processed first, regardless of what kind it is. The exception is when a 2-entry structural element contains nonstructural elements; the structural element is active rather than the first nonstructural one.

21. Complete Ending Rule: If a 0 occurs as the first entry of any kind of element, it is deleted. Zeroes occurring in other places in nonstructural elements are not well defined. If they appear anywhere else in a structural element, the element is replaced with the trivial element {X}.

22. Complete Breakout Rule: If a 1 occurs anywhere except the first entry in any array of a nonstructural element, the element is replaced with the trivial element ([1]).

(Note:

\[\circledast[1](X) = X\]

which is why rule 22 is equivalent to all other forms of the breakout rule)

23. Trivial Element Rule: If the active element is trivial, it is reduced to a single number.

24a. Nonstructural Delegation Rule: If the active element is nonstructural, it is processed by the rules of PRBN.

(Note: This is kind of a copout, so rules 24b and 24c should be used instead, as the way structural elements are converted to nonstructural elements means a simplified ruleset for nonstructural elements will be sufficient)

24b. Simplified Nonstructural Rule I: If the active nonstructural element contains no subblocks, the first entry is reduced by 1 and the second entry is replaced by the entire function with the second entry of that nonstructural element reduced by 1.

24c. Simplified Nonstructural Rule II: If the active nonstructural element contains a subblock, it is replaced by a number of entries all equal to X, equal to the number in the subblock.

(Note: Subblocks will only contain a single entry; a structural element representing a nonstructural element containing a subblock with an array in it would be 3-entry and contain a smaller nonstructural element, and so could never be active according to rule 20)

25. If the active element is structural and 4+-entry, or 3-entry if the first entry is greater than 1, the first entry is reduced by 1 and the second entry is replaced with the structural element itself with the second entry reduced by 1 and the third entry replaced by X.

(Note: Or, more neatly:

\[\circledast[\{C,B,A\}](X) = \circledast[\{C-1,\{C,B-1,X\},A\}](X)\]

which is actually exactly the same as rule 14.)

26. If the active element is structural and contains two entries, the entries are reversed and concatenated and the element becomes nonstructural.

\[\{([\&B]),([\&A])\} \rightarrow ([\&A,\&B])\]

\[\circledast[\{([\&B]),([\&A])\}] = \circledast[\&A,\&B]\]

27. If the active element is structural and 3-entry and the first entry is 1, it is replaced with a nonstructural element consisting of the second entry followed by a subblock of the first entry.

\[\{1,B,A\} \rightarrow ([A,[B]])\]

\[\circledast[\{1,B,A\}](X) = \circledast[A,[B]]\]

These rules make up the first stage of PCLBN, and can successfully duplicate anything expressible in PRBN using only structural elements with four or fewer entries.

\[\circledast[\{X,X,X\}](X) = X \circledcirc X\]

\[\circledast[\{1,2,3,4\}](X) = \circledast[3]_{[X]_{4,X}[4]}[4]\]


Results

\[\circledast[100,100](100) = \mbox{Yareox}\]

\[\circledast[1,100,100](100) = \mbox{Grand Yareox}\]

\[\circledast[2,100,100](100) = \mbox{Grand Grand Yareox}\]

\[\circledast[100,100,100](100) = \mbox{Axstox}\]

\[\circledast[1,100,100,100](100) = \mbox{Grand Axstox}\]

\[\circledast[100,100,100,100](100) = \mbox{Acaciox}\]

\[\circledast[100,[4]](100) = \mbox{Vvyngox}\]

\[\circledast[100,[5]](100) = \mbox{Alteraox}\]

\[\circledast[100,[6]](100) = \mbox{Celeriox}\]

\[\circledast[100,[7]](100) = \mbox{Trancox}\]

\[\circledast[100,[8]](100) = \mbox{Chaozox}\]

\[\circledast[100,[9]](100) = \mbox{Eleveriox}\]

\[\circledast[100,[10]](100) = \mbox{Ciraiox}\]

\[\circledast[100,[100]](100) = \mbox{Azerdus}\]

\[\circledast[100,[1,100]](100) = \mbox{Grand Azerdus}\]

\[\circledast[100,[2,100]](100) = \mbox{Grand Grand Azerdus}\]

\[\circledast[100,[100,100]](100) = \mbox{Azeryaredus}\]

\[\circledast[100,[100,100,100]](100) = \mbox{Azeraxsdus}\]

\[\circledast[100,[100,100,100,100]](100) = \mbox{Azeracacidus}\]

\[\circledast[100,[100,[4]]](100) = \mbox{Azervvyndus}\]

\[\circledast[100,[100,[5]]](100) = \mbox{Azeralterdus}\]

\[\circledast[100,[100,[6]]](100) = \mbox{Azerceleridus}\]

\[\circledast[100,[100,[7]]](100) = \mbox{Azertrandus}\]

\[\circledast[100,[100,[8]]](100) = \mbox{Azerchaodus}\]

\[\circledast[100,[100,[9]]](100) = \mbox{Azereleveridus}\]

\[\circledast[100,[100,[10]]](100) = \mbox{Azerciraidus}\]

\[\circledast[100,[100,[100]]](100) = \mbox{Drazerdus}\]

\[\circledast[100,[100,[1,100]]](100) = \mbox{Grand Drazerdus}\]

\[\circledast[100,[100,[100,100]]](100) = \mbox{Drazeryaredus}\]

\[\circledast[100,[100,[100,100,100]]](100) = \mbox{Drazeraxsdus}\]

\[\circledast[100,[100,[100,100,100,100]]](100) = \mbox{Drazeracacidus}\]

\[\circledast[100,[100,[100,[4]]]](100) = \mbox{Drazervvyndus}\]

\[\circledast[100,[100,[100,[5]]]](100) = \mbox{Drazeralterdus}\]

\[\circledast[100,[100,[100,[6]]]](100) = \mbox{Drazerceleridus}\]

\[\circledast[100,[100,[100,[7]]]](100) = \mbox{Drazertrandus}\]

\[\circledast[100,[100,[100,[8]]]](100) = \mbox{Drazerchaodus}\]

\[\circledast[100,[100,[100,[9]]]](100) = \mbox{Drazereleveridus}\]

\[\circledast[100,[100,[100,[10]]]](100) = \mbox{Drazerciraidus}\]

\[\circledast[100,[100,[100,[100]]]](100) = \mbox{Trazerdus}\]

\[\circledast[4]_2[100](100) = \mbox{Tetrazerdus}\]

\[\circledast[5]_2[100](100) = \mbox{Pentazerdus}\]

\[\circledast[6]_2[100](100) = \mbox{Hexazerdus}\]

\[\circledast[7]_2[100](100) = \mbox{Heptazerdus}\]

\[\circledast[8]_2[100](100) = \mbox{Octazerdus}\]

\[\circledast[9]_2[100](100) = \mbox{Ennazerdus}\]

\[\circledast[10]_2[100](100) = \mbox{Decazerdus}\]

\[\circledast[100]_2[100](100) = \mbox{Aleaxyss}\]

\[\circledast[1,100]_2[100](100) = \mbox{Grand Aleaxyss}\]

\[\circledast[2,100]_2[100](100) = \mbox{Grand Grand Aleaxyss}\]

\[\circledast[100,100]_2[100](100) = \mbox{Aleaxyaryss}\]

\[\circledast[100,100,100]_2[100](100) = \mbox{Aleaxaxsyss}\]

\[\circledast[100,100,100,100]_2[100](100) = \mbox{Aleaxacacyss}\]

\[\circledast[100,[4]]_2[100](100) = \mbox{Aleaxvvyngyss}\]

\[\circledast[100,[5]]_2[100](100) = \mbox{Aleaxalteryss}\]

\[\circledast[100,[100]]_2[100](100) = \mbox{Aleaxazerdyss}\]

\[\circledast[100,[1,100]]_2[100](100) = \mbox{Grand Aleaxazerdyss}\]

\[\circledast[100,[100,100]]_2[100](100) = \mbox{Aleaxazeryaredyss}\]

\[\circledast[100,[100,100,100]]_2[100](100) = \mbox{Aleaxazeracacidyss}\]

\[\circledast[100,[100,[3]]]_2[100](100) = \mbox{Aleaxazervvyndyss}\]

\[\circledast[100,[100,[100]]]_2[100](100) = \mbox{Aleaxdrazerdyss}\]

\[\circledast[100,[100,[1,100]]]_2[100](100) = \mbox{Grand Aleaxdrazerdyss}\]

\[\circledast[100,[100,[100,100]]]_2[100](100) = \mbox{Aleaxdrazeryaredyss}\]

\[\circledast[100,[100,[100,100,100]]]]_2[100](100) = \mbox{Aleaxdrazeraxsdyss}\]

\[\circledast[3]_2[100]_2[100](100) = \mbox{Aleaxtrazeraxsdyss}\]

\[\circledast[4]_2[100]_2[100](100) = \mbox{Aleaxtetrazeraxsdyss}\]

\[\circledast[5]_2[100]_2[100](100) = \mbox{Aleaxpentazeraxsdyss}\]

\[\circledast[100]_2[100]_2[100](100) = \mbox{Draleaxyss}\]

\[\circledast[1,100]_2[100]_2[100](100) = \mbox{Grand Draleaxyss}\]

\[\circledast[100,100]_2[100]_2[100](100) = \mbox{Draleaxyaryss}\]

\[\circledast[100,[100]]_2[100]_2[100](100) = \mbox{Draleaxazerdyss}\]

\[\circledast[100,[100,100]]_2[100]_2[100](100) = \mbox{Draleaxazeryaredyss}\]

\[\circledast[2]_2[100]_2[100]_2[100](100) = \mbox{Draleaxdrazeryaredyss}\]

\[\circledast[3]_2[100]_2[100]_2[100](100) = \mbox{Draleaxtrazeryaredyss}\]

\[\circledast[3]_3[100](100) = \mbox{Traleaxyss}\]

\[\circledast[4]_3[100](100) = \mbox{Tetraleaxyss}\]

\[\circledast[5]_3[100](100) = \mbox{Pentaleaxyss}\]

\[\circledast[6]_3[100](100) = \mbox{Hexaleaxyss}\]

\[\circledast[7]_3[100](100) = \mbox{Heptaleaxyss}\]

\[\circledast[8]_3[100](100) = \mbox{Octaleaxyss}\]

\[\circledast[9]_3[100](100) = \mbox{Ennaleaxyss}\]

\[\circledast[10]_3[100](100) = \mbox{Decaleaxyss}\]

\[\circledast[100]_3[100](100) = \mbox{Aleaxytryss}\]

\[\circledast[1,100]_3[100](100) = \mbox{Grand Aleaxytryss}\]

\[\circledast[100,100]_3[100](100) = \mbox{Aleaxyarytryss}\]

\[\circledast[100,[100]]_3[100](100) = \mbox{Aleaxazertryss}\]

\[\circledast[2]_2[100]_3[100](100) = \mbox{Aleaxdrazertryss}\]

\[\circledast[3]_2[100]_3[100](100) = \mbox{Aleaxtrazertryss}\]

\[\circledast[100]_2[100]_3[100](100) = \mbox{Aleaxizmorysstryss}\]

\[\circledast[1,100]_2[100]_3[100](100) = \mbox{Grand Aleaxizmorysstryss}\]

\[\circledast[100,100]_2[100]_3[100](100) = \mbox{Aleaxyarizmorysstryss}\]

\[\circledast[100,[100]]_2[100]_3[100](100) = \mbox{Aleaxazerdizmorysstryss}\]

\[\circledast[2]_2[100]_2[100]_3[100](100) = \mbox{Aleaxdrazerdizmorysstryss}\]

\[\circledast[3]_2[100]_2[100]_3[100](100) = \mbox{Aleaxtrazerdizmorysstryss}\]

\[\circledast[100]_2[100]_2[100]_3[100](100) = \mbox{Aleaxfijysstryss}\]

\[\circledast[1,100]_2[100]_2[100]_3[100](100) = \mbox{Grand Aleaxfijysstryss}\]

\[\circledast[100,100]_2[100]_2[100]_3[100](100) = \mbox{Aleaxyarefijysstryss}\]

\[\circledast[100,[100]]_2[100]_2[100]_3[100](100) = \mbox{Aleaxazerdyfijysstryss}\]

\[\circledast[2]_2[100]_2[100]_2[100]_3[100](100) = \mbox{Aleaxdrazerdyfijysstryss}\]

\[\circledast[3]_3[100]_3[100](100) = \mbox{Aleaxaecrysstryss}\]

\[\circledast[4]_3[100]_3[100](100) = \mbox{Aleaxerlfurysstryss}\]

\[\circledast[5]_3[100]_3[100](100) = \mbox{Aleaxemtwysstryss}\]

\[\circledast[6]_3[100]_3[100](100) = \mbox{Aleaxymusthysstryss}\]

\[\circledast[7]_3[100]_3[100](100) = \mbox{Aleaxouvcosysstryss}\]

\[\circledast[8]_3[100]_3[100](100) = \mbox{Aleaxexrysstryss}\]

\[\circledast[9]_3[100]_3[100](100) = \mbox{Aleaxyfaezysstryss}\]

\[\circledast[10]_3[100]_3[100](100) = \mbox{Aleaxyseritysstryss}\]

\[\circledast[100]_3[100]_3[100](100) = \mbox{Draleaxytryss}\]

\[\circledast[1,100]_3[100]_3[100](100) = \mbox{Grand Draleaxytryss}\]

\[\circledast[100,100]_3[100]_3[100](100) = \mbox{Draleaxyarytryss}\]

\[\circledast[100,[100]]_3[100]_3[100](100) = \mbox{Draleaxazertryss}\]

\[\circledast[2]_2[100]_3[100]_3[100](100) = \mbox{Draleaxdrazertryss}\]

\[\circledast[100]_2[100]_3[100]_3[100](100) = \mbox{Draleaxizmorysstryss}\]

\[\circledast[1,100]_2[100]_3[100]_3[100](100) = \mbox{Grand Draleaxizmorysstryss}\]

\[\circledast[100,100]_2[100]_3[100]_3[100](100) = \mbox{Draleaxyarizmorysstryss}\]

\[\circledast[100,[100]]_2[100]_3[100]_3[100](100) = \mbox{Draleaxazerdizmorysstryss}\]

\[\circledast[100]_2[100]_2[100]_3[100]_3[100](100) = \mbox{Draleaxyfijysstryss}\]

\[\circledast[3]_3[100]_3[100]_3[100](100) = \mbox{Draleaxaecrysstryss}\]

\[\circledast[4]_3[100]_3[100]_3[100](100) = \mbox{Draleaxerlfurysstryss}\]

\[\circledast[100]_3[100]_3[100]_3[100](100) = \mbox{Traleaxytryss}\]

\[\circledast[4]_4[100](100) = \mbox{Tetraleaxytryss}\]

\[\circledast[5]_4[100](100) = \mbox{Pentaleaxytryss}\]

\[\circledast[6]_4[100](100) = \mbox{Hexaleaxytryss}\]

\[\circledast[100]_4[100](100) = \mbox{Aleaxytetryss}\]

\[\circledast[1,100]_4[100](100) = \mbox{Grand Aleaxytetryss}\]

\[\circledast[100,100]_4[100](100) = \mbox{Aleaxyarytetryss}\]

\[\circledast[100,[100]]_4[100](100) = \mbox{Aleaxazerdytetryss}\]

\[\circledast[100]_2[100]_4[100](100) = \mbox{Aleaxizmorysstetryss}\]

\[\circledast[1,100]_2[100]_4[100](100) = \mbox{Grand Aleaxizmorysstetryss}\]

\[\circledast[100,100]_2[100]_4[100](100) = \mbox{Aleaxyarizmorysstetryss}\]

\[\circledast[100,[100]]_2[100]_4[100](100) = \mbox{Aleaxazerdizmorysstetryss}\]

\[\circledast[100]_2[100]_2[100]_4[100](100) = \mbox{Aleaxyfijysstetryss}\]

\[\circledast[3]_3[100]_4[100](100) = \mbox{Aleaxaecrysstetryss}\]

\[\circledast[100]_3[100]_4[100](100) = \mbox{Aleaxizmorytrysstetryss}\]

\[\circledast[1,100]_3[100]_4[100](100) = \mbox{Grand Aleaxizmorytrysstetryss}\]

\[\circledast[100,100]_3[100]_4[100](100) = \mbox{Aleaxyarizmorytrysstetryss}\]

\[\circledast[100,[100]]_3[100]_4[100](100) = \mbox{Aleaxazerdizmorytrysstetryss}\]

\[\circledast[100]_2[100]_3[100]_4[100](100) = \mbox{Aleaxizmoryssizmorytrysstetryss}\]

\[\circledast[100]_3[100]_3[100]_4[100](100) = \mbox{Aleaxyfijytrysstetryss}\]

\[\circledast[3]_4[100]_4[100](100) = \mbox{Aleaxaecrytrysstetryss}\]

\[\circledast[4]_4[100]_4[100](100) = \mbox{Aleaxerlfurytrysstetryss}\]

\[\circledast[100]_4[100]_4[100](100) = \mbox{Draleaxytetryss}\]

\[\circledast[1,100]_4[100]_4[100](100) = \mbox{Grand Draleaxytetryss}\]

\[\circledast[100,100]_4[100]_4[100](100) = \mbox{Draleaxyarytetryss}\]

\[\circledast[100,[100]]_4[100]_4[100](100) = \mbox{Draleaxazerdytetryss}\]

\[\circledast[100]_2[100]_4[100]_4[100](100) = \mbox{Draleaxizmorysstetryss}\]

\[\circledast[100]_3[100]_4[100]_4[100](100) = \mbox{Draleaxizmortrysstetryss}\]

\[\circledast[100]_4[100]_4[100]_4[100](100) = \mbox{Traleaxytetryss}\]

\[\circledast[4]_5[100](100) = \mbox{Tetraleaxytetryss}\]

\[\circledast[5]_5[100](100) = \mbox{Pentaleaxytetryss}\]

\[\circledast[100]_5[100](100) = \mbox{Aleaxypentyss}\]

\[\circledast[1,100]_5[100](100) = \mbox{Grand Aleaxypentyss}\]

\[\circledast[100,100]_5[100](100) = \mbox{Aleaxyarypentyss}\]

\[\circledast[100[100]]_5[100](100) = \mbox{Aleaxazerdypentyss}\]

\[\circledast[100]_2[100]_5[100](100) = \mbox{Aleaxizmorysspentyss}\]

\[\circledast[100]_3[100]_5[100](100) = \mbox{Aleaxizmorytrysspentyss}\]

\[\circledast[100]_4[100]_5[100](100) = \mbox{Aleaxizmorytetrysspentyss}\]

\[\circledast[100]_5[100]_5[100](100) = \mbox{Draleaxypentyss}\]

\[\circledast[3]_6[100](100) = \mbox{Traleaxypentyss}\]

\[\circledast[100]_6[100](100) = \mbox{Aleaxyhexyss}\]

\[\circledast[100]_7[100](100) = \mbox{Aleaxyheptyss}\]

\[\circledast[100]_8[100](100) = \mbox{Aleaxoctyss}\]

\[\circledast[100]_9[100](100) = \mbox{Aleaxennyss}\]

\[\circledast[100]_10[100](100) = \mbox{Aleaxydecyss}\]

\[\circledast[100]_100[100](100) = \mbox{Kryglon}\]

\[\circledast[\{([1,100]),100,100\}](100) = \mbox{Grand Kryglon}\]

\[\circledast[\{([100,100]),100,100\}](100) = \mbox{Kryyarglon}\]

\[\circledast[\{([1,100,100]),100,100\}](100) = \mbox{Grand Kryyarglon}\]

\[\circledast[\{([100,100,100]),100,100\}](100) = \mbox{Kryxstglon}\]

\[\circledast[\{\{1,100,100\},100,100\}](100) = \mbox{Kryzerglon}\]

\[\circledast[\{\{1,([1,100]),100\},100,100\}](100) = \mbox{Grand Kryzerglon}\]

\[\circledast[\{\{1,([100,100]),100\},100,100\}](100) = \mbox{Kryzeryarglon}\]

\[\circledast[\{\{1,([1,100,100]),100\},100,100\}](100) = \mbox{Grand Kryzeryarglon}\]

\[\circledast[\{\{1,([100,100,100]),100\},100,100\}](100) = \mbox{Kryzerxstglon}\]

\[\circledast[\{\{1,\{1,100,100\},100\},100,100\}](100) = \mbox{Krydrazerglon}\]

\[\circledast[\{\{2,100,100\},100,100\}](100) = \mbox{Kryyssglon}\]

\[\circledast[\{\{2,([1,100]),100\},100,100\}](100) = \mbox{Grand Kryyssglon}\]

\[\circledast[\{\{2,([100,100]),100\},100,100\}](100) = \mbox{Kryyaryssglon}\]

\[\circledast[\{\{2,([1,100,100]),100\},100,100\}](100) = \mbox{Grand Kryyaryssglon}\]

\[\circledast[\{\{2,\{1,100,100\},100\},100,100\}](100) = \mbox{Kryzeryssglon}\]

\[\circledast[\{\{2,\{1,([1,100]),100\},100\},100,100\}](100) = \mbox{Grand Kryzeryssglon}\]

\[\circledast[\{\{2,\{1,([100,100]),100\},100\},100,100\}](100) = \mbox{Kryzeryaryssglon}\]

\[\circledast[\{\{2,\{1,\{1,100,100\},100\},100\},100,100\}](100) = \mbox{Krydrazeryssglon}\]

\[\circledast[\{\{2,\{1,\{1,([1,100]),100\},100\},100\},100,100\}](100) = \mbox{Grand Krydrazeryssglon}\]

\[\circledast[\{\{2,\{2,100,100\},100\},100,100\}](100) = \mbox{Kryfijyssglon}\]

\[\circledast[\{\{2,\{2,([1,100]),100\},100\},100,100\}](100) = \mbox{Grand Kryfijyssglon}\]

\[\circledast[\{\{2,\{2,([100,100]),100\},100\},100,100\}](100) = \mbox{Kryyarfijyssglon}\]

\[\circledast[\{\{2,\{2,\{1,100,100\},100\},100\},100,100\}](100) = \mbox{Kryzerfijyssglon}\]

\[\circledast[\{\{3,4,100\},100,100\}](100) = \mbox{Kryerlfuryssglon}\]

\[\circledast[\{\{3,100,100\},100,100\}](100) = \mbox{Krytryssglon}\]

\[\circledast[\{\{3,([1,100]),100\},100,100\}](100) = \mbox{Grand Krytryssglon}\]

\[\circledast[\{\{3,([100,100]),100\},100,100\}](100) = \mbox{Kryyartryssglon}\]

\[\circledast[\{\{3,\{1,100,100\},100\},100,100\}](100) = \mbox{Kryzertryssglon}\]

\[\circledast[\{\{3,\{2,100,100\},100\},100,100\}](100) = \mbox{Kryysstryssglon}\]

\[\circledast[\{\{3,\{3,100,100\},100\},100,100\}](100) = \mbox{Kryfijytryssglon}\]

\[\circledast[\{\{3,\{3,\{3,100,100\},100\},100\},100,100\}](100) = \mbox{Kryaecrytryssglon}\]

\[\circledast[\{\{4,100,100\},100,100\}](100) = \mbox{Krytetryssglon}\]

\[\circledast[\{\{100,100,100\},100,100\}](100) = \mbox{Kryglonduexis}\]

\[\circledast[\{\{([1,100]),100,100\},100,100\}](100) = \mbox{Grand Kryglonduexis}\]

\[\circledast[\{\{([100,100]),100,100\},100,100\}](100) = \mbox{Kryyarglonduexis}\]

\[\circledast[\{\{\{1,100,100\},100,100\},100,100\}](100) = \mbox{Kryzerglonduexis}\]

\[\circledast[\{\{\{1,([1,100]),100\},100,100\},100,100\}](100) = \mbox{Grand Kryzerglonduexis}\]

\[\circledast[\{\{\{1,([100,100]),100\},100,100\},100,100\}](100) = \mbox{Kryzeryarglonduexis}\]

\[\circledast[\{\{\{1,\{1,100,100\},100\},100,100\},100,100\}](100) = \mbox{Krydrazerglonduexis}\]

\[\circledast[\{\{\{1,\{1,\{1,100,100\},100\},100\},100,100\},100,100\}](100) = \mbox{Krytrazerglonduexis}\]

\[\circledast[\{\{\{2,100,100\},100,100\},100,100\}](100) = \mbox{Kryyssglonduexis}\]

\[\circledast[\{\{\{2,([1,100]),100\},100,100\},100,100\}](100) = \mbox{Grand Kryyssglonduexis}\]

\[\circledast[\{\{\{2,([100,100]),100\},100,100\},100,100\}](100) = \mbox{Kryyaryssglonduexis}\]

\[\circledast[\{\{\{2,\{1,100,100\},100\},100,100\},100,100\}](100) = \mbox{Kryzeryssglonduexis}\]

\[\circledast[\{\{\{2,\{2,100,100\},100\},100,100\},100,100\}](100) = \mbox{Kryfijyssglonduexis}\]

\[\circledast[\{\{\{2,\{2,([1,100]),100\},100\},100,100\},100,100\}](100) = \mbox{Grand Kryfijyssglonduexis}\]

\[\circledast[\{\{\{2,\{2,([100,100]),100\},100\},100,100\},100,100\}](100) = \mbox{Kryyarfijyssglonduexis}\]

\[\circledast[\{\{\{2,\{2,\{1,100,100\},100\},100\},100,100\},100,100\}](100) = \mbox{Kryzerfijyssglonduexis}\]

\[\circledast[\{\{\{2,\{2,\{2,100,100\},100\},100\},100,100\},100,100\}](100) = \mbox{Kryaecryssglonduexis}\]

\[\circledast[\{\{\{3,100,100\},100,100\},100,100\}](100) = \mbox{Krytryssglonduexis}\]

\[\circledast[\{\{\{4,100,100\},100,100\},100,100\}](100) = \mbox{Krytetryssglonduexis}\]

\[\circledast[\{\{\{100,100,100\},100,100\},100,100\}](100) = \mbox{Kryglontrixis}\]

\[\circledast[\{1,4,100,100\}](100) = \mbox{Kryglonvurlex}\]

\[\circledast[\{1,5,100,100\}](100) = \mbox{Kryglonfiefin}\]

\[\circledast[\{1,6,100,100\}](100) = \mbox{Kryglonsiesu}\]

\[\circledast[\{1,7,100,100\}](100) = \mbox{Kryglonziewer}\]

\[\circledast[\{1,8,100,100\}](100) = \mbox{Kryglonahgdu}\]

\[\circledast[\{1,9,100,100\}](100) = \mbox{Kryglonienenne}\]

\[\circledast[\{1,10,100,100\}](100) = \mbox{Kryglonzehrdur}\]

\[\circledast[\{1,100,100,100\}](100) = \mbox{Hyperior Kryglon}\]

\[\circledast[\{1,([1,100]),100,100\}](100) = \mbox{Hyperior Grand Kryglon}\]

\[\circledast[\{1,([100,100]),100,100\}](100) = \mbox{Hyperior Kryyarglon}\]

\[\circledast[\{1,\{1,100,100\},100,100\}](100) = \mbox{Hyperior Kryzerglon}\]

\[\circledast[\{1,\{2,100,100\},100,100\}](100) = \mbox{Hyperior Kryyssglon}\]

\[\circledast[\{1,\{3,100,100\},100,100\}](100) = \mbox{Hyperior Krytryssglon}\]

\[\circledast[\{1,\{100,100,100\},100,100\}](100) = \mbox{Hyperior Kryglonduexis}\]

\[\circledast[\{1,\{\{100,100,100\},100,100\},100,100\}](100) = \mbox{Hyperior Kryglontrixis}\]

\[\circledast[\{2,100,100,100\}](100) = \mbox{Dihyperior Kryglon}\]

\[\circledast[\{3,100,100,100\}](100) = \mbox{Trihyperior Kryglon}\]

\[\circledast[\{4,100,100,100\}](100) = \mbox{Tetrahyperior Kryglon}\]

\[\circledast[\{5,100,100,100\}](100) = \mbox{Pentahyperior Kryglon}\]

\[\circledast[\{6,100,100,100\}](100) = \mbox{Hexahyperior Kryglon}\]

\[\circledast[\{7,100,100,100\}](100) = \mbox{Heptahyperior Kryglon}\]

\[\circledast[\{8,100,100,100\}](100) = \mbox{Octahyperior Kryglon}\]

\[\circledast[\{9,100,100,100\}](100) = \mbox{Enneahyperior Kryglon}\]

\[\circledast[\{10,100,100,100\}](100) = \mbox{Decahyperior Kryglon}\]

\[\circledast[\{100,100,100,100\}](100) = \mbox{Kryglas}\]

\[\circledast[\{([1,100]),100,100,100\}](100) = \mbox{Grand Kryglas}\]

\[\circledast[\{([100,100]),100,100,100\}](100) = \mbox{Kryyarglas}\]

\[\circledast[\{\{1,100,100\},100,100,100\}](100) = \mbox{Kryzerglas}\]

\[\circledast[\{\{2,100,100\},100,100,100\}](100) = \mbox{Kryyssglas}\]

\[\circledast[\{\{3,100,100\},100,100,100\}](100) = \mbox{Krytryssglas}\]

\[\circledast[\{\{100,100,100\},100,100,100\}](100) = \mbox{Kryglonglas}\]

\[\circledast[\{\{([1,100]),100,100\},100,100,100\}](100) = \mbox{Grand Kryglonglas}\]

\[\circledast[\{\{([100,100]),100,100\},100,100,100\}](100) = \mbox{Kryyarglonglas}\]

\[\circledast[\{\{\{1,100,100\},100,100\},100,100,100\}](100) = \mbox{Kryzerglonglas}\]

\[\circledast[\{\{\{2,100,100\},100,100\},100,100,100\}](100) = \mbox{Kryyssglonglas}\]

\[\circledast[\{\{\{100,100,100\},100,100\},100,100,100\}](100) = \mbox{Kryglonduexiglas}\]

\[\circledast[\{\{1,3,100,100\},100,100,100\}](100) = \mbox{Kryglontrixiglas}\]

\[\circledast[\{\{1,4,100,100\},100,100,100\}](100) = \mbox{Kryglonverleglas}\]

\[\circledast[\{\{1,100,100,100\},100,100,100\}](100) = \mbox{Kryhyperiglonglas}\]

\[\circledast[\{\{2,100,100,100\},100,100,100\}](100) = \mbox{Krydihyperiglonglas}\]

\[\circledast[\{\{3,100,100,100\},100,100,100\}](100) = \mbox{Krytrihyperiglonglas}\]

\[\circledast[\{\{3,100,100,100\},100,100,100\}](100) = \mbox{Krytrihyperiglonglas}\]

\[\circledast[\{\{4,100,100,100\},100,100,100\}](100) = \mbox{Krytetrahyperiglonglas}\]

\[\circledast[\{\{100,100,100,100\},100,100,100\}](100) = \mbox{Kryglasduexis}\]

\[\circledast[\{\{([1,100]),100,100,100\},100,100,100\}](100) = \mbox{Grand Kryglasduexis}\]

\[\circledast[\{\{([100,100]),100,100,100\},100,100,100\}](100) = \mbox{Kryyarglasduexis}\]

\[\circledast[\{\{\{1,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryzerglasduexis}\]

\[\circledast[\{\{\{2,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryyssglasduexis}\]

\[\circledast[\{\{\{100,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryglonglasduexis}\]

\[\circledast[\{\{\{([1,100]),100,100\},100,100,100\},100,100,100\}](100) = \mbox{Grand Kryglonglasduexis}\]

\[\circledast[\{\{\{([100,100]),100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryyarglonglasduexis}\]

\[\circledast[\{\{\{\{1,100,100\},100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryzerglonglasduexis}\]

\[\circledast[\{\{\{\{100,100,100\},100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryglonduexiglasduexis}\]

\[\circledast[\{\{\{1,3,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryglontrixiglasduexis}\]

\[\circledast[\{\{\{1,100,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryhyperiglonglasduexis}\]

\[\circledast[\{\{\{2,100,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Krydihyperiglonglasduexis}\]

\[\circledast[\{\{\{3,100,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Krytrihyperiglonglasduexis}\]

\[\circledast[\{\{\{100,100,100,100\},100,100,100\},100,100,100\}](100) = \mbox{Kryglastrixis}\]

\[\circledast[\{1,4,100,100,100\}](100) = \mbox{Kryglasvurlex}\]

\[\circledast[\{1,5,100,100,100\}](100) = \mbox{Kryglasfiefin}\]

\[\circledast[\{1,100,100,100,100\}](100) = \mbox{Hyperior Kryglas}\]

\[\circledast[\{1,([1,100]),100,100,100\}](100) = \mbox{Hyperior Grand Kryglas}\]

\[\circledast[\{1,([100,100]),100,100,100\}](100) = \mbox{Hyperior Kryyarglas}\]

\[\circledast[\{1,\{1,100,100\},100,100,100\}](100) = \mbox{Hyperior Kryzerglas}\]

\[\circledast[\{1,\{2,100,100\},100,100,100\}](100) = \mbox{Hyperior Kryyssglas}\]

\[\circledast[\{1,\{100,100,100\},100,100,100\}](100) = \mbox{Hyperior Kryglonglas}\]

\[\circledast[\{1,\{1,2,100,100\},100,100,100\}](100) = \mbox{Hyperior Kryglonduexiglas}\]

\[\circledast[\{1,\{1,3,100,100\},100,100,100\}](100) = \mbox{Hyperior Kryglontrixiglas}\]

\[\circledast[\{1,\{1,100,100,100\},100,100,100\}](100) = \mbox{Hyperior Kryhyperiglonglas}\]

\[\circledast[\{1,\{2,100,100,100\},100,100,100\}](100) = \mbox{Hyperior Krydihyperiglonglas}\]

\[\circledast[\{1,\{100,100,100,100\},100,100,100\}](100) = \mbox{Hyperior Kryglasduexis}\]

\[\circledast[\{2,100,100,100,100\}](100) = \mbox{Dihyperior Kryglas}\]

\[\circledast[\{3,100,100,100,100\}](100) = \mbox{Trihyperior Kryglas}\]

\[\circledast[\{4,100,100,100,100\}](100) = \mbox{Tetrahyperior Kryglas}\]

\[\circledast[\{5,100,100,100,100\}](100) = \mbox{Pentahyperior Kryglas}\]

\[\circledast[\{6,100,100,100,100\}](100) = \mbox{Hexahyperior Kryglas}\]

\[\circledast[\{100,100,100,100,100\}](100) = \mbox{Krygler}\]

\[\circledast[\{([1,100]),100,100,100,100\}](100) = \mbox{Grand Krygler}\]

\[\circledast[\{([100,100]),100,100,100,100\}](100) = \mbox{Kryyargler}\]

\[\circledast[\{\{1,100,100\},100,100,100,100\}](100) = \mbox{Kryzergler}\]

\[\circledast[\{\{100,100,100\},100,100,100,100\}](100) = \mbox{Kryglongler}\]

\[\circledast[\{\{1,2,100,100\},100,100,100,100\}](100) = \mbox{Kryglonduexigler}\]

\[\circledast[\{\{1,100,100,100\},100,100,100,100\}](100) = \mbox{Kryhyperiglongler}\]

\[\circledast[\{\{2,100,100,100\},100,100,100,100\}](100) = \mbox{Krydihyperiglongler}\]

\[\circledast[\{\{100,100,100,100\},100,100,100,100\}](100) = \mbox{Kryglasgler}\]

\[\circledast[\{\{([1,100]),100,100,100\},100,100,100,100\}](100) = \mbox{Grand Kryglasgler}\]

\[\circledast[\{\{([100,100]),100,100,100\},100,100,100,100\}](100) = \mbox{Kryyarglasgler}\]

\[\circledast[\{\{\{1,100,100\},100,100,100\},100,100,100,100\}](100) = \mbox{Kryzerglasgler}\]

\[\circledast[\{\{\{100,100,100\},100,100,100\},100,100,100,100\}](100) = \mbox{Kryglonglasgler}\]

\[\circledast[\{\{\{1,2,100,100\},100,100,100\},100,100,100,100\}](100) = \mbox{Kryglonduexiglasgler}\]

\[\circledast[\{\{\{1,100,100,100\},100,100,100\},100,100,100,100\}](100) = \mbox{Kryhyperiglonglasgler}\]

\[\circledast[\{\{\{2,100,100,100\},100,100,100\},100,100,100,100\}](100) = \mbox{Krydihyperiglonglasgler}\]

\[\circledast[\{\{1,2,100,100,100\},100,100,100,100\}](100) = \mbox{Kryglasduexigler}\]

\[\circledast[\{\{1,100,100,100,100\},100,100,100,100\}](100) = \mbox{Kryhyperiglasgler}\]

\[\circledast[\{\{2,100,100,100,100\},100,100,100,100\}](100) = \mbox{Krydihyperiglasgler}\]

\[\circledast[\{\{100,100,100,100,100\},100,100,100,100\}](100) = \mbox{Kryglerduexis}\]

\[\circledast[\{1,3,100,100,100,100\}](100) = \mbox{Kryglertrixis}\]

\[\circledast[\{1,100,100,100,100,100\}](100) = \mbox{Hyperior Krygler}\]

\[\circledast[\{2,100,100,100,100,100\}](100) = \mbox{Dihyperior Krygler}\]

\[\circledast[\{3,100,100,100,100,100\}](100) = \mbox{Trihyperior Krygler}\]

\[\circledast[\{100,100,100,100,100,100\}](100) = \mbox{Kryglem}\]

\[\circledast[\{([1,100]),100,100,100,100,100\}](100) = \mbox{Grand Kryglem}\]

\[\circledast[\{([100,100]),100,100,100,100,100\}](100) = \mbox{Kryyarglem}\]

\[\circledast[\{\{1,100,100\},100,100,100,100,100\}](100) = \mbox{Kryzerglem}\]

\[\circledast[\{\{100,100,100\},100,100,100,100,100\}](100) = \mbox{Kryglonglem}\]

\[\circledast[\{\{100,100,100,100\},100,100,100,100,100\}](100) = \mbox{Kryglasglem}\]

\[\circledast[\{\{100,100,100,100,100\},100,100,100,100,100\}](100) = \mbox{Kryglerglem}\]

\[\circledast[\{1,2,100,100,100,100,100\}](100) = \mbox{Kryglemduexis}\]

\[\circledast[\{1,100,100,100,100,100,100\}](100) = \mbox{Hyperior Kryglem}\]

\[\circledast[\{100,[6]\}](100) = \mbox{Kryglusth}\]

\[\circledast[\{100,[7]\}](100) = \mbox{Kryglouv}\]

\[\circledast[\{100,[8]\}](100) = \mbox{Kryglex}\]

\[\circledast[\{100,[9]\}](100) = \mbox{Kryglaez}\]

\[\circledast[\{100,[10]\}](100) = \mbox{Kryglerit}\]

\[\circledast[\{100,[100]\}](100) = \mbox{Erytyryum}\]

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