User blog comment:Tetramur/Pentational arrays and beyond - comparisons/@comment-37993808-20200108130039/@comment-39541634-20200108140409

There are extensions of BEAF (made by others) which are well-defined.

The problem is that all of them ignore Bowers' own statement that the stucture f(X) & X must have f(X) entries.

For example, a Triakulus is supposed to be 3 & 3 & 3 = {3,3,3} & 3 = 3^^^3. According to Bowers, this should be a pentational array of 3^^^3 threes, yet none of the BEAF extensions work in this manner. So none of these extensions can be considered "what Bowers intended" even if he himself claims otherwise. They are simply not compatible with the basics of his own idea.

It should also be noted that the very concept of BEAF arrays isn't compatible with Saibian's climbing method. The intuitive basis for Saibian's method is that we're dealing with infinitely-high structures. To Saibian, X^^^X is an infinite tower of X's and the climibng method arises from climbing up this infinite tower.

Bowers array, on the other hand, are finite arrays. A 3^^^3 pentational array is supposed to be some kind of vastly complex 3x3x3x...x3 "pentational" cube. Whatever that means, it is a finite "geometrical" object with a finite number of elements (exactly 3^^^3 elements). This array has to be expanded one element at time, without the whole thing losing its meaning. Conceptually, this is a complete antithesis to what Saibian was doing.

So even if we manage to crack the (very difficult) problem of defining pentational arrays in a way that's true to Bowers original framework, it is highly unlikely that such a definition would be able to take advantage of Saibian's ideas. I know Bowers himself is big proponent of the BEAF-BSC approach, but he doesn't seem to be aware of this contradiction.