User blog comment:Tetramur/Pentational arrays and beyond - comparisons/@comment-37993808-20200108154849/@comment-39541634-20200109160454

"I understood that p↑↑↑p is not the resulting value, but I am not certain about how it works in that strong way. If the resulting structure does not include repetition of X↑↑, I have no idea on how it corresponds to ζ_0."

It pretty much does include a repetition of X↑↑.

To be precise, p↑↑↑p is equivalent to p↑↑p↑↑p↑↑...p↑↑p (with p repeating p times) which resides in a space whose structure is X↑↑↑p=X↑↑X↑↑...X↑↑X (with X repeating p times). Our structure can then expand to enormous size within the confines of this space, so we kinda have the entire power of an X↑↑↑p space at our disposal.

Notice that I've said "kinda", though. The catch is that we are only allowed to work with finite subspaces of X↑↑↑p and we have to construct these in a step-by-step manner.

None of the above, by the way, is particular to pentational arrays. You can see the entire concept in action with much a simpler array, like (say) X2 & 3 (Dutritri). The structure here is a simple 3x3 square containing 9 threes, and by experimenting you'll be able to see how it's geometrical shape evolves as it expands.