User blog:GamesFan2000/Extreme Condensed Multi-Chaining Function (ECMCF)

Hello! The following function is one I’ve theorized about in my head all day today. I call this the Extreme Condensed Multi-Chaining Function, or ECMCF.

General Definition
This function takes the form of n[m], where n is the base and m is the recursor. It also represents the simplest part of this function.

n[0]
The first case is n[0]. This is equal to n→nn. a→cb=a→c-1a...a→c-1a, which has b terms in it. Also, I’m making this right-associative, meaning that something like 3→33→23 is legal. That expression becomes 3→³(3→3→3). Making this right-associative puts the growth limit at ωω, compared to ω³ if I used the Hurford rules. 4[0]=4→⁴4, 5[0]=5→⁵5, etc.

n[1]
Next is n[1]. n[1] is equal to n[0][0][0][0]...[0], with (n[0])+1 [0]’s, solved left-to-right. The only trivial example is 2[1], which becomes 2[0][0][0][0][0]. Beyond that, it’s impossible to write out any of the expressions in an expanded format.

n[m]
For n[m] where m>0, n[m]=n[m-1][m-1]...[m-1], with (n[m-1])+1 [m-1]’s. Remember, this is left-associative, so you would expand out the first [m-1] into [m-2]’s and so on until you’ve reached the next [m-1]. 2[2]=2[1][1][1]...(2[1])+1 [1]’s...[1]=2[0][0][0][0][0][1][1][1]...2[1] [1]’s...[1].

I’ll leave this here for now. I have much more in store for this function.