User blog comment:ArtismScrub/Hyper-Alpha Notation Part III: Full Circle/@comment-30754445-20180307182140

I haven't yet tried to analyze your notation, but I would like to comment on this:

"I thought you guys said making a functional SGH-FGH map was hard!"

Not exactly.

The thing which is said to be impossibly hard, is to do it in a way that's compatible with Bower's notion of an array notation.

Defining "10↑↑↑10 & 10" as a ζ₀-level number isn't too difficult. The tricky part is getting "10↑↑↑10 & 10" to refer to an actual array structure that contains 10↑↑↑10 entries.

More precisely, the tricky part is how to track the decreasing polyponent properly when we're expanding pentational arrays, while keeping the base (10) constant. This is why most extensions of this sort use a transfinite symbol (like your A) instead of using explicit numbers. Saibian's Hyper-E extensions also use a similar mechanism (he uses the symbol "#" instead of A).

Again, I haven't thoroughly analyzed your notation, but It looks like you've independently discovered the same trick. Getting it to work all the way up to ϑ(Ωω) on your own is a hugely impressive feat, though.