User blog comment:Emlightened/Hypernomials (Γ₀)/@comment-5529393-20160113233321/@comment-5529393-20160113234032

whoops, I just noticed that your family always uses x as the base for the Knuth arrows. This should make the problem substantially easier. Still a lot of work to prove it though. There are published papers proving that the same family using only exponentiation and tetration have order type $$\varphi(2,0)$$