User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 2/@comment-5529393-20130527105642

Hold on. [1,1,...,2]w/[2] is nowhere near \(\theta(\omega+1)\). Remember if f(n) is at level \(\alpha\), then even level \(\alpha+1\) is f(f(f(...f(n)), with n n's. So if f(n) = [1,...,1,2] with n 1's is level \(\theta(\omega)\), then even level \(\theta(\omega) + 1 \) is [f(f(...f(n)...), in other words, [1,...,1,2] w/ [1,...,1,2] w/ [1,...,1,2] ... w/ [1,...,1,2] with n 1's, which is what you are calling [1(1)2].  So 1[(1)2] is just at level \(theta(\omega) + 1\), not \(\Gamma(0)\).  And \(\theta(\omega + 1)\) is WAY bigger than \(\theta(\omega) + 1\), it is \(\theta(\theta(...\theta(0,0)...,0),0)\).

So I'm afraid this throws the whole analysis way off, there's no way this notation reaches the Bachmann-Howard ordinal.