Original[]
\(\psi(0) = 1\)
\(\psi(\alpha+1)[n] = \psi(\alpha)n\)
\(\psi(\alpha)[n] = \psi(\alpha[n])\) if \(\alpha\) is limit.
If \(\alpha\) has cofinality \(\Omega\) and \(\alpha\) is the output of some function \(f\), \(\psi(\alpha)[0] = \psi(f(0))\) and \(\psi(\alpha)[n+1] = \psi(f(\psi(\alpha)[n]))\).
What went wrong[]
The function \(f\) is undefined, so the output can vary wildly defined on the choice of \(f\). That is the main problem.
How I fixed it[]
A hyperoperator is either the successor function, or the iteration of a previous hyperoperator via the rule: \(H_{n+1}(a,b) = H_n(a,H_n(a,...H_n(a,a))))\) w/ \(b\) copies. \(H_n(a,b)\) with limit \(b\) is \(\textrm{sup}\{H_n(a, c): c < b\}\) and \(H_n(a,b)\) with limit \(a\) is \(\textrm{sup}\{H_n(c, b): c < a\}\).
A hyperoperator combination is a function which is essentially, a combination of hyperoperators. Formally, a function \(f\) is a hyperoperator combination \(a_0 h_1 a_1 h_2 ... h_n a_n\), where \(h_i\) are hyperoperators. \(a_0\) is the base and \(a_n\) is the peak. For example, \(10 \uparrow^2 2 \uparrow 3\) is a hyperoperator combination, with base 10 and peak 3. Then, I say that \(f\) must be a hyperoperator combination with base \(\Omega\), peak \(\alpha\) and the peak is the only place where \(\alpha\) is allowed to appear (so \(\lambda x.\Omega^{x^x}\) would not be allowed but \(\lambda x.\Omega^{\Omega^x}\)). For example:
Examples[]
In the original definition, \(\psi(\Omega^2)\) is ambiguous, as \(f\) could be many things, such as \(\lambda x.x^2\) or \(\lambda x.\Omega \cdot x\). However, with my formulation, we are restricted to hyperoperator combinations with base \(\Omega\) and peak \(x\), so (theorem) the latter would be allowed, and therefore \(\psi(\Omega^2)\) is the first fixed point of \(\alpha \mapsto \psi(\Omega \cdot \alpha)\).
Another example: \(\psi(\Omega^{\Omega^\Omega})\) is ambiguous, as \(f\) could be many things, such as \(\lambda x.x \uparrow^2 3\) or \(\lambda x. \Omega^{\Omega^x}\). The latter is a hyperoperator combination with base \(\Omega\) and peak \(x\), so (theorem) only the latter would be allowed, and therefore \(\psi(\Omega^{\Omega^\Omega})\) is the first fixed point of \(\alpha \mapsto \psi(\Omega^{\Omega^\alpha})\).
I'm pretty sure that the theorems are true, but I haven't been able to find a way to prove it mathematically. Therefore, I may have missed something: take the examples with a pinch of salt.
This starts to fall apart at \(BHO\) level, but I think I fixed the first bit of UNOCF.