Mini-definitions:
Here, \(x\) in the definition of \(C\) is the input of \(\gamma\).
\(\begin{align}
C(0)&=\{0,1,\omega,x\}\cup\bigcup_{k\lt x}\{\gamma(k)\}\\
C(n+1)&=\{\alpha+1,\alpha+\beta,\alpha\beta,\alpha^\beta,\gamma(\eta):\alpha,\beta\in C(n),\eta\lt x\cap C(n)\}\cup C(n)\\
C(n)&=\bigcup_{k\lt n}C(k)\cup\{\sup(\bigcup_{k\lt\omega}\{C(n[k])\})\}\iff\lim?(n)\\
\gamma(0)&=\omega\\
\gamma(x+1)&=\min(\{k:k\not\in C(\gamma(x))\})\\
\gamma(x)&=\min(\{k:\forall n[n\lt x\Rightarrow k\ge\gamma(n)]\})\iff\lim?(x)
\end{align}\)
Some values:
\(\begin{align}
\gamma(0)&=\omega\\
\gamma(1)&=\varepsilon_0\\
\gamma(2)&=\varepsilon_{\varepsilon_0}\\
\gamma(\omega)&=\zeta_0\\
\gamma(\omega+1)&=\varepsilon_{\zeta_0+1}\\
\gamma(\omega2)&=\zeta_1\\
\gamma(\omega^2)&=\zeta_\omega\\
\gamma(\omega^\omega)&=\zeta_{\omega^\omega}\\
\gamma(\varepsilon_0)&=\zeta_{\varepsilon_0}\\
\gamma(\zeta_0)&=\zeta_{\zeta_0}\\
\gamma(\eta_0)&=\eta_0\\
\gamma(\alpha)&=\alpha\iff\alpha\mapsto\zeta_\alpha
\end{align}\)
Let \(\mathbb{S}(n)\) be set of ordinals which satisfy \(\gamma(n,\alpha)=\alpha\) and \(\mathbb{S}(n)[\alpha]\) be the \(\alpha\)th smallest ordinal(starts from \(0\)th) in \(\mathbb{S}(n)\).
Let \(\mathbb{C}(n)\) be set of ordinals which satisfy \(\min(\{k:k\not\in C(n,\alpha)\})=\alpha\) and \(\mathbb{C}(n)[\alpha]\) be the \(\alpha\)th smallest ordinal(starts from \(0\)th) in \(\mathbb{C}(n)\).
\(\begin{align}
C(0,0)&=\{0,1,\omega,x\}\cup\bigcup_{a\lt x}\bigcup_{b\lt\mathbb{S}(\max(\{k:k\lt a\}))[y]}\{\gamma(a,b)\}\\
C(m+1,0)&=\bigcup_{a\le m}\bigcup_{b\lt\mathbb{C}(a)[0]}\{C(a,b)\}\\
C(m,0)&=\bigcup_{k\lt m}C(k,0)\cup\{\sup(\bigcup_{k\lt\omega}\{C(m[k],0)\})\}\iff\lim?(m)\\
C(m,n+1)&=\{\alpha+1,\alpha+\beta,\alpha\beta,\alpha^\beta,\gamma(\eta,\alpha):\alpha,\beta\in C(m,n),\eta\lt x\cap C(m,n)\}\cup C(m,n)\\
C(m,n)&=\bigcup_{k\lt n}C(k)\cup\{\sup(\bigcup_{k\lt\omega}\{C(m,n[k])\})\}\iff\lim?(n)\\
\gamma(0,0)&=\omega\\
\gamma(x+1,0)&=\mathbb{S}(x)[0]\\
\gamma(x,0)&=\sup(\bigcup_{k\lt\omega}\gamma(x[k],0))\iff\lim?(x)\\
\gamma(x,y+1)&=\min(\{k:k\not\in C(x,\gamma(x,y))\})\\
\gamma(x,y)&=\min(\{k:\forall n[n\lt y\Rightarrow k\ge\gamma(x,n)]\})\iff\lim?(y)
\end{align}\)
The definitions of the 2 arguments \(C\) and \(\gamma\) is equivalent to the 1 argument \(C\) and \(\gamma\) if the first argument is \(0\).
\(s\) is a string.
Let \(\mathbb{S}(n,s;k)\) be set of ordinals which satisfy \(\gamma(n,s,\alpha,\text{string of k 0's})=\alpha\) and \(\mathbb{S}(n,s;k)[\alpha]\) be the \(\alpha\)th smallest ordinal(starts from \(0\)th) in \(\mathbb{S}(n,s;k)\).
Let \(\mathbb{C}(n,s;k)\) be set of ordinals which satisfy \(\min(\{k:k\not\in C(n,s,\alpha,\text{string of k 0's})\})=\alpha\) and \(\mathbb{C}(n,s;k)[\alpha]\) be the \(\alpha\)th smallest ordinal(starts from \(0\)th) in \(\mathbb{C}(n,s;k)\).
\(i\) is the input. \(i[n]\) is the \(n\)th of inputs, right to left, starting from \(0\)th.
\(l(a)\) is the length of \(a\).
\(r\) is string of \(0\)'s with length\(\ge1\).
\(\gamma*(\cdots)\) is "small" \(\gamma\). If the leftmost inputs(could be none) are equal to the leftmost inputs for the \(\gamma\), then next is smaller than the corresponding input of \(\gamma\)(this must exist), then it is equal to \(\gamma\) function with equal input. Else, it is equal to \(0\).
\(\begin{align}
C(0,s)&=C(s)\\
C(0)&=\{0,1,\omega,i[l(i)-1]\}\cup\bigcup_{a\lt i[l(i)-1]}\bigcup_{b\lt\mathbb{S}(\max(\{k:k\lt a\});l(i)-2)[i[l(i)-2]]}\cdots\{\gamma(a,b,\cdots)\}\\
C(s,m+1,r)&=\bigcup_{a\le m}\bigcup_{b\lt\mathbb{C}(a;l(r)-1)[0]}\cdots\{C(a,b,\cdots;0)\}\\
C(s,m,r)&=\bigcup_{k\lt m}C(k,0)\cup\{\sup(\bigcup_{k\lt\omega}\{C(s,m[k],r)\})\}\iff\lim?(m)\\
C(s,n+1)&=\{\alpha+1,\alpha+\beta,\alpha\beta,\alpha^\beta,\gamma*(\chi_0,\chi_1,\cdots,\chi_{l(i)-1}):\alpha,\beta,\chi_0,\cdots,\chi_{l(i)-1}\in C(s,n)\}\cup C(s,n)\\
C(s,n)&=\bigcup_{k\lt n}C(k)\cup\{\sup(\bigcup_{k\lt\omega}\{C(s,n[k])\})\}\iff\lim?(n)\\
\gamma(0,s)&=\gamma(s)\\
\gamma(0)&=\omega\\
\gamma(s,x+1,r)&=\mathbb{S}(s,x;l(r)-1)[0]\\
\gamma(s,x,r)&=\sup(\bigcup_{k\lt\omega}\gamma(s,x[k],r))\iff\lim?(x)\\
\gamma(s,x+1)&=\min(\{k:k\not\in C(s,\gamma(s,x))\})\\
\gamma(s,x)&=\min(\{k:\forall n[n\lt x\Rightarrow k\ge\gamma(s,n)]\})\iff\lim?(x)
\end{align}\)