User blog comment:MilkyWay90/My ordinal function/@comment-35470197-20180630144403/@comment-35470197-20180710044053

For the fixed version, you have \begin{eqnarray*} F(\beta_1,\ldots,\beta_i,a(n)) = F(\beta_1 + \cdots + \beta_i,a(n)) \end{eqnarray*} as I mentioned before. Therefore you obtain \begin{eqnarray*} f_{\alpha} \uparrow^2 2^{2^{\beta_1 + \cdots + \beta_i - 2}} < F(\beta_1,\ldots,\beta_i,f_{\alpha}) < f_{\alpha} \uparrow^2 2^{2^{\beta_1 + \cdots + \beta_i - 1}} \end{eqnarray*} and hence \begin{eqnarray*} f_{\omega + 2^{2^{\beta_1 + \cdots + \beta_i - 2}}}(n) < F(\beta_1,\ldots,\beta_i,a(n)) < f_{\omega + 2 * 2^{2^{\beta_1 + \cdots + \beta_i - 1}}}. \end{eqnarray*}

What is \(\varphi\) in your definition? Transfinitary Veblen hierarchy? If so, it does not make sense because it does not return ordinal numbers.