User blog comment:SuperJedi224/Question/@comment-5029411-20141124132603/@comment-4297047-20141126173019

$$\varphi_{\Omega^\omega}(\alpha)$$ does enumerate those ordinals that are fixed points of $$\gamma\rightarrow\varphi(\gamma,\underset{\delta}{\underbrace{0,0,...,0}})$$ for all $$\delta<\omega$$, I know that much. It was the equivalence with the theta function that was just a guess.