Where $\mu, \beta, \gamma, \alpha,$ and $\kappa$ are ordinals and $\alpha, \kappa > 0$ does there exist a function $\phi$ such that $\phi(\alpha) = (\mu, \beta, \gamma)$ where:
1) $\mu, \beta, \gamma < \alpha$, and
2) $\alpha \neq \kappa \implies \phi(\alpha) \neq \phi(\kappa)$
We can restrict all variables to being less than $\omega_1$ (or any arbitrary regular uncountable cardinal) so as to apply Fodor's lemma if that helps, though we are able to consider $\phi$ as a class function too.
Where there are $\omega_1^3$ triplets $(\mu, \beta, \gamma)$ that can be comprised from the elements of $\omega_1$ given $\mu, \beta, \gamma < \omega_1$, we generally have $\phi : \omega_1 \rightarrow \omega_1^3$ (or, if a class function, $\phi: Ord \rightarrow Ord^3$). This is a little different than Fodor's lemma, which considers only $\phi : \omega_1 \rightarrow \omega_1$. That is not to say Fodor's lemma may not prove valuable in answering the question, as the cardinality of $\omega_1$ and $\omega_1^3$ are equal, etc.
I personally am asking because I am working with different models for what I have been calling $T$ sequences (sequences generated by starting with some initial finite segment of ordinals and then considering all the triplets, quadruplets, quintuplets, and so on, that can be made from the initial segment so as to generate rules that add additional ordinals to the sequence based on the availability of the triplets, quadruplets, etc., over $\omega$ initial segments generated using an iterative process).