The version of this question that does not assume CH is located here:
Question Involving Strength of Fodor's Lemma, No CH Assumption
Question:
Based on the definitions below for function $t$ and the functions $(\phi_{\alpha})_{2\leq\alpha<\omega_1}$ and assuming CH, must there be a minimum element $\kappa \in \omega_1$ where $\kappa \in \phi_{\omega}(\kappa)$?
Note that there is some $\kappa \in \omega_1$ where $\kappa \in \phi_{\alpha}(\kappa)$ for each $\alpha < \omega$ due Fodor's lemma (Fodor would have each element of $\{ \kappa : \kappa \in \phi_{\alpha}(\kappa) \}$ map to a constant for each $\alpha$ in $\omega$ should $\phi_{\alpha}$ be fully regressive instead of just almost regressive. See the comments below for clarification on the use of those terms here).
Define $t(\alpha)$ and $t^{-1}(\alpha)$ for any ordinal $\alpha \geq 2$:
Let $t(\alpha)$ equal a doublet of ordinals $(a,b)$ if $\alpha = 2$, a triplet of ordinals $(a,b,c)$ if $\alpha = 3$, a quadruplet of ordinals $(a,b,c,d)$ if $\alpha = 4$, and so on, for any ordinal $\alpha$. Similarly, let $t^{-1}(x)$ equal $2$ for any doublet of ordinals $x$, $3$ for any triplet of ordinals $x$, $4$ for any quadruplet of ordinals $x$, and so on, as determined by the order type of $x$.
Use of set builder notation:
Consider the set of all doublets of ordinals such that each element of each doublet is a member of $\omega_1$. It will become helpful to use set-builder notation in the following manner to define such a set: $$\{t(2) : a,b \in t(2) \implies a,b \in \omega_1 \} = \{ (a,b) : a,b \in \omega_1 \} = \{(0,0),(0,1),(1,0),(a \in \omega_1,b \in \omega_1),\dots\}$$
Define the functions $(\phi_{\alpha})_{2 \leq \alpha < \omega_1}$:
Let each element of $(\phi_{\alpha})_{2 \leq \alpha < \omega_1}$ be almost regressive such that:
1) $$\phi_{\alpha} : \omega_1 \setminus \{0\} \rightarrow \{ t(\alpha) : a,b,c,\dots \in t(\alpha) \implies a,b,c,\dots < \omega_1 \} \text{ is bijective},$$
2) $$a,b,c,\dots \leq \kappa \, \text{for each } a,b,c,\dots \in \phi_{\alpha}(\kappa), \, \text{and}$$
3) $$\zeta < \alpha \implies min\{ \phi_{\zeta}^{-1}(b) : \exists k \in b \text{ where } k \geq \phi_{\zeta}^{-1}(b)\} < min\{ \phi_{\alpha}^{-1}(b) : \exists k \in b \text{ where } k \geq \phi_{\alpha}^{-1}(b)\}.$$
Background as to question: I use the functions $(\phi_{\alpha})_{2\leq\alpha<\omega_1}$ to classify the ordinals less than $\omega_1$ in a paper I've written. I am looking to confirm my understanding. I am also curious if I need to (or should) refine my definition of function $t$ given that my notation may be less than ideal.