\[\textbf{Abstract}\]

Define $\mathcal{Q}(\alpha)$ as a function that returns the minimal element of the set of all Gödel numbers $\ulcorner \mathcal{M} \urcorner$ of Turing machines $\mathcal{M}$ such that $\mathcal{M}$ computes a computable well-ordering $\prec_s$ on a recursive subset $s \subseteq \mathbb{N}$, and $(s, \prec_s)$ has the same order type as the recursive ordinal $\alpha$. Define $\mathcal{W}(i)$ as the function such that $\mathcal{W}(\mathcal{Q}(\alpha)) = \alpha$. Now define a fundamental sequence

\[

\mathcal{E}(i,j) =

\begin{cases}

    i & \text{if } \mathcal{W}(i) = 0 \\

    \mathcal{W}^{-1}(\operatorname{maxel}(\mathcal{W}(i))) & \text{if } \exists \alpha : \mathcal{W}(i) = \alpha+1 \\

    \mathcal{W}^{-1}(\{0\} \cup \mathrm{maxel}(\{\gamma \in \mathcal{W}(i) \mid \mathcal{W}^{-1}(\gamma) < j\}) + j) & \text{if } \neg \exists \alpha : \mathcal{W}(i) = \alpha+1

\end{cases}

\]

where \(\textrm{maxel}(\alpha) & = & \beta \text{ such that } \beta \in \alpha \land \forall x \in \alpha(x \leq \beta)\)

Then, define a function

\[

\mathcal{R}(i,j) =

\begin{cases}

    i & \text{if } \mathcal{W}(i) = 0 \\

    \sum_{k=0}^{j} \mathcal{R}^k(\mathcal{E}(i,k),k) & \text{if } \mathcal{W}(i) \neq0 \\

\end{cases}

\]

Finally, \(\textrm{13960 : s2x2qb2d4jv ; 5.3.0-6-g76ae}\) is equal to \[\max\{\mathcal{R}(k,12^{13960}) \mid k \in \mathbb{N} \ \land k \leq i\}\]