## 3 Results
To this end, assume the mapping \(\|\cdot\|_{\star}:\bar{\mathfrak{x}}_{\star}\to\mathfrak{R}\), \((f,\Lambda_{\bar{\mathfrak{x}}},[\cdot]_{\mathfrak{R}}^{\star})\mapsto(\|f\|_{ \star},\Lambda_{\mathfrak{R}},[\cdot]_{\mathfrak{R}})\) is computable and let \((\mathbf{x}_{k})_{k\in\mathbb{N}}\) be a sequence that satisfies Lemma 1, Lemma 2, respectively, for some recursively enumerable _nonrecursive_ set \(\Omega\subset\mathbb{N}\). Then, there exists a computable mapping  
\[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]5  
such that for all \(k\in\mathbb{N}\), the pair \(((s_{n,m}(k))_{n,m\in\mathbb{N}},\xi_{k})\) determines \(\mathbf{x}_{k}\) in the sense of Definition 5, and we have  
\[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]6  
If the mapping \(\|\cdot\|_{\star}:\bar{\mathfrak{x}}_{\star}\to\mathfrak{R}\), \((f,\Lambda_{\bar{\mathfrak{x}}},[\cdot]_{\mathfrak{R}}^{\star})\mapsto(\|f\|_{ \star},\Lambda_{\mathfrak{R}},[\cdot]_{\mathfrak{R}})\) is indeed computable, there must also exist a computable mapping \(((s_{n,m}(k))_{n,m\in\mathbb{N}},\xi_{k})\mapsto((r_{m}(k))_{m\in\mathbb{N}},\xi_{k}^{\prime}),k\in\mathbb{N}\) such that, for all \(k\in\mathbb{N}\), we have  
\[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]7By concatenation, we conclude that \(k\mapsto\|f_{k}\|_{\star}\), \(k\in\mathbb{N}\) is computable as well. For all \(k\in\mathbb{N}\), we define  
\[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]8  
Then, \(r_{<}(k)\) is a rational number, and the mapping \(k\mapsto r_{<}(k)\), \(k\in\mathbb{N}\), is computable. For all \(k\in\mathbb{N}\), we further have \(r_{<}(k)<\lim_{m\to\infty}r_{m}(k)<r_{<}(k)+1\) by construction, and thus, \(k\in\Omega\Leftrightarrow r_{<}(k)>0\) by the requirements of Lemma 1, Lemma 2, respectively. We define  
\[g:\mathbb{N}\to\mathbb{N},k\mapsto g(k):= \begin{cases}1,&\text{if }r_{<}(k)>0,\\ 0,&\text{otherwise},\end{cases} \]  
and observe that, we have \(g=\mathds{1}_{\Omega}\). By the SMN-Theorem (c.f. Section 2), \((k\mapsto g(k))\) is computable, i.e., \(g\) is a \(\mu\)-recursive function. Accordingly, \(\Omega\) is recursive, which contradicts the prerequisite of \(\Omega\) being _nonrecursive_. We have \(\mathfrak{F}_{\star}\succ\mathfrak{X}_{\star}\). In the sense of Section 2, the inequality is strict. Proof. : Denote by \(\mathrm{Id}_{\mathfrak{X},\mathfrak{F}}^{\star}:\mathfrak{X}_{\star}\to \mathfrak{F}_{\star}\) and \(\mathrm{Id}_{\mathfrak{F},\mathfrak{X}}^{\star}:\mathfrak{F}_{\star}\to \mathfrak{X}_{\star}\) the relevant identity mappings in the sense of Section 2. That is, we have  
\[\mathrm{Id}_{\mathfrak{X},\mathfrak{F}}^{\star}\big{(}f,\Lambda_{ \mathfrak{X}},[\cdot]_{\mathfrak{X}}^{\star}\big{)} =\big{(}f,\Lambda_{\mathfrak{F}},[\cdot]_{\mathfrak{F}}^{\star} \big{)},\] \[\mathrm{Id}_{\mathfrak{F},\mathfrak{X}}^{\star}\big{(}f,\Lambda_{ \mathfrak{F}},[\cdot]_{\mathfrak{F}}^{\star}\big{)} =\big{(}f,\Lambda_{\mathfrak{X}},[\cdot]_{\mathfrak{X}}^{\star} \big{)}\]  
for all \(\boldsymbol{f}\in\mathcal{CB}(\star)\). We divide the proof in two parts: first, we prove that \(\mathrm{Id}_{\mathfrak{X},\mathfrak{F}}^{\star}\) is _not_ computable; second, we prove that \(\mathrm{Id}_{\mathfrak{F},\mathfrak{X}}^{\star}\) is computable.