## 3 Results
In essence, the first part is a corollary of Theorems 2 and 3, which follows by contradiction. Assume \(\mathrm{Id}_{\mathfrak{X},\mathfrak{F}}^{\star}\) is computable. By Theorem 2, the mapping \(\|\cdot\|_{\star}:\mathfrak{F}_{\star}\to\mathfrak{F}\), \((f,\Lambda_{\mathfrak{F}},[\cdot]_{\mathfrak{F}}^{\star})\mapsto(\|f\|_{\star},\Lambda_{\mathfrak{F}},[\cdot]_{\mathfrak{F}})\), is computable. Hence, by concatenation, we obtain the computable mapping  
\[\big{\|}\mathrm{Id}_{\mathfrak{X},\mathfrak{F}}^{\star}(\cdot)\big{\|}_{ \star}:\mathfrak{X}_{\star}\to\mathfrak{F},\ (f,\Lambda_{\mathfrak{X}},[\cdot]_{\mathfrak{X}}^{\star})\mapsto(\|f\|_{\star},\Lambda_{\mathfrak{F}},[\cdot]_{\mathfrak{F}}),\]  
contradicting Theorem 3. Thus, \(\mathrm{Id}_{\mathfrak{X},\mathfrak{F}}^{\star}\) cannot be computable. The second part is a consequence of the continuity of the sampling operator \(S_{\star}:\mathcal{B}(\star)\to\ell(\star)\). Particularly, there exist constants \(C_{\star}\in\{q\in\mathbb{Q}:\log_{2}(q)\in\mathbb{Z}\}\) such that \(\|S_{\star}f\|_{\star}<C_{\star}\|f\|_{\star}\) holds true for all \(\boldsymbol{f}\in\mathcal{B}(\star)\). Let \([(s^{\prime}_{n,m})_{n,m\in\mathbb{N}},\xi^{\prime\prime}]_{\mathfrak{F}}^{ \star}=T_{\star}\boldsymbol{x}\) be arbitrary. For all \(M\in\mathbb{N}\), we have  
\[\frac{1}{2^{M}}>\left\|T_{\star}\boldsymbol{x}-\sum_{n=0}^{\xi^{\prime}(M)}s ^{\prime}_{n,M}\cdot\operatorname{sinc}_{v(n)}\right\|_{\star}\geq\frac{1}{C _{\star}}\left\|\boldsymbol{x}-\sum_{n=0}^{\xi^{\prime}(M)}s^{\prime}_{n,M} \cdot\delta_{v(n)}\right\|_{\star}.\]  
Define \(K_{\star}:=\log_{2}C_{\star},\xi:\mathbb{N}\to\mathbb{N},M\mapsto\xi^{\prime }(M+K_{\star})\), and \((s_{n,m})_{n,m\in\mathbb{N}}:=(s^{\prime}_{n,m+K_{\star}})_{n,m\in\mathbb{N}}\), and observe that \((s_{n,m})_{n,m\in\mathbb{N}}\) is a computable double sequence of rational-complex numbers and \(\xi\) is a \(\mu\)-recursive function. For all \(M\in\mathbb{N}\), we have  
\[\left\|\boldsymbol{x}-\sum_{n=0}^{\xi(M)}s_{n,M}\cdot\delta_{v(n)}\right\|_{ \star}=\left\|\boldsymbol{x}-\sum_{n=0}^{\xi^{\prime}(M+K_{\star})}s^{\prime}_ {n,M+K_{\star}}\cdot\delta_{v(n)}\right\|_{\star}<\frac{C_{\star}}{2^{M+K_{ \star}}}=\frac{C_{\star}}{C_{\star}}\frac{1}{2^{M}}=\frac{1}{2^{M}}.\]  
Thus, the pair \(((s_{n,m})_{n,m\in\mathbb{N}},\xi)\) determines \(\boldsymbol{x}\) in the sense of Definition 5, and we have \([(s_{n,m})_{n,m\in\mathbb{N}},\xi]_{\mathfrak{X}}^{\star}=T_{\star} \boldsymbol{x}\). Further, by the SMN-Theorem (c.f. Section 2), the mapping \(((s^{\prime}_{n,m})_{n,m\in\mathbb{N}},\xi^{\prime})\mapsto((s_{n,m})_{n,m\in \mathbb{N}},\xi)\) is computable, which concludes the proof. For the languages \(\mathfrak{F}_{\star}\) and \(\mathfrak{X}_{\star}\), Theorem 4 corresponds to the Case 1 of the distinction made in Section 2: descriptions in the language \(\mathfrak{F}_{\star}\) contain more information than descriptions in the language \(\mathfrak{X}_{\star}\). In Section 2, we also indicated a link between the relationship of \(\mathfrak{F}_{\star}\) and \(\mathfrak{X}_{\star}\), i.e., the inequality \(\mathfrak{F}_{\star}\succ\mathfrak{X}_{\star}\), and the computability of the operators \(\mathcal{S}_{\star}\) and \(\mathcal{T}_{\star}\). In turn, whether \(\mathcal{S}_{\star}\) and \(\mathcal{T}_{\star}\) are computable is the formal rephrasing of whether the generalized Shannon equivalence holds true on the algorithmic level. Consider the set  
\[\mathcal{S}_{\star}(\mathcal{C}\mathcal{B}(\star)):=\big{\{}\boldsymbol{x} \in\mathcal{C}\ell(\star):\boldsymbol{x}=\boldsymbol{S}_{\star}\boldsymbol{ f}\text{ for some }\boldsymbol{f}\in\mathcal{C}\mathcal{B}(\star)\big{\}}\subset \mathcal{C}\ell(\star),\]  
i.e., \(\mathcal{S}_{\star}(\mathcal{C}\mathcal{B}(\star))\) consists of those sequences \(\boldsymbol{x}\in\mathcal{C}\ell(\star)\) that equate to some computable signal \(\boldsymbol{f}\in\mathcal{C}\mathcal{B}(\star)\) under the action of \(\mathcal{T}_{\star}\).