## 3 Results
Observe that (4) is a purely analytical relation, describing the requirement that \(G_{\text{dt}}\big{(}\mathfrak{a}_{t}\big{)}\) and \(G_{\text{ct}}\big{(}\mathfrak{a}_{t}\big{)}\) at each time \(t\in\mathbb{N}\) provide a sufficiently accurate approximation to the instantaneous properties of the physical channel. The \"true\" sequence \((\boldsymbol{f}_{\text{ph},t})_{t\in\mathbb{N}}\) of channel characteristics does _not_ need to consist of computable components. In the design process of a digital-twin system, such as that shown in Figure 1, the responsible engineer has to prove--based on mathematical modeling--that, during the system's operation, the sequence \((G_{\text{dt}}(\mathfrak{a}_{t}))_{t\in\mathbb{N}}\), \((G_{\text{ct}}(\mathfrak{a}_{t}))_{t\in\mathbb{N}}\), respectively, will satisfy (4). Recall that, by definition, both \(\mathfrak{X}_{1}\) and \(\mathfrak{F}_{1}\) are machine-readable languages for the set \(\mathcal{CB}^{1}_{\pi}\). Hence, according to Section 2, \(G_{\text{dt}}\) and \(G_{\text{ct}}\) each induce a mapping \(G_{\text{dt}}:\mathcal{A}\supseteq\to\mathcal{CB}^{1}_{\pi}\), \(G_{\text{ct}}:\mathcal{A}\supseteq\to\mathcal{CB}^{1}_{\pi}\), respectively. In the following, we assume \(G_{\text{dt}}\) and \(G_{\text{ct}}\) to be the same, and we denote \(G_{\text{dt}}(a_{t})=G_{\text{ct}}(a_{t})=:\boldsymbol{f}_{t}\). The system's design process will then include the choice between implementing \((\boldsymbol{f}_{t})_{t\in\mathbb{N}}\) using \((G_{\text{dt}}(\mathfrak{a}_{t}))_{t\in\mathbb{N}}\)--i.e., approximating \((\boldsymbol{f}_{\text{ph}})_{t\in\mathbb{N}}\) through discrete-time descriptions--or using \((G_{\text{ct}}(\mathfrak{a}_{t}))_{t\in\mathbb{N}}\)--i.e., approximating \((\boldsymbol{f}_{\text{ph}})_{t\in\mathbb{N}}\) through continuous-time descriptions--the implications of which we will analyze subsequently. Motivated by the generalized Shannon equivalence, the textbook approach considers discrete-time descriptions of bandlimited signals. As indicated before, the evident advantage of this paradigm consists of computational \"convenience\". In a simplified manner, the standard (abstract) engineering model--that is, without considering questions of computability, yet--for the wireless communication (sub)system from Figure 1 may look as follows. At time \(t\in\mathbb{N}\), the robot aims to transmit one of \(M\in\mathbb{N}\) messages to the receiving end, for which he employs an encoding scheme \(\mathcal{E}_{t}:\{1,\ldots,M\}\to\text{dom}(\boldsymbol{T}_{\infty})\), \(m\mapsto\mathcal{E}_{t}(m):=\boldsymbol{y}_{t,m}\). For reasons of simplicity, we summarize processes such as encoding (in the sense of information theory), channel precoding, modulation, and pulse shaping in this step. Setting \(x_{\text{ph},t}:=S_{1}f_{\text{ph},t}\), the signal at the receiving end is of the form  
\[y_{\text{re},t}:=\mathbf{w}+\left(x_{\text{ph},t}*y_{t,m}\right),\]  
where \(\mathbf{w}\in\ell_{0}^{\infty}\) is a sequence of additive noise-like disturbances, and \(x_{\text{ph},t}*y_{t,m}\) denotes the convolution of \(x_{\text{ph},t}\) and \(y_{t,m}\). The receiving end then employs a decoding scheme \(\mathcal{D}_{t}:\ell_{0}^{\infty}\to\{1,\dots,M\}\), \(y_{\text{re},t}\mapsto\mathcal{D}_{t}(y_{\text{re},t})\). Again, for reasons of simplicity, we summarize processes such as demodulation, filtering, and decoding (in the sense of information theory) in this step. Region-based decoding is a common way of implementing \(\mathcal{D}_{t}\), in which case \(\mathcal{D}_{t}\) is of the form  
\[\mathcal{D}_{t}(y_{\text{re},t}):=D\big{(}\arg\min_{n\in\{1,\dots,N\}}\big{\|} y_{\text{re},t}-y_{\text{de},t,n}\big{\|}_{\infty}\big{)}\]  
where \(y_{\text{de},t,1},\dots,y_{\text{de},t,N}\in\ell_{0}^{\infty},\ N\in\mathbb{N}\), is a list of reference signals and \(D:\{1,\dots,N\}\to\{1,\dots,M\}\) is a mapping that assigns each reference signal an inferred message.