## 1 Introduction
We employ the theory of [PERSON] machines and effective analysis, classifying our results in terms of digital twinning and the article's fundamental problem statement. Particularly, we provide formal definitions of the terms machine-readable languages and machine-readable descriptions, and discuss formal examples of relevant properties. After the mathematical part of the article, we provide a brief subsumption and interpretation of  
Figure 1: Application of digital twinning as discussed in [10]. (**Left**) A robot (physical entity) is moving on the floor of a laboratory environment. It is sequentially measuring its position relative to a fixed coordinate system and transmitting the relevant data through wireless communication to a receiving end. (**Middle**) The receiving end tracks the position and forms a virtual representation (digital twin) of the robot inside the room. It updates the virtual representation to match the physical robot whenever new information becomes available. The depicted image is thus a visualization of the robot’s instantaneous machine-readable description. (**Right**) Using the robot’s virtual representation, the receiving end computes the impulse response of the wireless communication link (the depicted implementation uses a ray-tracing approach), i.e., a sequence of samples representing a bandlimited signal. In essence, the impulse response forms a digital twin of the communication link. our results, together with some prospects of how they affect near-future digital information-processing technology. The remainder of the article is structured as follows. In Section 2, we provide some mathematical background on SSI, introducing the signal spaces \(\ell_{0}^{\infty}\), \(\ell_{0}^{1}\), \(\mathcal{B}_{0,\pi}^{\infty}\), and \(\mathcal{B}_{\pi}^{1}\), and formally establishing the (generalized) Plancherel-Polya Theorem in terms of the Banach-space operators \(\mathcal{S}_{\star}:\mathcal{B}(\star)\to\ell(\star)\) and \(\mathcal{T}_{\star}:=\mathcal{S}_{\star}^{-1}\). Applying the theory of Turing computability and effective analysis, we continue to develop a framework of machine-readable languages for \(\mathcal{B}_{0,\pi}^{\infty}\) and \(\mathcal{B}_{\pi}^{1}\). This framework formalizes the traditional theory of digital signal processing for communications engineering based on a mathematically rigorous notion of computability. Particularly, we define the machine-readable languages \(\mathfrak{X}_{1}\) and \(\mathfrak{X}_{\infty}\), which mirror the implicit quasi-standard in digital signal processing, and the machine-readable languages \(\mathfrak{F}_{1}\) and \(\mathfrak{F}_{\infty}\), which take the relevant signal's continuous-time behavior into account. In Section 3, we provide (for didactic purposes) a mathematical model of digital-twinning systems such as the one shown in Figure 1, marking the Banach-space norms \(\|\cdot\|_{\infty}\) and \(\|\cdot\|_{1}\) as a relevant property of signals \(\boldsymbol{f}\in\mathcal{B}_{0,\pi}^{\infty}\), \(\boldsymbol{f}\in\mathcal{B}_{\pi}^{1}\), respectively. Guided by the exemplary application case, we establish our main results: The (generalized) Plancherel-Polya Theorem does _not_ hold true on the algorithmic level. Depending on which of the established machine-readable languages we choose, we either can or cannot compute \(\|\boldsymbol{f}\|_{\infty}\), \(\|\boldsymbol{f}\|_{1}\), respectively, despite all languages determining the relevant signals uniquely in the (analytic) sense of the (generalized) Plancherel-Polya Theorem. Finally, Section 4 discusses several other signal properties our theory can analyze and closes the article by interpreting our results as indicated above.