# The New CAP Theorem on Blockchain Consensus Systems
## 5 Discussion
### Revealing the Intrinsic Constraints  
Building on these concepts, in this work we introduce the _new CAP theorem_ for distributed Blockchain systems. Through a uniform distributed consensus system consideration, we formally demonstrate that Consensus achievement (C), Autonomy (A), and entropic Performance (P) cannot be optimized simultaneously in the generic case. Our findings are further validated by the empirical benchmarking data from the prominent large-scale Blockchain systems, presented in Section 5.2. In this work, the theorem is formally proved through two inherent mutual exclusions:  
1. Between Autonomy (A) and Consensus achievement (C) In an attempt to optimize one of two, the other is sacrificed. This derives from Equations (4)-(6) in Section 3.1 and is demonstrated in Figure 4, Section 4.2. An attempt to optimize Autonomy, i.e., moving \(A\to 1\), would leave the nodes without any resources to serve the community: \(W\) moves close to 0, leading Consensus achievement to minimum, and the system degrades down to a set of isolated nodes. Again, trying to optimize Consensus (\(C\to 1\)), we would have to withhold resources from the atoms to serve the system, sacrificing Autonomy. This intrinsic constraint was revealed in this work by starting from two apparently independent starting points: while we define Autonomy in the micro-scale of the node, Consensus is defined in macro, with respect to the properties of the realm. This strengthens even more our initial hypothesis for the existence of an inherent intrinsic constraint among (\(A\)) and (C). 2. Between Consensus achievement (C) and entropic Performance (P) Consensus achievement is an energy-consuming process: it relies on data transmission, processing, and storage, all of which are known to be energy-consuming tasks. The constraint between (C) and (\(P\)) is revealed in Section 3.1 through Equations (5) and (10) and is demonstrated in Figure 5, Section 4.3. Again, trying to optimize one of the two, the other is forced away from optimal. Our consideration for the two properties has independent starting points as well: while \(C\) is defined with respect to the macroscopic traits of the system, \(P\) is defined based on Shannon information entropy principles. This further strengthens the finding of the inherent constraint among \(C\) and \(P\). The _entropic traits_ of distributed consensus systems are studied extensively in [14]. In this work, we extend the scope of [14] by applying the notion of energy consumption to our considerations. We treat _Consensus achievement_ as a distinct, self-contained process, and we exploit its entropic features to introduce a new metric on its efficiency, as the entropy reduction it induces to the system per unit of consumed energy. This paves the way for further research and development aimed at Blockchain optimization. Throughout this work, we acknowledge event witnessing as the core feature of the consensus process in its entirety [2; 8; 14]. Across the different architectures, event witnesses take various forms and names, i.e., validators, solvers, verifiers, pruned or full nodes, etc. Regardless of the terminology, the witnessing process underlies every consensus mechanism, reflecting the Aristotelian notions of _true_ and _existent_[1]. In this work, we exploit the architectural neutrality of the IoT micro-Blockchain framework to formalize (A), (C), and (P) with respect to the primitive traits of the distributed autonomous systems. Throughout this work, we prioritized simplicity and generality, focusing on fundamental universal principles rather than on the implementation details of the various systems. To keep the focus, in several points we opted for the simpler applicable approach. In this work, the _new CAP theorem_ is formally proved through the thorough analysis of a universal consensus system and validated through real-life systems' empirical data benchmarking.