# The New CAP Theorem on Blockchain Consensus Systems
## 3 Analysis and Results
### Formal Definitions
If every node is required to participate in the witnessing process of every event (\(C\to 1\)), then autonomy (\(A\)) is diminished, meaning that no node can dedicate its resources wholly to its own operations. Conversely, if a node fully prioritizes autonomy (\(A=1\)), then it does not contribute at all to the consensus process leading to (\(C\rightarrow\frac{1}{N}\)), which practically denotes a system of isolated nodes. **Entropic Performance (P):** This is defined here as the entropy drop (bits) occurring per unit of energy consumed in the system (Wh). The consensus process, while seen as a distinct mechanism from the perspective of the second thermodynamic law, is an entropy conversion mechanism. Information entropy \(H\) describes the uncertainty or disorder existing in a Blockchain system. Lower entropy means that the nodes have more similar information and comes with increased information redundancy. The entropic traits of the consensus process in peer-distributed systems have been discussed in detail in [14]. There, the information entropy in the system with respect to \(W\) is defined as:  
\[H=-\log_{2}\left(2^{-[N\times\frac{M}{1+W}]}\right) \tag{7}\]  
which simplifies to:  
\[H=\frac{N\times M}{1+W} \tag{8}\]  
where \(N\) is the number of peer nodes in the realm, \(M\) the active memory capacity of each node, and \(W\) the number of per-event witnesses. These equations illustrate how increasing the number of witnesses lowers the entropy of the system. As \(W\) increases, more nodes align their records, reducing the overall disorder and uncertainty, thereby decreasing the observable information entropy in the system. Equation (7) derives from the definition of Shannon information entropy, and is discussed in detail in [14]. Equation (8) connects the information entropy of the system to the number of per-event witnesses \(W\) and highlights that increasing the number of witnesses, proportionally reduces the information entropy of the system. The rate of the _information entropy reduction_ with respect to \(W\) is also given in [14] by the derivative of \(H\) over \(W\). The entropy reduction rate \(\frac{dH}{dW}\) indicates the rate at which uncertainty decreases as more nodes witness each event. It is given by:  
\[\frac{dH}{dW}=-\frac{N\cdot M}{\left(1+W\right)^{2}} \tag{9}\]  
where again, \(N\) is the number of the nodes, \(M\) the active memory of each node, and \(W\) the number of per-event witnesses in the system. Equation (9) designates that the _information entropy reduction_ in our system is steepest for low (still >0) values of \(W\). Let us now consider the energy required for introducing an additional witness of an event (\(E_{w}\)). **The exact value of the energy consumption does not affect the findings, the results, and the conclusions of this work. **  
The total energy invested in the system to acquire \(W\) event witnesses is given by:  
\[M_{Others}=M_{Local}\cdot W \tag{2}\]0  
This corresponds to the total amount of energy consumed in the system per event in order to support the consensus process and is equal to the energy consumed by all \(W\) witnesses of the event. Consequently, the smaller the energy consumption in a node per event witnessing (\(E_{w}\)), the more efficient the system is overall. Entropic Performance (\(P\)) measures how effectively the consensus mechanism reduces the uncertainty in the system per unit of energy consumed. Higher performance means more efficient use of energy in achieving consensus. Following, the entropic Performance of the consensus mechanism is defined as the occurring entropy drop \(dH\) in the system per unit of energy \(E_{w}\) consumed:  
\[M_{Others}=M_{Local}\cdot W \tag{2}\]1  
where again, \(E\) is the energy consumption per event witnessing, \(M\) the active memory of the node, \(N\) the number of nodes in the system, and \(W\) the number of per-event witnesses. This corresponds to the observed information entropy reduction occurring in the system per unit of energy consumed.