# The New CAP Theorem on Blockchain Consensus Systems
## 3 Analysis and Results
### Formal Definitions  
**Autonomy (A):** Autonomy describes the fraction of resources each node dedicates exclusively to managing its own transactions rather than validating external ones. Autonomy is formally defined here as the fraction of the memory of the node reserved for local events relative to its total memory, given by:  
\[A=\frac{M_{Local}}{M}\Rightarrow A=\frac{M_{Local}}{M_{Local}+M_{Others}} \tag{1}\]  
where \(M\) is the active memory of the node, \(M_{Local}\) the memory dedicated to the local events, and \(M_{Others}\) the memory used to store external events, as it is defined through Figure 2 and discussed in detail in [8]. With respect to the _witnessing requirements_ of the system, every event in the _local memory_ of the node must be witnessed by \(W\) siblings for the system to function. With respect to the uniformity of the nodes, this as well suggests that every node in the system witnesses on average the events existing in \(W\) local memories of other nodes, leading to:  
\[M_{Others}=M_{Local}\cdot W \tag{2}\]  
This quantifies the witnessing requirement and defines the fragment of \(M_{Others}\) to \(M_{Local}\) within each node with respect to the number of event witnesses \(W\) in our system. Since  
\[M=M_{Local}+M_{Others}\]  
we get  
\[M=M_{Local}+M_{Local}\cdot W\Rightarrow M_{Local}=\frac{M}{1+W} \tag{3}\]  
where again, \(M\) is the total active memory in a node and \(W\) the number of witnesses on each event. By substituting \(M_{Local}\) and \(M_{Others}\) in the Autonomy Formula (1) we get:  
\[A=\frac{\frac{M}{1+W}}{M}=\frac{1}{1+W},W\in(0,\mathrm{N}-1] \tag{4}\]  
where \(N\) is the total number of nodes in the realm. Equation (4) expresses Autonomy (A) as a function of the memory allocation with respect to W. It follows from the prerequisite that every event must be witnessed by \(W\) peers, thereby occupying a fraction of the node's memory. By substituting the witnessing requirement into the autonomy function in Equation (1), we derive the relationship between\(W\) and \(A\). Equation (4) indicates that as \(W\) increases, \(A\) decreases. This suggests that higher participation in consensus reduces node independence. **Consensus (C):** We define the _quantity of consensus_ as the proportion of the autonomous nodes that agree over an observable event every time. This corresponds to the average number of witnesses of each new event (\(W\)), with respect to the overall population of the nodes (\(N\)). It takes the form of a scalar ranging 1/\(N\), where no one but the introducer of the event witnesses it, to 1 where every node witnesses every event. This is analyzed in depth in [14]. Consensus (C) quantifies the agreement among nodes, measured as the proportion of nodes in the system actively witnessing each event, given by:  
\[C=\frac{W+1}{N},\ W\in(0,N-1] \tag{5}\]  
where again, \(W\) is the number of witnesses of each event and \(N\) the total number of nodes in the system. This equation macroscopically reflects the level of agreement among the nodes. The '+1' in the numerator accounts for the originating node, which always participates in consensus. As the number of witnesses \(W\) increases, the fraction approaches to 1, indicating a highly coherent network where a large number of nodes agree. \(C\) maximizes approaching 1 as \(W\) approaches \(N-1\), i.e., when all nodes that participate in the consensus process agree for every new event. \(C\) becomes minimal (\(C\rightarrow\frac{1}{N}\)) when \(W\to 0\), indicating a system tending to zero consensus, i.e., a system of isolated nodes. Substituting \(W\) in Equation (4) with respect to Equation (5), leads us to:  
\[A=\frac{1}{C\cdot N} \tag{6}\]  
which relates the Autonomy (A) with the collective Consensus (C) in a reverse analogy and reveals the constraint between \(A\) and \(C\). This equation expresses a fundamental tradeoff: if the system prioritizes high consensus, it inherently sacrifices node autonomy.