# The New CAP Theorem on Blockchain Consensus Systems
## 2 Materials and Methods
### Consensus, Autonomy, and Entropic Performance  
The basic traits of Consensus (C), Autonomy (A), and entropic Performance (P) in the consideration of a universal consensus mechanism are given here to facilitate their formal definition later in Section 3.  
* _Consensus (C):_ Constitutes a collective attribute and can thus be defined only within a system of nodes. In its essence, consensus is a binary function: in the end, each node may either _agree_ or _disagree_ with _the others_ over an _observable event_.  
In its general form, consensus constitutes a metric of the awareness of \"how close\" the separate perceptions of the atoms in a realm are upon an _observable event_. If only infinitesimal events are considered, i.e., single-bit events like the opening of a door, the crossing of a temperature threshold, or the validity of a monetary transaction, then it is easy to identify consensus as \"a count of \"1's\". Consensus is identified as the fraction of the population that witnesses a specific event, meaning they are _aware of_ and _agree on_ it. In every case though, the outcome of consensus remains a binary agreement/disagreement flag: e.g., \"Does the hash produced in block X with nonce Y start with 5 zeros in a row?\" (yes/no). This is analyzed in [14].  
* _Autonomy (A):_ Autonomy is a metric of the independence of the node. It constitutes an atomic trait, defined here as the fraction of its memory each node can dedicate to serving itself. Atomic traits refer to properties defined in each individual node within the system. As mentioned earlier, at a fundamental level Autonomy generally reflects the fraction of the resources dedicated to processing and storing the node's local events.  
At an abstract level, in a peer-distributed system, every peer node operates in _conformity_ with its siblings. To achieve this, part of its resources is contributed to serving the others, the community. Conceptually seen, the existence and the coherency of the community itself rely on this tribute from the part of each peer node.  
In a peer, collective-consensus environment, autonomy is expected to increase when the required number of per event witnesses (\(W\)) is reduced. A fully independent node should operate consistently without the necessity of any event-sharing with _the others_.  
In a fully autonomous system, as witnessing tends to zero (\(W\)- \(>0\)), the node tends towards absolute autonomy (\(A\)- \(>1\)) allocating all its resources to serve itself. Still, to comply with the notion of _system_, \(W\), however small, can never be equal to zero, since this would lead to a system of totally isolated nodes with zero external awareness: a set of unconnected, lonely nodes.  
* _Entropic Performance (P):_ At the cost of running the consensus mechanism, the observed information entropy of an isolated autonomous system decreases as \(W\) increases. As the number of witnesses increases, nodes store more uniform information, reducing the overall system entropy. This effect is strongest when the number of witnesses is low, meaning that even a small increase in witnesses leads to a sharp drop in entropy as discussed in detail in [14].  
In this work, we introduce the _entropic performance_ of the system as the ratio of the observed _information entropy reduction_ to the overall _energy consumed_ to achieve it, with respect to \(W\).  
A literature review of the relevant works is presented in the next section.