Description:
Bitcoin realised price function \(\mathbf{r}\) could be presented as a linear composition:
$$\mathbf{r} = \sum_{i \in I}w_ir_i,$$
where \(w_i\) - weight of class \(A_i\), \(r_i\) - realised price for class \(A_i\), \(i \in I=\{-3,-2,\ldots,14\}\). Multiplying two parts of equation with a circulating supply we get a composition for realised cap. A set \(A\) of all bitcoin addresses presents a union of disjoint classes
\(A=\cup_{i \in I}A_i \), where \(A_{-3}=\{a \in A: b(a) \in [0,0,125)\}\), \(A_{i}=\{a \in A: b(a) \in [2^{i-1}, 2^{i})\}\), \(i \in I \setminus \{-3,14\}\), \(A_{14}=\{a \in A: b(a) \in [2^{13},21 \cdot 10^{6}]\},\), \(b(a)\) is a balance of address \(a \in A.\)
Union \(\cup_{i=-3,-2,-1,0}A_i \) is a set of sat addresses (their balance is less than one bitcoin). The weight \(w_i\) of the class \(A_i, i \in I,\) is a ratio of the sum of balances of addreses in class \(A_i\) to a circulating supply. The weights are in the
[
chart],
the supply distrubution (weights multiplied with the circulating supply) in \(A_i\) classes is in the
[
chart].