$$ \newcommand \Stake {\mathrm{Stake}} \newcommand \Units {\mathrm{Units}} \newcommand \floor [1]{\left \lfloor #1 \right \rfloor } \newcommand \MinBalance {b_{\min}} $$

Reward State

The reward state consists of three 64-bit unsigned integers:

• The total amount of money distributed to each earning unit since the genesis state \( T_r \),

• The amount of money to be distributed to each earning unit at the next round \( R_r\),

• The amount of money left over after distribution \( B^\ast_r \).

The reward state depends on:

• The address of the incentive pool \( I_\mathrm{pool} \),

• The functions \( \Stake(r, I_\mathrm{pool}) \)

• \( \Units(r) \).

These are defined as part of the Account State.

Informally, every \( \omega_r \) rounds, the rate \( R_r \) is updated such that rewards given over the next \( \omega_r \) rounds will drain the incentive pool, leaving it with the minimum balance \( \MinBalance \).

The rewards residue \( B^\ast_r \) is the amount of leftover rewards that should have been given in the previous round but could not be evenly divided among all reward units. The residue carries over into the rewards to be given in the next round.

The actual draining of the incentive pool account is described in the Validity and State Changes section.

More formally, let \( Z = \Units(r) \).

Given a reward state \( (T_r, R_r, B^\ast_r) \), the new reward state is \( (T_{r+1}, R_{r+1}, B^\ast_{r+1}) \), where:

• \( R_{r+1} = \floor{\frac{\Stake(r, I_{pool}) - B^\ast_r - \MinBalance}{\omega_r}} \) if \(R_r \equiv 0 \bmod \omega_r \) or \( R_{r+1} = R_r \) otherwise, and

• \( T_{r+1} = T_r + \floor{\frac{R_r}{Z}} \) if \( Z \neq 0 \) or \( T_{r+1} = T_r \) otherwise, and

• \( B^\ast_{r+1} = (B^\ast_r + R_r) \bmod Z \) if \(Z \neq 0\) or \( B^\ast_{r+1} = B^\ast_r \) otherwise.

A valid block’s reward state matches the expected reward state.