papers/lux-validator-economics/sections/02-system-model.tex

\section{System Model}
\label{sec:model}

\subsection{Validators and Delegators}

Let $\mathcal{V} = \{v_1, \ldots, v_n\}$ be the set of validators and $\mathcal{D} = \{d_1, \ldots, d_m\}$ the set of delegators. Each validator $v_i$ has:
\begin{itemize}
    \item Self-stake $s_i^{\text{self}} \geq s^{\min}_{\text{val}} = 2{,}000$ LUX
    \item Delegated stake $s_i^{\text{del}} = \sum_{d \in \mathcal{D}_i} s_d$
    \item Total stake $s_i = s_i^{\text{self}} + s_i^{\text{del}}$
    \item Commission rate $c_i \in [c^{\min}, c^{\max}] = [0.05, 0.25]$
    \item Uptime factor $u_i \in [0, 1]$
    \item Appchain set $\mathcal{K}_i \subseteq \mathcal{K}$ (appchains validated)
    \item Hardware cost $h_i$ per epoch
\end{itemize}

Each delegator $d$ has stake $s_d \geq s^{\min}_{\text{del}} = 25$ LUX delegated to exactly one validator.

\begin{definition}[Stake Cap]
Validator $v_i$'s effective stake is capped at:
\begin{equation}
    s_i^{\text{eff}} = \min(s_i, \omega \cdot s_i^{\text{self}})
\end{equation}
where $\omega = 15$ is the maximum delegation multiplier. This prevents a validator with minimal self-stake from attracting disproportionate delegation.
\end{definition}

\subsection{Epoch Structure}

Time is divided into epochs of $E = 8{,}640$ blocks (approximately one day at 10-second block times). Rewards are computed and distributed at epoch boundaries. Validator set changes (joins, exits, stake updates) take effect at the next epoch boundary after a registration delay of one epoch.

\subsection{Primary Network and Appchains}

The primary network validates the P-Chain (platform), X-Chain (exchange), and C-Chain (contracts). Appchains are independent chains that share validators with the primary network. A validator must stake on the primary network to validate any appchain.

\begin{definition}[Multi-Appchain Validator]
A validator $v_i$ validating appchains $\mathcal{K}_i$ earns rewards from each:
\begin{equation}
    r_i^{\text{total}} = r_i^{\text{primary}} + \sum_{k \in \mathcal{K}_i} r_{i,k}^{\text{appchain}}
\end{equation}
where $r_i^{\text{primary}}$ is the primary network reward and $r_{i,k}^{\text{appchain}}$ is the reward from appchain $k$.
\end{definition}