papers/lux-tokenomics/sections/02-token-supply-model.tex

\section{Token Supply Model}
\label{sec:supply}

\subsection{Initial Distribution}

The Lux token (LUX) has a genesis supply of $S_0 = 500{,}000{,}000$ tokens distributed according to the allocation in Table~\ref{tab:distribution}.

\begin{table}[H]
\centering
\caption{Genesis token distribution for Lux Network.}
\label{tab:distribution}
\begin{tabular}{lrrp{5cm}}
\toprule
\textbf{Category} & \textbf{Allocation (\%)} & \textbf{Tokens} & \textbf{Vesting} \\
\midrule
Validator Rewards & 35.0 & 175,000,000 & Released per epoch over 10 years \\
Foundation Reserve & 15.0 & 75,000,000 & 4-year linear with 1-year cliff \\
Core Contributors & 15.0 & 75,000,000 & 4-year linear with 1-year cliff \\
Ecosystem Fund & 12.0 & 60,000,000 & Governed by DAO \\
Public Sale & 10.0 & 50,000,000 & 25\% at TGE, 12-month linear \\
Strategic Partners & 8.0 & 40,000,000 & 3-year linear with 6-month cliff \\
Community Airdrops & 5.0 & 25,000,000 & Distributed over 24 months \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Supply Dynamics}

Let $S_t$ denote the circulating supply at epoch $t$, $R_t$ the rewards emitted, $B_t$ the tokens burned from fees, and $V_t$ the tokens vested from locked allocations. The supply evolves as:
\begin{equation}\label{eq:supply}
    S_{t+1} = S_t + R_t + V_t - B_t
\end{equation}

\begin{definition}[Net Emission Rate]
The net emission rate at epoch $t$ is defined as $\eta_t = (R_t + V_t - B_t) / S_t$. The network is deflationary when $\eta_t < 0$.
\end{definition}

The vesting schedule $V_t$ is deterministic and decays to zero after all allocations unlock. The reward emission $R_t$ follows a decreasing schedule defined in Section~\ref{sec:staking}. The burn $B_t$ depends on transaction volume. The design target is $\eta_t < 0$ once vesting completes and network utilization exceeds a threshold.

\subsection{Maximum Supply and Asymptotic Behavior}

\begin{theorem}[Bounded Supply]\label{thm:bounded}
Under the reward schedule defined in equation~\eqref{eq:reward_schedule} and assuming transaction volume grows at rate $g > 0$ per epoch, the total supply $S_t$ is bounded above by $S_{\max} = 720{,}000{,}000$ LUX and converges to a stable equilibrium $S^* < S_{\max}$ as $t \to \infty$.
\end{theorem}

\begin{proof}
The reward emission $R_t = R_0 \cdot \alpha^t$ with decay factor $\alpha \in (0,1)$ forms a geometric series with sum $R_0 / (1 - \alpha)$. Vesting $V_t$ is finite and terminates at epoch $T_v$. The cumulative maximum supply is:
\begin{equation}
    S_{\max} = S_0 + \sum_{t=0}^{\infty} R_t + \sum_{t=0}^{T_v} V_t = S_0 + \frac{R_0}{1-\alpha} + V_{\text{total}}
\end{equation}
With $R_0 = 5{,}000{,}000$, $\alpha = 0.97$, and $V_{\text{total}} = 175{,}000{,}000$, we get $S_{\max} \approx 720{,}000{,}000$. Since $B_t > 0$ for all $t$ with positive transaction volume, the actual supply $S_t < S_{\max}$ for all $t$. For the convergence, observe that $R_t \to 0$ while $B_t$ grows with transaction volume, so there exists $t^*$ such that $B_t > R_t + V_t$ for all $t > t^*$, after which $S_t$ is strictly decreasing and bounded below by zero. The sequence converges by the monotone convergence theorem.
\end{proof}