papers/lux-omnichain-yield/sections/03-share-based-accounting.tex

\section{Share-Based Accounting}
\label{sec:share}
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\subsection{Token Model}

The Teleport yield system issues three classes of bridge tokens per underlying asset:

\begin{definition}[Bridge Token Classes]
For an underlying asset $A$:
\begin{enumerate}[nosep]
  \item \textbf{LA} (e.g., LETH): 1:1 wrapped token. Share price fixed at 1.0. No yield.
  \item \textbf{yLA} (e.g., yLETH): Yield-bearing token. Share price appreciates as yield accrues. Non-rebasing.
  \item \textbf{syLA} (e.g., syLETH): Shariah-compliant yield token. Same mechanics as yLA but restricted to halal-classified strategies~\cite{luxteleport}.
\end{enumerate}
\end{definition}

The yield-bearing variants follow the ERC-4626 tokenized vault standard~\cite{erc4626}, adapted for cross-chain operation where the vault (source chain) and the share token (destination chain) reside on different chains.

\subsection{Share Price Dynamics}

\begin{definition}[Share Price]
Let $B(t)$ denote the total backing assets on the source chain at time $t$, and $S(t)$ denote the total outstanding shares (yield-bearing bridge tokens) on all destination chains. The share price is:
\begin{equation}\label{eq:shareprice}
  p(t) = \frac{B(t)}{S(t)}
\end{equation}
\end{definition}

The backing $B(t)$ evolves as:
\begin{equation}\label{eq:backing}
  B(t) = B(t_0) + \underbrace{\int_{t_0}^{t} Y(\tau)\,d\tau}_{\text{yield accrued}} - \underbrace{\int_{t_0}^{t} F(\tau)\,d\tau}_{\text{fees extracted}} - \underbrace{\int_{t_0}^{t} W(\tau)\,d\tau}_{\text{withdrawals}}
\end{equation}
where $Y(\tau)$ is the instantaneous yield rate, $F(\tau)$ is the performance fee rate, and $W(\tau)$ is the withdrawal rate.

The share supply evolves as:
\begin{equation}\label{eq:supply}
  S(t) = S(t_0) + \underbrace{D(t_0, t)}_{\text{new deposits}} - \underbrace{R(t_0, t)}_{\text{redemptions}}
\end{equation}
where $D$ and $R$ are cumulative deposits and redemptions respectively.

\begin{theorem}[Share Price Monotonicity]\label{thm:monotone}
If the net yield rate after fees is non-negative, i.e., $Y(\tau) - F(\tau) \geq 0$ for all $\tau \in [t_0, t]$, and no withdrawals reduce backing below the collateralization threshold, then for any $t_1 < t_2$:
\begin{equation}
  p(t_2) \geq p(t_1)
\end{equation}
\end{theorem}

\begin{proof}
New deposits of amount $a$ at time $t$ simultaneously increase $B(t)$ by $a$ and increase $S(t)$ by $a/p(t)$ shares, leaving $p(t)$ unchanged. Redemptions of $s$ shares decrease $S(t)$ by $s$ and decrease $B(t)$ by $s \cdot p(t)$, also leaving $p(t)$ unchanged. Between deposits and redemptions, only yield and fees affect the price:
\begin{equation}
  \frac{dp}{dt} = \frac{Y(t) - F(t)}{S(t)} \geq 0
\end{equation}
Since $S(t) > 0$ (the token has nonzero supply) and $Y(t) - F(t) \geq 0$ by assumption, $dp/dt \geq 0$, so $p$ is monotonically non-decreasing.
\end{proof}

\subsection{Conversion Functions}

Following ERC-4626 semantics with virtual offset for first-depositor protection:

\begin{align}
  \texttt{convertToShares}(a) &= \left\lfloor \frac{a \cdot (S(t) + V_S)}{B(t) + V_A} \right\rfloor \label{eq:toShares}\\[4pt]
  \texttt{convertToAssets}(s) &= \left\lfloor \frac{s \cdot (B(t) + V_A)}{S(t) + V_S} \right\rfloor \label{eq:toAssets}
\end{align}

where $V_S = V_A = 10^8$ (\texttt{VIRTUAL\_SHARES} and \texttt{VIRTUAL\_ASSETS}) provide production-grade first-depositor inflation attack protection. The matched pair ensures the initial \texttt{pricePerShare} and \texttt{exchangeRate} start at 1:1 with virtual offset applied consistently across all share price calculations. Rounding is always in favor of the vault (floor for shares on deposit, floor for assets on withdrawal), preventing rounding-based extraction attacks~\cite{erc4626}. Per-operation \texttt{forceApprove} (approve to zero then to exact amount) eliminates infinite approval risks; each strategy interaction approves only the exact amount needed for that operation.

\subsection{Cross-Chain State Synchronization}

The share price on destination chains is updated through MPC-attested backing reports (Section~\ref{sec:attestation}). Between attestations, the share price on the destination chain is stale by at most one attestation interval $\Delta$:

\begin{equation}
  |p_{\text{dest}}(t) - p_{\text{source}}(t)| \leq \frac{Y_{\max} \cdot \Delta}{S(t)}
\end{equation}

where $Y_{\max}$ is the maximum yield rate across all active strategies. For $\Delta = 1$ hour and $Y_{\max}$ corresponding to 30\% APY, the maximum staleness on a \$100M vault is approximately 0.0034\%, which is negligible relative to bridge fees.

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