papers/lux-liquid-staking/sections/04-validator-selection-and-abstraction.tex

\section{Validator Selection and Abstraction}
\label{sec:validators}

\subsection{Validator Set Management}

The StakingPool delegates to a curated set of validators selected by governance:

\begin{algorithm}[H]
\caption{Validator Selection and Rebalancing}
\label{alg:validator}
\begin{algorithmic}[1]
\Require Candidate validators $\mathcal{V}_{\text{candidates}}$, selection criteria
\State $\mathcal{V}_{\text{active}} \gets$ top $K$ validators by score
\State Score $= w_u \cdot \text{uptime} + w_c \cdot (1 - \text{commission}) + w_d \cdot \text{decentralization}$
\For{each epoch}
    \State Compute target delegation per validator: $\delta_i = \mathsf{totalPooled} / |\mathcal{V}_{\text{active}}|$
    \For{each validator $v_i \in \mathcal{V}_{\text{active}}$}
        \State $\Delta_i \gets \delta_i - \text{currentDelegation}(v_i)$
        \If{$|\Delta_i| > \mathsf{rebalanceThreshold}$}
            \State Delegate or undelegate $\Delta_i$ LUX
        \EndIf
    \EndFor
\EndFor
\end{algorithmic}
\end{algorithm}

\subsection{Decentralization Constraints}

\begin{definition}[Delegation Cap]
No single validator receives more than $\mathsf{maxDelegationShare} = 10\%$ of the total staking pool:
\begin{equation}
    \forall\, v_i : \text{delegation}(v_i) \leq 0.10 \cdot \mathsf{totalPooled}
\end{equation}
\end{definition}

With a minimum of 10 validators, this ensures that compromise of any single validator affects at most 10\% of the pool.

\begin{proposition}[Decentralization Preservation]
If the LLS staking pool holds $P$ of total stake $S$, the Nakamoto coefficient satisfies:
\begin{equation}
    \mathsf{NC} \geq \left\lceil \frac{S/3}{0.10 \cdot P + \max_{j \notin \text{LLS}} s_j} \right\rceil
\end{equation}
\end{proposition}

For $P = 0.3S$ (LLS holds 30\% of stake): an attacker needs the entire LLS pool plus additional stake, requiring compromise of governance (multi-sig + timelock) rather than a single validator.