papers/hanzo-hmm/sections/10-liquidity-provision.tex

\section{Liquidity Provision}

\label{sec:liquidity}

\subsection{LP Shares}

Liquidity providers deposit resource-credit pairs $(\Delta q_i, \Delta p_i)$ and receive pool shares:
\begin{equation}
s = \left(\prod_{i=1}^n \left(\frac{\Delta q_i}{q_i}\right)^{w_i}\right) \cdot S_{\text{total}},
\end{equation}
where $S_{\text{total}}$ is the total outstanding share supply. This weighted geometric mean ensures that LPs receive shares proportional to their contribution across all resource dimensions.

\subsection{Fee Distribution}

Fees accrue to LPs proportionally to their share of the pool:
\begin{equation}
\text{Fee}_{\text{LP}} = \frac{s}{S_{\text{total}}} \cdot \sum_{\text{swaps}} f(\Delta\bm{q}).
\end{equation}

We implement a time-weighted fee distribution that rewards long-term LPs:
\begin{equation}
\text{Fee}_{\text{LP}}^{\text{tw}} = \frac{s \cdot \tau}{S_{\text{total}} \cdot \bar{\tau}} \cdot \sum_{\text{swaps}} f(\Delta\bm{q}),
\end{equation}
where $\tau$ is the LP's holding duration and $\bar{\tau}$ is the average across all LPs.

\subsection{Expected Free Energy Routing}

We route liquidity toward high-information-gain tasks using an active inference framework~\cite{friston2017active}:
\begin{equation}
\eta_\pi = \frac{\exp(\beta \cdot \text{EFE}(\pi))}{\sum_{\pi'} \exp(\beta \cdot \text{EFE}(\pi'))},
\end{equation}
where $\text{EFE}(\pi) = \mathbb{E}[\Delta I + \Delta U - \lambda_c \cdot \text{cost}]$ is the expected free energy of policy $\pi$, $\Delta I$ is information gain, $\Delta U$ is utility, and $\lambda_c$ is cost sensitivity. This biases the market toward funding compute for tasks that maximize information gain per unit cost.