papers/hanzo-hmm/sections/16-proof-of-theorem-ref-thm-il.tex

\section{Proof of Theorem~\ref{thm:il}
}
\label{app:il_proof}

Consider an LP depositing at time $t_0$ with reserves $(q_0, p_0)$ such that $\mathcal{H}(q_0, p_0) = \kappa$. At time $t_1$, the price ratio is $r = \pi_1/\pi_0$.

For CPMM ($\lambda = 0$): $qp = \kappa$, $\pi = p/q$, so $q_1 = q_0/\sqrt{r}$, $p_1 = p_0\sqrt{r}$. Pool value $V_{\text{pool}} = q_1\pi_1 + p_1 = 2q_0\sqrt{r}\pi_0$. Hold value $V_{\text{hold}} = q_0\pi_1 + p_0 = q_0\pi_0(r + 1)$. Thus $\text{IL}_{\text{CPMM}} = 2\sqrt{r}/(1+r) - 1$.

For HMM ($\lambda > 0$): The curvature term penalizes deviations from $\bar{q}$, reducing the effective price impact. The reserve at $t_1$ satisfies:
\begin{equation}
q_1 = \bar{q} + (q_0 - \bar{q}) \cos(\omega t) + \text{lower order},
\end{equation}
where $\omega = \sqrt{\lambda/\kappa}$. The oscillatory correction reduces the net displacement from $q_0$, yielding the bound in Theorem~\ref{thm:il}. \qed