papers/hanzo-hmm/sections/12-security-analysis.tex

\section{Security Analysis}

\label{sec:security}

\subsection{Flash Loan Attacks}

HMM's continuous-time dynamics, discretized via symplectic integrators, prevent atomic swaps from exploiting price manipulation. The minimum block time of 2 seconds ensures that the integrator completes at least one full step between any two swaps by the same address. Risk fees scale superlinearly with swap size, making large flash-loan-funded trades prohibitively expensive.

\begin{theorem}[Flash Loan Resistance]
For a flash loan of size $L$ and HMM with risk fee coefficient $f_r$, the maximum extractable profit is bounded by:
\begin{equation}
\Pi_{\text{flash}} \le \frac{L^2}{4\|\bm{q}\|^2} \cdot \left(\frac{1}{1 + f_r \cdot L/\|\bm{q}\|}\right) - 2f_m \cdot L,
\end{equation}
which is negative for $L > L^* = \|\bm{q}\| \cdot (1 - 2f_m)/(2f_r)$.
\end{theorem}

\subsection{Oracle Manipulation}

By design, HMM uses no external price feeds for core pricing. The invariant $\mathcal{H} = \kappa$ determines prices endogenously from reserve states. Optional TWAP oracles are used only for cross-chain settlement and employ a 30-minute averaging window that makes manipulation economically infeasible.

\subsection{Sybil Resistance}

Workers must stake collateral proportional to their declared capacity. The staking requirement is:
\begin{equation}
\text{Stake}_j = \gamma \cdot \sum_i q_{ij}^{\text{raw}} \cdot \pi_i,
\end{equation}
where $\gamma = 1.5$ is the collateralization ratio. Slashing conditions include: failed attestation ($q_j < 0.5$ for 7 consecutive epochs), timeout (no heartbeat for 10 minutes), and malicious behavior (detected by PoAI cross-validation).

\subsection{Formal Verification}

We verified the following properties using the Dafny formal verification language:
\begin{enumerate}[leftmargin=1.4em]
    \item \textbf{Invariant preservation}: $\forall$ valid swaps, $|\mathcal{H}' - \kappa| < \epsilon$.
    \item \textbf{Non-negative reserves}: $q_i > 0 \wedge p_i > 0$ after every operation.
    \item \textbf{Fee positivity}: $f(\Delta\bm{q}) > 0$ for all non-trivial swaps.
    \item \textbf{Monotonic shares}: LP shares increase monotonically with deposit size.
\end{enumerate}