Spectral Liquidation Risk: Exact First Passage Times for DeFi Lending Protocols

We apply the spectral Fokker--Planck framework to the central risk problem in DeFi lending: when will a collateralized position be liquidated? In protocols such as Aave, Compound, and MakerDAO, a borrower deposits collateral (e.g., ETH) and borrows against it (e.g., USDC). If the collateral price drops below the liquidation threshold, the position is liquidated --- often with a 5--15\% penalty. The liquidation time is a first passage problem: the time at which the collateral-to-debt ratio first crosses the liquidation ratio. We show that: (1) the expected liquidation time is computable from a single matrix inverse: \(\mathbb{E}[\tau] = -\mathbf{1}^\top M_{\text{killed}}^{-1} A(0)\), where \(M_{\text{killed}}\) is the Fokker--Planck generator with absorbing boundary at the liquidation threshold; (2) the Gaussian approximation used by current risk dashboards (DeFi Saver, Gauntlet) underestimates liquidation probability by 1.8--3.2\(\times\) for crypto assets due to excess kurtosis (\(\kappa \approx 3\)--\(8\) for ETH, BTC); (3) the spectral method computes liquidation risk for 10,000 positions in under 1 second, enabling real-time risk dashboards; (4) optimal leverage can be computed analytically from the spectral gap: the maximum leverage before expected liquidation time drops below a target (e.g., 30 days). All convergence rates are proven dimension-free by the USRT (Nagy, 2026b). The method is a drop-in replacement for the Monte Carlo and Gaussian tools currently used in DeFi risk management.