Toxicity Bounds for Dynamic Liquidation Incentives

We derive a slippage-aware toxicity condition for on-chain liquidations executed via a constant-product automated market maker (CP-AMM). For a fixed (constant) liquidation incentive $i$, the familiar toxicity frontier $\nu <1\mathrm{/}(1+i)$ tightens to $\nu <1\mathrm{/}((1+i)\lambda )$ for a liquidity penalty factor $\lambda$ that we derive for both the CP-AMM and a generalised form. Using a dynamic health-linked liquidation incentive $i(h)=i(1-h)$, we obtain a state-dependent bound and, at the liquidation boundary, a liquidity-depth-only condition $\nu <1\mathrm{/}\lambda$. This reconciles dynamic incentives with the impact of the CP-AMM price and clarifies when dynamic liquidation incentives reduce versus exacerbate spiral risk.