Constant Product Market Makers use fees that are typically fixed proportions of trade size. When these fees are automatically reinvested into the pool, as in Uniswap~V2 and some designs of Uniswap V4, the final state after a trade can depend on how the trade is split into smaller transactions. This path dependence complicates the risk assessment for liquidity providers and affects composability guarantees. We characterize the functional class of fee structures that ensure path independence: the combined fee factor must depend only on the current pool invariant k=xy. For this class, we derive a system of ordinary differential equations governing pool dynamics and obtain a closed-form integral exchange formula. Within this class, we construct a parametric family of fee functions that achieve zero Impermanent Loss for a given initial pool state, and prove that no universal fee function can eliminate Impermanent Loss for all initial states simultaneously. We analyze implications for arbitrage windows and slippage, and validate our theory through controlled simulations. Our framework provides protocol designers with a principled approach to fee optimization that aligns liquidity provider and trader incentives while preserving composability.

翻译：恒定乘积做市商通常采用交易规模固定比例的费用。当这些费用被自动再投资到流动性池中时（如Uniswap V2及Uniswap V4的部分设计），交易完成后的最终状态可能取决于该笔交易被拆分为多笔子交易的方式。这种路径依赖性增加了流动性提供者的风险评估难度，并影响组合性保证。我们刻画了能确保路径无关性的费用函数类：组合费用因子必须仅依赖于当前池的恒定乘积 k=xy。针对该函数类，我们推导出支配池动态的常微分方程组，并得到闭合形式的积分交换公式。在此函数类中，我们构造了一个参数化费用函数族，能在给定初始池状态下实现零无常损失，并证明不存在能同时消除所有初始状态下无常损失的通用费用函数。我们分析了该框架对套利窗口和滑点的影响，并通过受控仿真验证了理论结果。本研究为协议设计者提供了在保持组合性的同时协调流动性提供者与交易者激励的费用优化原则性方法。