Three basic axioms of fairness and logic make the math of decentralized finance inevitable.
April 23, 2026
Original Paper
From Swap Axioms to Weighted Geometric Means: A Characterization of AMMs
arXiv · 2604.16898
The Takeaway
Automated market makers like Uniswap use specific formulas to trade billions of dollars. Most people assume these equations were just clever inventions by programmers. This proof shows that these forms are the only possible outcome of three simple rules of logic. Validity invariance and Pareto efficiency force the math into these specific shapes. It reveals a hidden mathematical necessity behind the structure of modern financial markets. Finance is moving away from human whim and toward inescapable geometric laws.
From the abstract
Many automated market makers can be understood through the geometry of their trading orbits, the sets of states reachable from one another through swaps. In prominent designs, this geometry is captured by a simple closed-form invariant such as the constant product $xy$ in Uniswap or a weighted geometric mean $x^w y^{1-w}$ in Balancer.This paper explains why these forms arise by deriving them from three basic assumptions: validity invariance (swaps preserve the validity of states), Pareto efficie