Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers

Abstract: In Temporal Function Market Making (TFMM), a dynamic weight AMM pool rebalances from initial to final holdings by creating a series of arbitrage opportunities whose total cost depends on the weight trajectory taken. We show that the per-step arbitrage loss is the KL divergence between new and old weight vectors, meaning the Fisher--Rao metric is the natural Riemannian metric on the weight simplex. The loss-minimising interpolation under the leading-order expansion of this KL cost is SLERP (Spherical Linear Interpolation) in the Hellinger coordinates $η_i = \sqrt{w_i}$, i.e.\ a geodesic on the positive orthant of the unit sphere traversed at constant speed. The SLERP midpoint equals the (AM+GM)/normalise heuristic of prior work (Willetts & Harrington, 2024), so the heuristic lies on the geodesic. This identity holds for any number of tokens and any magnitude of weight change; using this link, all dyadic points on the geodesic can be reached by recursive AM-GM bisection without trigonometric functions. SLERP's relative sub-optimality on the full KL cost is proportional to the squared magnitude of the overall weight change and to $1/f^2$, where $f$ is the number of interpolation steps.