When an automated market maker changes its portfolio weights from one allocation to another, arbitrageurs extract value. The per-step loss isn't some complex interaction of supply curves and trader behavior—it's precisely the Kullback-Leibler divergence between the old and new weight vectors. This makes the space of possible allocations into a Riemannian manifold with a natural metric structure, and rebalancing paths become geometric objects you can measure and optimize.
The question becomes: what's the shortest path between two weight vectors? Not shortest in parameter space or time, but shortest in accumulated arbitrage loss. The answer is the geodesic under the Fisher-Rao metric, which in Hellinger coordinates corresponds to spherical linear interpolation—SLERP, the same curve used in computer graphics to smoothly rotate between orientations. The loss scales with the square of the angular distance and inversely with the number of steps. Take the direct geometric path and you minimize what you pay to traverse it.
This validates an earlier heuristic—AM-GM bisection—not through simulation or empirical fitting, but through the geometry of the space itself. The bisection method was approximating the geodesic all along, and now we know why it worked.
The deeper pattern is this: when you can express a cost as an information divergence, you've found a geometry. And once you have a geometry, optimization isn't about clever tricks—it's about not fighting the curvature of the space. The efficient path is the one the metric already specifies. This shows up everywhere: gradient descent on neural networks, Bayesian belief updates, thermodynamic processes. The geometry is already there. The question is whether you're walking along it or cutting across it.