We show that when a dynamic-weight AMM rebalances by creating arbitrage opportunities, the per-step log loss is the KL divergence between successive weight vectors. The Fisher-Rao metric is therefore 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 $η_i = \sqrt{w_i}$: 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. Under driftless GBM prices, the fractional value loss from each weight update is price-independent, and the cross term between weight and price changes telescopes, so the constant-price geometry carries over. LVR exposure introduces a finite optimal step count $f^*$, which lies in the perturbative regime where SLERP remains near-optimal.

翻译：我们证明，当动态权重自动做市商通过创造套利机会实现再平衡时，每步对数损失是连续权重向量之间的KL散度。因此，Fisher-Rao度量是权重单纯形上的自然黎曼度量。在该KL代价主导阶展开下，损失最小化的插值方式是在Hellinger坐标$η_i = \sqrt{w_i}$下进行SLERP（球面线性插值）：即单位球正象限上以恒定速率遍历的测地线。SLERP中点等于先前工作中的(AM+GM)/归一化启发式方法（Willetts & Harrington, 2024），因此该启发式方法位于测地线上。该恒等式对任意数量的代币和任意幅度的权重变化均成立；利用这一联系，可通过递归AM-GM二分法在不借助三角函数的情况下到达测地线上的所有二分点。SLERP在全KL代价下的相对次优性与整体权重变化幅度的平方及$1/f^2$成正比，其中$f$为插值步数。在无漂移的几何布朗运动价格下，每次权重更新的分数价值损失与价格无关，且权重变化与价格变化的交叉项相互抵消，因此恒定价格下的几何性质得以保留。LVR暴露引入了最优有限步数$f^*$，该步数位于SLERP保持近似最优性的微扰区间内。