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.

翻译：在时序函数市场做市（TFMM）中，动态权重自动做市商（AMM）池通过创建一系列套利机会从初始持仓再平衡至最终持仓，其总成本取决于所采用的权重轨迹。我们证明每步套利损失是新旧权重向量之间的KL散度，这意味着Fisher-Rao度量是权重单纯形上自然的黎曼度量。在此KL成本主导阶展开下的损失最小化插值，即为Hellinger坐标$η_i = \sqrt{w_i}$中的球面线性插值（SLERP），亦即在单位球面正卦限上以恒定速度行进的一条测地线。SLERP中点等于先前研究（Willetts & Harrington, 2024）中（算术平均+几何平均）/归一化的启发式方法，因此该启发式解位于测地线上。此恒等式对任意数量的代币及任意幅度的权重变化均成立；利用这一关联，测地线上所有二分点均可通过递归的算术-几何平均二分法实现，无需三角函数运算。SLERP在完整KL成本上的相对次优性，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。