The Value of Foresight in iid Trading: Sharp Inventory Asymptotics and Gaussian Occupation

We determine the sharp first-order value of foresight in iid trading with limited inventory. A trader starts with k free units, has capacity B ≥ k, and observes nonnegative buying and selling quotes that may be arbitrarily dependent within a date. The worst optimal online-to-prophet ratio is 1 − C/√k + o(k^(-1/2)), uniformly over all excess capacities. The coefficient is a convex-Hamiltonian variational value. Its normalized generalized maximizer is unique, and the corresponding optimal drift has standard Gaussian expected time occupation. Pure-selling instances realize every finite Hamiltonian at prescribed superlinear horizons; more generally, iid multiple selection has the exact sampling-density coefficient C√(1 − k/T). We also characterize the necessary limits of actual worst-case quote sequences: replenishment disappears at the fluid scale, the normalized shadow price tends to 1/2, and excess capacity has only o(√k) value. Analytic bounds, an effective finite approximation, and continuous numerical certificate formulas complete the quantitative description.