The economic warehouse lot scheduling problem is a foundational inventory-theory model, capturing computational challenges in dynamically coordinating replenishment decisions for multiple commodities subject to a shared capacity constraint. Even though this model has generated a vast body of literature over the last six decades, our algorithmic understanding has remained surprisingly limited. Indeed, for general problem instances, the best-known approximation guarantees have remained at a factor of $2$ since the mid-1990s. These guarantees were attained by the now-classic work of Anily [Operations Research, 1991] and Gallego, Queyranne, and Simchi-Levi [Operations Research, 1996] via the highly-structured class of "stationary order sizes and stationary intervals" (SOSI) policies, thereby avoiding direct competition against fully dynamic policies. The main contribution of this paper resides in developing new analytical foundations and algorithmic techniques that enable such direct comparisons, leading to the first provable improvement over the $2$-approximation barrier. Leveraging these ideas, we design a constructive approach that allows us to balance cost and capacity at a finer granularity than previously possible via SOSI-based methods. Consequently, given any economic warehouse lot scheduling instance, we present a polynomial-time construction of a random capacity-feasible dynamic policy whose expected long-run average cost is within factor $2-\frac{17}{5000} + ε$ of optimal.