In periodic-review stochastic inventory control, replenishment policies must balance holding and backlogging costs while responding to uncertain demands and evolving forecasts that may be arbitrarily correlated and nonstationary. In this general information setting, Levi, P{á}l, Roundy, and Shmoys (Mathematics of Operations Research 2007) established that the fully adaptive optimum can be approximated within factor $2$, which had remained the best-known performance guarantee since then. Their marginal cost accounting and dual-balancing framework made this comparison possible without computing optimal decisions across future trajectories, opening a broad research program on approximation algorithms for stochastic inventory systems. We break this long-standing barrier for periodic-review stochastic inventory control, showing how to compete against the fully adaptive benchmark within factor $\frac{ e }{ e-1 } \approx 1.582$ while retaining general demand correlations and evolving forecasts. We attain these advances by discovering a structural connection between marginal inventory accounting and weighted stopping problems, where each unit's conditional holding charge forms a supermartingale, while postponing its order incurs a history-dependent backlog charge. To leverage the random thresholding approach of Ma and Tzamos (ICML 2023) in this context, we develop a potential-function analysis, leading to an $\frac{ e }{ e-1 }$-approximation with respect to any unit's stopping problem. Subsequently, our common-threshold policy coordinates these marginal decisions into a global replenishment policy without additional loss.

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