Forecasting many sparse series requires two choices: how much of each series' own past to retain, and how much to borrow from other series. We show that when one exponential recency operator is applied to item- and group-level statistics alike in a hierarchical empirical-Bayes hurdle model, the two become one decision: forgetting preserves the leverage of a shared prior while raising the noise floor against which fine cross-series structure must be resolved. We formalize this two-sided effect as a fitting-window diagnostic, a credibility bound and a resolution condition for candidate pools, and derive from it an ex ante screen for regimes where refinement cannot pay. On five public intermittent-demand panels comprising over 19,000 series, where item-level evidence is sparsest, the diagnostic is right in both directions. Where truncation lowers credibility the shared prior pays: switching it off costs $5\%$ and $12\%$ at the shortest histories on the two panels it helps most, against about $1\%$ at full length, and the gain sits in the occurrence block. On four panels the diagnostic reports at most $0.6\%$ of room for refinement and realized gains are no larger; on the one panel with room the learned mixture does not collect it, which locates the open problem in the partition objective. The resulting forecaster, EBB, stays within about $3\%$ of a per-series Gaussian-process specialist at fixed origin and within $1\%$ under walk-forward evaluation where both run, at two to three orders of magnitude lower cost.

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