Beyond Peak Backlog: Conditional Energy and Temporal Geometry in Capacity-Constrained Delayed Bandit Optimization

What is the right delay complexity when a learner can track only C pending feedback items and discarded feedback is permanently lost? Existing one-point bandit convex optimization guarantees in this model pay √Tσmax, where σmax is the peak backlog, although unlimited tracking admits the sharper √dtot dependence on total delay. We introduce a scheduler-side conditional-energy interface that separates rate adaptation from the one-point perturbation filtration and handles the dependent importance weights created by randomized admission. Under the same semi-clairvoyant oracle and pathwise hard-capacity contract, this yields an untuned learner whose delay term scales as O(√EC dtot), with only an explicit restart factor EC; a public constant-factor peak bound removes this factor while dtot remains unknown. Under strong convexity, the same interface yields the temporal cost HA(d)=sumt σt/(A+t). Two delay vectors with identical delay multisets, dtot, σmax, and capacity can nevertheless have polynomially different minimax regret, showing that timing matters under curvature even when aggregate delay summaries agree. Finally, a continuous hard family converts tracking capacity into a zeroth-order query budget and gives a complementary capacity-starvation lower endpoint. The upper bounds require C≥ ln T+1 and do not constitute a complete capacity minimax characterization.