Discrepancy-Rounded Fair Bandits with Static and Time-Varying Exposure Floors

Minimum-exposure constraints arise in recommendation, content curation, and regulated allocation when each provider, arm, or group must receive guaranteed exposure inside a period rather than only in aggregate. We study stochastic bandits with exact exposure floors and show that the right object is a rounding problem: a fractional fair schedule is realized as integral pulls, and the exposure error is exactly a discrepancy vector. The main contribution is a blockwise model with time-varying floors. BDQ-UCB satisfies every block floor deterministically and has fair regret governed by the nonmandatory budget R, not the horizon T, with high-probability regret O(\sqrt{KR\log(KT)}). A MOSS residual variant attains O(\sqrt{KR}), and a matching lower bound gives the minimax rate Θ(\sqrt{KR}), even with positive mandatory exposure; a kl-UCB^{++} residual rule adds instance-dependent optimality. The formulation becomes essential for overlapping group floors: per-arm rounding can violate a group constraint by Ω(s) in the group size, whereas Beck--Fiala null-space rounding meets every group floor within the block budget with violation below the arm degree t, and composes with UCB at the same R-parametrized regret. For learned group plans, we close disjoint systems at \widetildeΘ(\sqrt{KT}), give a dual-ledger decomposition explaining why naive index rules fail under overlap, and prove a plan-sampling rule that is pathwise feasible under an initial cover-slack condition and attains a conditional \widetilde O(\sqrt{KT}) guarantee, leaving the condition-free overlap rate open. Experiments on synthetic floors, MovieLens-100k genre exposure, and deployment stress tests show exact feasibility without penalty tuning and regret competitive with tuned Lagrangian baselines.