KakeyaShare: Incidence-Structured Query Sharing for Simultaneous Directional Risk Certification

Estimating local model risk along several directions can require many black-box evaluations. Overlapping neighborhoods offer reusable observations, yet exploiting them while controlling every directional error requires more than pooling queries. We introduce KakeyaShare, a simultaneous auditor for bounded loss averages over tubes in a three-dimensional input chart. The method groups locations by their exact tube memberships, so one estimated intersection mean contributes to all member directions. A geometry-only criterion selects a budget-feasible core, residual regions preserve the remaining mass, and an exact integer allocation distributes evaluations across the resulting strata. Independent volume measurements and weighted concentration yield simultaneous finite-sample intervals, while a finite-incidence bound relates the ideal sharing gain to mean tube multiplicity. At the main geometry, interpolation gives about 33.5% fewer queries than independent Monte Carlo at matched radius and 95% simultaneous confidence. Independent reference evaluation shows that median dataset RMSE decreases from 0.00619 to 0.00484; controlled ablations and cross-predictor execution expose the effects of allocation, geometry uncertainty, and rejection cost. These results show how finite tube incidence supports query reuse in simultaneous certification of local risk averages.