Exact service-risk frontiers for sequential inspection with a single hidden fault

We study sequential information acquisition before one irreversible recovery deployment when exactly one of finitely many suppliers has an unknown fault. Perfect inspections have a common fee and a common delay. The objective combines expected capped outage loss and actual inspection expenditure, subject to an unconditional service-failure probability bound. Every adaptive policy is dominated for these two upper budgets by a mixture of ordered inspection lists: a positive report triggers deployment, while an all-negative path ends with an uninspected fallback. For a fixed planned depth, inspection reach probabilities make the expected fee separable by position. This yields an exact weighted-assignment reduction, an O(n^4) arithmetic scalarization oracle, and a randomized budget-frontier linear program with O(n^3) variables and O(n^2) structural constraints, apart from nonnegativity bounds. Each feasible LP point is implemented by time-zero randomization independent of the fault. With equal deployment times, one decreasing-prior order simultaneously optimizes both criteria at each depth, reducing the complete frontier to at most n stopping points. A positive-fee, positive-delay example shows why an unprofitable one-step inspection does not justify stopping. AI tools assisted with mathematical derivation, proof checking, and manuscript preparation.