arXiv:2611.07567v1 Announce Type: new Abstract: We study the cost of certifying a scalar target driven by a product of centered Bernoulli means when an unobserved terminal gate leaves irreducible uncertainty. Each factor can be queried independently at a price. We derive the exact identified interval and determine when a requested accuracy is feasible. At every fixed strictly feasible interior instance, the optimal high-confidence expected cost is $(1+o(1))\log(1/\delta)/D^*$, where $D^*$ is an answer-and-allocation information game; a globally honest procedure attains this benchmark. At product cancellation, we solve the game through a one-dimensional inverse-KL equation and explicit cost weights. For two tickets, quartic projections and rational bounds give checkable global optimality gaps. Joint confidence--precision limits expose a failure of the analyzed fixed pilot: it can cost asymptotically at least twice the optimum. A charged, precision-aware pilot with checkpoint certification restores optimal expected cost in exact arithmetic when $\log\{e+\log(1/g)\}=o(\log(1/\delta))$ for effective precision $g$, including exponential and certain double-exponential precision scales. Pooled rescue preserves validity and cannot increase observation cost on coupled paths. Implemented auditors verify returns by rational inequalities; reproducible experiments show both savings and finite-confidence overhead. These guarantees concern the specified independent-reference model and its scalar target.