The Cost of Exact Batch Decisions with Partial Evidence

We study the cost of reproducing a fixed batch decision from paid evidence. Each item has a hidden result and a finite query interface. The output must be correct for every result vector compatible with the purchased responses. We compare its expected cost with full evaluation. A local query policy saves work but leaves uncertainty in the batch count. We price these two quantities together. This gives an additive comparison with the optimum over all legal queries. The error separates missing local policies, insufficient repeated items, and threshold geometry. For a single threshold, bounded representation and supply losses give a bounded remainder. No variance lower bound is needed. We then let the shared response probabilities be unknown. Observed support endpoints provide learning feedback without reading every retained result. Under declared interfaces and supplied completion bounds, the extra fee is the square root of the remaining opportunity scale. It is sharply of order $N^{1/4}$ at a critical threshold. All acquisition and final completion queries are charged.