Active-Learning Query Synthesis for Artificial Intelligence Agriculture Soil-Nutrient Inference Under Laboratory Assay Budget Caps: A Computer Information-Gain Regret Bound

Soil-nutrient inference from spectral and proximal sensing is constrained less by model capacity than by the cost of laboratory assays used to label training queries. the present analysis formalises budget-capped active learning for soil-nutrient regression as a sequential decision problem and derives a finite-horizon information-gain regret bound for a query-synthesis policy that selects assay locations by maximising expected reduction in posterior predictive entropy per unit assay cost. The bound decomposes cumulative regret into an information-gain term, a cost-normalisation term and a misspecification residual, and it predicts that regret grows as the square root of the assay budget rather than linearly in the number of candidate locations. The derivation is connected to the computer-science literature on active learning, Gaussian-process bandits and cost-sensitive information acquisition, and its assumptions are stated explicitly so that the bound can be falsified by future field evaluation. The contribution is a formal, testable statement of when cost-aware query synthesis is provably preferable to uniform or random assay allocation under a fixed laboratory budget, together with the boundary conditions under which the guarantee fails.