Higher-Order Conditional Attainability: Opacity, Observation Order, and Decision Guarantees Beyond Local Views

Local measurements are often treated as if enough of them must eventually reveal a system’s global capability. That inference can fail even when every bounded-order view is known exactly. We develop Higher-Order Conditional Attainability (HCAT), a mathematical framework that turns this local-to-global gap into a quantitative object. For a complete operational envelope, HCAT groups all globally admissible envelopes that agree with observations through order k, measures their reconstruction radius and pairwise opacity, and defines the minimum observation order required for identification. We prove nested-class monotonicity, radius–diameter bounds, a maximum-opacity parity hierarchy, and a decision theorem connecting unresolved attainability ambiguity to worst-case regret under Lipschitz utility. Exact enumeration of all 65,535 nonempty relations on a four-bit universe verifies the theory across parity, threshold, sparse-XOR, one-hot, and hypergraph constructions. In 250 random relations, HCAT’s robust action improves certified worst-case utility at observation orders two and three while not claiming universal average-regret dominance. A structural audit of 569 Wisconsin Diagnostic Breast Cancer records shows that lower-order views leave thousands of globally compatible supports, with exact identification only at fourth order; this rank is stable in 200 bootstrap resamples. Open-source software reproduces all reported results, figures, and theorem audits in fully reproducible finite settings.