Beyond Causal Identifiability: Order-Indexed Ambiguity, Decision Sufficiency, and Consequence-Directed Intervention Design

Causal discovery often stops when the remaining graphs form an observational or interventional equivalence class, yet class size alone does not measure how consequentially different the surviving models are. We introduce Order-Indexed Causal Ambiguity, a Higher-Order Conditional Attainability extension that measures structural, interventional, and task-relative diameters of causal models compatible with bounded evidence. The framework defines decision-sufficiency ranks, proves that compatibility-class cardinality and task ambiguity are incomparable, and derives a robust regret certificate controlled by compatibility-class radius and diameter. We further propose consequence-directed intervention design, selecting experiments that minimize residual task ambiguity rather than graph uncertainty alone. Exact finite constructions verify the separation theorems. A controlled finite-sample synthetic benchmark then compares task-ambiguity, graph-oriented, class-size, and random intervention policies across 432 matched instances per policy. At ε = 0.30, task-directed selection reached decision sufficiency in 65.7% of instances versus 47.9% for the graph-oriented proxy, with the all-run capped stopping score lower by 0.93 and final task ambiguity lower by 0.072; paired bootstrap intervals excluded zero for both differences. The graph-oriented policy retained a smaller structural-diameter proxy, while regret differences were not significant. The results therefore support a task-versus-structure tradeoff without claiming universal dominance or external-benchmark superiority.