The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning

Manufacturing lines, service journeys, supply chains, and AI reasoning chains share a common challenge: attributing a terminal outcome to the intermediate stage that caused it. We establish an information-theoretic barrier to this credit assignment problem: the signal connecting early steps to final outcomes decays exponentially with depth, creating a critical horizon beyond which reliable learning from endpoint data alone requires exponentially many samples. We prove four results. First, a Signal Decay Bound: sample complexity for attributing outcomes to early stages grows exponentially in the number of intervening steps. Second, Width Limits: parallel rollouts provide only logarithmic relief, with correlation capping the effective number of independent samples. Third, an Objective Mismatch: additive reward aggregation optimizes the wrong quantity when sequential validity requires all steps to be correct. Fourth, Optimal Inspection Design: uniform checkpoint spacing is minimax-optimal under homogeneous signal attenuation, while a greedy algorithm yields optimal non-uniform schedules under heterogeneous attenuation. Together, these results provide a common analytical foundation for inspection design in operations and supervision design in AI.