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[2602.09394] The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning
Summary
This article presents an information-theoretic analysis of credit assignment in multi-stage operations, highlighting the challenges of attributing outcomes to earlier stages and proposing optimal inspection design principles.
Why It Matters
Understanding the credit assignment problem is crucial for improving efficiency in manufacturing, service operations, and AI systems. The insights provided can lead to better inspection designs, enhancing operational effectiveness and decision-making processes across various fields.
Key Takeaways
- Signal decay complicates outcome attribution in multi-stage processes.
- Sample complexity increases exponentially with the number of stages.
- Optimal inspection design can mitigate challenges posed by signal attenuation.
Statistics > Machine Learning arXiv:2602.09394 (stat) [Submitted on 10 Feb 2026 (v1), last revised 13 Feb 2026 (this version, v2)] Title:The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning Authors:Seyed Morteza Emadi View a PDF of the paper titled The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning, by Seyed Morteza Emadi View PDF HTML (experimental) Abstract: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 ...
