Metacognition as L1 Constraint Minimization

This paper introduces a formal and computable theory of metacognition based on structural inconsistency minimization in graph-structured reasoning systems. Rather than defining metacognition in terms of confidence or calibration, it is modeled as the capacity of a system to detect and reduce its own internal inconsistency using the L1 coboundary norm. A reasoning process is represented as a section over a domain graph, with local inconsistencies captured by a defect field. The total defect admits a decomposition: I (s) = Phi + R (s) where Phi is the irreducible (cohomological) component and R is the repairable component attributable to model error. This decomposition enables a quantitative distinction between intrinsic domain ambiguity and model-induced hallucination. The paper establishes the following: A set of structural axioms (locality, faithfulness, monotonicity, and additivity) under which an L1-based defect signal provides a natural measure of metacognitive evaluation A computable metacognitive distance metric dₘeta (M, D) that quantifies deviation from ideal self-monitoring behavior A convergence framework in which iterative repair dynamics reduce removable defect and isolate irreducible structure A formal definition of hallucination via the hallucination ratio eta = R / I Empirical experiments demonstrating that L1-based structural signals outperform confidence-based metrics in predicting reasoning errors The framework yields falsifiable predictions, including characteristic scaling behavior for defect under L2 projection and measurable bounds on metacognitive performance in existing models. It also provides a practical pathway for augmenting AI systems with explicit verification layers based on L1 optimization. This work connects ideas from graph theory, cohomology, convex optimization, and cognitive science to provide a unified structural account of metacognition in artificial systems.