Verifiable AI via the ℓ¹ Coboundary Norm

This paper introduces a mathematical framework for quantifying, decomposing, and repairing structural inconsistency in AI system outputs using the ℓ¹ coboundary norm on graph-structured domains. A knowledge domain is modeled as a finite directed graph whose vertices represent concepts and whose edges encode constraints via restriction maps. An AI system output defines a section over this graph, and its total inconsistency is measured by the ℓ¹ coboundary norm. Within a class of defect measures satisfying locality, additivity, faithfulness, and symmetry, the ℓ¹ norm provides a canonical choice that preserves the structural visibility of inconsistencies. This enables a deterministic hallucination score for any output relative to a specified domain graph. The framework provides the following components: A computable total defect I(s) measuring structural inconsistency A decomposition into repairable error and irreducible domain complexity via ℓ¹ Hodge decomposition A linear programming formulation for computing the irreducible cohomological component Φ A convergent repair algorithm (Lift Gauge Iterator) that projects outputs to minimal-defect configurations A hallucination certificate (I, Φ, R, η, C) enabling auditable and contract-level guarantees Unlike probabilistic or statistical approaches to hallucination detection, this framework is deterministic, model-agnostic, and operates directly on structured domain constraints. It separates AI error from inherent domain inconsistency and provides explicit bounds on residual defect. The approach is complementary to retrieval, calibration, and fact-checking methods, adding a structural consistency layer with provable guarantees and polynomial-time computability. The mathematical foundations are developed in a preceding series of papers; this work applies those results to AI verification and outlines a practical implementation pathway.