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AI systems increasingly shape high-stakes decisions in healthcare, law, defense, and education, yet existing governance paradigms—AI Ethics, AI Safety, and AI Alignment—share a common limitation: they evaluate outcomes rather than verifying the reasoning process itself. This paper introduces AI Integrity—a state in which the Authority Stack of an AI system (its layered hierarchy of values, epistemological standards, source preferences, and data selection criteria) is protected from corruption, contamination, manipulation, and bias, and maintained in a verifiable manner. We distinguish AI Integrity from the three existing paradigms, define the Authority Stack as a 4-layer cascade model (Normative, Epistemic, Source, and Data Authority), characterize the distinction between legitimate cascading and Authority Pollution, and identify Integrity Hallucination as the central measurable threat to value consistency. Unlike normative frameworks that prescribe which values are correct, AI Integrity is a procedural concept: it requires that the path from evidence to conclusion be transparent and auditable, regardless of which values a system holds. This shift from outcome evaluation to process verification opens a new approach to AI governance grounded in empirical measurement. The companion measurement framework (PRISM: Profile-based Reasoning Integrity Stack Measurement) and empirical dataset (113,400 forced-choice value judgment responses across 10 AI models) are published separately; see Related Works.