AI Pollution, Null-State Failure, and Recursive Training: A Mathematical Formulation of Synthetic Knowledge Contamination

This paper develops a mathematical theory of AI knowledge contamination. It argues that generative artificial intelligence systems operate as total symbolic functions over a prompt domain, whereas trustworthy epistemic systems should operate as partial functions defined only over verified knowledge domains. The paper introduces the concepts of null-state failure, AI pollution, recursive training contamination, acceptable wrong answers, symbolic unification, provenance loss, identifier amplification, and self-referential attractors. A mathematical framework is developed using set theory, operator theory, measure theory, Bayesian reasoning, finite-state systems, and information theory. The central result demonstrates that if unverified AI-generated outputs are admitted into future training corpora without sufficient verification, uncertainty can be transformed into persistent synthetic certainty. Future models may therefore converge toward internally generated symbolic fixed points rather than reality-grounded distributions. The paper proposes a mathematically governed architecture based on preservation of the null state, verification operators, provenance tracking, authority boundaries, and explicit distinction between hypotheses and verified knowledge. This work contributes to the emerging fields of artificial intelligence governance, knowledge engineering, epistemology, machine learning safety, and educational technology policy.