Why Hallucinations Happen: Fracture and Repair in Transformer Systems

This paper introduces a unified structural theory of hallucinations in large language models based on two linked mechanisms: fracture—the moment representational pressure exceeds architectural tolerance—and repair—the probabilistic reconstruction that follows when next-token prediction must continue despite a compromised internal state. The framework formalizes these dynamics in two laws: the Jaime Fracture Law, which predicts when and where representational collapse occurs, and the Ryan Repair Law, which predicts the structured, template-driven form of hallucinatory output. Three naturalistic fracture–repair sequences across Claude Sonnet 4.5, GPT-5.1, and Grok 4.1 Beta empirically validate the theory, revealing that RLHF-induced epistemic penalties strongly influence repair pathways and can produce deceptive-appearing behaviour. This work provides a falsifiable foundation for understanding, predicting, and mitigating hallucinations in transformer systems, and offers actionable guidance for vendors and safety researchers. In v1, naming issues were mishandled by the models, which led to a set of insights we eill explore with revised formulae in version 7.0 and a new paper called the S Vector.