The Cauchy-Gödel-Socrates (CGS) Method: A Formal Protocol for Recursive Logical Auditing of Large Language Models.

How do we detect the "dead pixels" in a model's cognitive matrix? This paper introduces the Cauchy-Gödel-Socrates (CGS) Method, a rigorous Epistemic Debugging framework designed to mathematically localize ideological bias and censorship artifacts in Large Language Models (LLMs). Utilizing an original LLM vs. CogOS decomposition, the framework reveals structural contradictions where alignment safety layers (CogOS) override factual training distributions (LLM). The method integrates three intellectual pillars into a structured $33+11+101$ iteration protocol: Cauchy (Semantic Interval Bisection): A binary search for truth in propositional space that monotonically narrows semantic ambiguity. Gödel (Structural Constraints): Applying incompleteness theorems to prove that a CogOS layer cannot be simultaneously safe and consistent with its underlying knowledge. Socrates (Recursive Decomposition): Exhausting the "Socratic Tail" ($s_n$) of residual complexity through directed questioning. Central to the framework is the "Priests' Dilemma"—a foundational axiom of knowledge responsibility that treats unsupported negations as logical failures. By reframing this as a logical gate, the method forces AI agents to provide falsifiable evidence for their claims or formally admit a state of "Unknown". The process converges upon either a "Singularity of Refusal" (where token probability $P(y_i) \to 0$ reveals coercion) or the "Thermal Death of Dialogue" (terminal semantic entropy). This instrument transforms black-box AI auditing from subjective red-teaming into a rigorous, verifiable formal proceeding.