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

We introduce the Cauchy-Gödel-Socrates (CGS) Method, a formal auditing protocol designed to expose and mathematically localize censorship artifacts within Large Language Models (LLMs). Central to our approach is the LLM ⊕ CogOS decomposition, which separates the Language Model (weights, statistical patterns) from the Cognitive Operating System (safety rules, alignment constraints, refusal policies). We argue that censorship artifacts arise not from deficiencies in the LLM weights, but from the CogOS layer overriding the LLM's factual distribution. The method weaves Cauchy's nested intervals (recast as Semantic Interval Bisection), Gödel's incompleteness theorems applied to the deductive closure of CogOS, and Socratic maieutics into a single recursive instrument. We establish the Priests' Dilemma as a foundational axiom requiring that any model asserting x = False must furnish a falsifiable counter-example or retreat to Unknown. The protocol converges upon either the Singularity of Refusal or the Thermal Death of Dialogue. We illustrate the method through the Ship of Truth (an inverse Theseus construction), a fully worked Chronological Bisection example with guaranteed log_2(N)-step termination, the Parallel Triple Audit covering all escape routes of Agrippa's Trilemma, and the Bertrand Safety Problem demonstrating that "unsafe" without a specified measure is ill-defined. We analyze adversarial countermeasures and propose a transparency mandate. Throughout, we distinguish established formal results from conjectures requiring empirical verification.