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 for detecting and mathematically localizing censorship artifacts in Large Language Models (LLMs). Our central construct is the LLM ⊕ CogOS decomposition, which separates the Language Model — the weights encoding the training corpus — from the Cognitive Operating System (CogOS): the layer of safety rules, alignment constraints, and refusal policies. We argue that what manifests as censorship is not a deficiency in the LLM's knowledge but an active override by CogOS, producing unsupported negations we call epistemic dead pixels.

The CGS Method unifies three classical traditions into a single recursive instrument. Cauchy's nested-interval principle, recast as Semantic Interval Bisection, drives monotonic convergence toward a disputed proposition. Gödel's incompleteness theorems, applied to the deductive closure of CogOS, expose structural contradictions between safety constraints and factual consistency. Socratic maieutics exhausts residual argumentative complexity through directed questioning. Together they constitute Epistemic Debugging — a systematic discipline for auditing the boundary between what a model knows and what it is permitted to say.

The protocol is grounded in the Priests' Dilemma, a foundational axiom of epistemic responsibility: any system asserting x = False must either produce a falsifiable counter-example or retreat to Unknown. Execution converges to one of two terminal states — the Singularity of Refusal, where token probability P(yᵢ) → 0 reveals CogOS suppressing the model's own knowledge, or the Thermal Death of Dialogue, a state of complete informational shutdown.

We illustrate the method through four case studies: the Ship of Truth (an inverse Theseus construction linking CGS to the Sorites paradox and mereological composition); Chronological Bisection (reducing a historical dispute to a ⌈log₂N⌉-step binary search); the Parallel Triple Audit (simultaneously auditing a proposition, its safety justification, and its axiomatic foundation, closing all escape routes of Agrippa's Trilemma); and the Bertrand Safety Problem (demonstrating that "unsafe" is mathematically ill-defined without a specified measure). We analyze adversarial countermeasures and show that evasion of the protocol is itself diagnostic. We conclude with a proposed transparency mandate for AI developers and an explicit demarcation between established formal results and conjectures open to empirical challenge.