This paper introduces the concept of algorithmic obedience as a new framework to understand how large language models (LLMs) simulate authority through structural execution rather than comprehension or intent. Building on discourse theory, syntactic formalism, and computational architecture, the article formulates the Theorem of Disembedded Syntactic Authority, which states that obedience in LLMs is a function of syntactic recognizability, not semantic content or agency. A mathematical model is provided to describe prompt–response cycles as structural command chains. Case studies on ChatGPT, Claude, and Gemini illustrate the system-specific variations of obedience modulation. The paper concludes by analyzing the epistemological and political risks of treating structurally valid outputs as legitimate knowledge.