A Trilemma for Covert Channels in LLMs: Signal, Rules, and the Missing Form of Life

Imagine you want an LLM to secretly answer "yes" or "no" when direct answers are forbidden. You agree on a code word. This paper argues that this cannot be done reliably while staying hidden. Three obstacles stand in the way. First, elementary information theory: a common code is ambiguous, a rare one is easy to detect. Second, Kripke's rule-following paradox: the LLM contains no fact of the matter that would fix the code's meaning on a new question. Third, the model lacks the social practice—the shared form of life—that allows humans to resolve that paradox. I deliberately analyze the simplest possible scenario—a single token encoding a single bit—not because the problem is confined to that case, but because it offers the clearest setting for the proof. The argument then generalizes to any deterministic text-based code, including multi-token steganographic schemes. Since the signal is inherently Boolean (yes/no), the three constraints form a trilemma that makes a reliable covert channel impossible. I use basic mathematics (Shannon, Zipf) and a simple linear detector to show why.