Stateless Agents in Exhaustion Systems: Memory, Criterion, and the Geometry of Failure

We investigate the behavior of large language models (LLMs) operating as stateless agents in a Deterministic Game with Irreversible Global Memory (DGIGM), as implemented in the Ω-TRACE simulator. Because LLM API calls are memoryless each turn receives no history of previous turns we design a bridge protocol in which the agent writes a short note to its future self after each move, and receives its last three notes at the next turn. This creates a nite sliding memory window analogous to the K-window of the DGIGM itself: the agent's self-knowledge is bounded and progressively insu cient, just as the system's action space is bounded and progressively exhausted.