A mathematical warning label for autonomy. If an agent must certify its own safety in an open world while paying nonzero upkeep each step under a fixed budget, PMTT shows a hard trilemma is unavoidable: you either weaken certification, lose fidelity, or scale complexity forever. We explicitly scope away closed-world controllers (thermostats/LQR/Kalman) and provide falsification targets—this is not universal doom, it’s a precise boundary for “self-authorizing” systems in persistent environments.