MINT: Certifying Queue Instability from Limited Commands and Hidden Modes

Can observation and command interfaces each pass a capacity test while together making stabilization impossible? We introduce MINT (Menu–Innovation No-go Test), a model-based certificate for controllers that send one of $K$ labels to a stateless executor with fixed action templates. MINT first averages each action's service over the unseen mode change, then finds the best service possible with $K$ fixed action templates, even granting the controller knowledge of the previous mode. If arrivals exceed this joint ceiling, every compliant controller using past departure feedback has growing expected backlog, regardless of its memory or training method. Separate command-only and prediction-only ceilings can both miss this obstruction. For every $m\ge 3$, we give an $m$-mode queue family and an open load interval where MINT detects it. Matching stable constructions establish exact minimum budgets of $m$ commands with departure history and $m-1$ with current-mode observation; two noncyclic instances show the same separation. In a four-mode task, reward-only search recovers finite controllers that are certified stable after training at a feasible budget.