In black-box large language model (LLM) services, response reliability is often only partially observable at decision time, while stronger inference pathways incur substantial computational cost, inducing a budgeted sequential decision problem: for each request, the system should decide whether the default low-cost response is sufficiently reliable or whether additional computation should be allocated to improve response quality. In this paper, we propose \textbf{Ver}ifiable \textbf{O}bservations for Risk-aware \textbf{I}nference \textbf{C}ontrol (\textsc{Veroic}), a framework for adaptive inference control in black-box LLM settings, which formulates request-time control as a \textit{partially observable Markov decision process} to capture partial observability and sequential budget coupling. It constructs a lightweight verifiable observation channel from the input-output pair by aggregating heterogeneous quality signals into a belief state over latent response reliability, which is then used by a budget-aware policy to decide whether to return the default output or trigger a higher-cost inference pathway. Experiments on diverse tasks show that \textsc{Veroic} achieves improved quality-cost trade-offs, stronger risk estimation and calibration, and more robust long-horizon inference control than competitive baselines.