When an AI system causes harm, the question of who is responsible is among the most consequential and least resolved questions in AI governance. Current frameworks define controls and monitoring requirements but do not specify when a system is sufficiently mature for responsibility to transfer from developer to deployer. This paper introduces the AI Maturity License Framework—a responsibility-assignment model that treats AI maturity as a prerequisite for accountability transfer. Drawing on analogies from human licensing systems, safety-critical system certification, and zero-trust security principles, the framework proposes a four-level maturity proof hierarchy—training data audit, synthetic scenario testing, shadow deployment, and graduated autonomy—through which AI systems must pass before responsibility transfers to the deployer. It further introduces automatic license revocation upon drift detection and a six-month renewal cycle that prevents the governance gap created by one-time certification. The framework aligns with the EU AI Act, NIST AI Risk Management Framework, and India's Digital Personal Data Protection Act and addresses three failure modes that existing frameworks do not resolve: premature responsibility transfer, under-trained models in high-risk environments, and model drift without re-evaluation. The paper concludes that responsibility should follow demonstrated competence—not ownership—and that this principle, if institutionalized, would fundamentally clarify the liability landscape for AI deployment.