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Artificial intelligence is increasingly embedded in human decision-making, where it can either enhance human reasoning or induce excessive cognitive dependence. This paper introduces a conceptual and mathematical framework for distinguishing cognitive amplification, in which AI improves hybrid human-AI performance while preserving human expertise, from cognitive delegation, in which reasoning is progressively outsourced to AI systems. To characterize these regimes, we define a set of operational metrics: the Cognitive Amplification Index (CAI*), the Dependency Ratio (D), the Human Reliance Index (HRI), and the Human Cognitive Drift Rate (HCDR). Together, these quantities provide a low-dimensional metric space for evaluating not only whether human-AI systems achieve genuine synergistic performance, but also whether such performance is cognitively sustainable for the human component over time. The framework highlights a central design tension in human-AI systems: maximizing short-term hybrid capability does not necessarily preserve long-term human cognitive competence. We therefore argue that human-AI systems should be designed under a cognitive sustainability constraint, such that gains in hybrid performance do not come at the cost of degradation in human expertise.