Artificial intelligence is increasingly embedded in human decision making. In some cases, it enhances human reasoning. In others, it fosters excessive cognitive dependence. This paper introduces a conceptual and mathematical framework to distinguish cognitive amplification, where AI improves hybrid human AI performance while preserving human expertise, from cognitive delegation, where reasoning is progressively outsourced to the AI system, risking long term atrophy of human capabilities. We define four operational metrics: the Cognitive Amplification Index, or CAI star, which measures collaborative gain beyond the best standalone agent; the Dependency Ratio, or D, and Human Reliance Index, or HRI, which quantify the structural dominance of the AI within the hybrid output; and the Human Cognitive Drift Rate, or HCDR, which captures the temporal erosion or maintenance of autonomous human performance. Together, these quantities characterize human AI systems in terms of both immediate hybrid performance and long term cognitive sustainability. We validate the framework through an agent based simulation in NetLogo across three reliance regimes and multiple dependency and atrophy configurations. The results distinguish degenerate AI dominated delegation, human preserving but weakly competitive interaction, and intermediate boundary regimes that approach the AI baseline while remaining structurally dependent. Across all tested configurations, no regime achieves genuine amplification. A constrained optimization over the atrophy parameter shows that reducing atrophy improves retained human capability, collaborative gain, and dependency structure, but even zero atrophy does not yield positive collaborative gain. The framework therefore provides a practical tool for evaluating whether human AI systems perform well in a way that also preserves human capability over time.