Artificial intelligence (AI) and renewable energy are increasingly being described as a \mbox{``power couple,''} based on the idea that rapid growth in AI will spur clean-energy investment. Yet growing AI demand could also deepen reliance on fossil power. We study when each outcome arises in a game in which renewable-capacity investment and AI scaling interact. The key is how the market value of greater AI capability grows relative to the energy needed to achieve it. When value grows at least as fast as energy use (market-led scaling), the developer pushes toward frontier capability even when additional electricity comes from fossil sources. Renewable investment can then enable further AI growth without eliminating fossil use. As climate damages increase, AI becomes more valuable for adaptation, strengthening incentives to sustain frontier capability despite the associated emissions. We call this the ``adaptation trap.'' When energy requirements grow faster than capability value (resource-led scaling), energy costs place greater limits on AI expansion. Renewable investment then makes additional capability less costly while also reducing emissions. As climate damages rise, the growing value of AI for adaptation can justify enough clean-capacity expansion to support AI entirely with renewable power. We call this the ``adaptation pathway.'' A calibrated case study shows that both mechanisms can arise at empirically plausible magnitudes. The results suggest that decarbonizing AI requires renewable capacity to keep pace with the growth of compute demand.

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