AI and renewable energy are increasingly framed as a "power couple," on the premise that surging AI demand will accelerate clean-energy investment, yet concerns persist that AI will entrench fossil-fuel carbon lock-in. We reconcile these views by modeling the equilibrium between AI growth and renewable investment. In a parsimonious game, a policymaker designs policies that guide investment in renewable capacity for AI, while an AI developer chooses its model's capability. The equilibrium depends on scaling regimes and market incentives. When the market payoff to capability is supermodular and performance gains are near-linear in compute (so the market rewards capability at least as fast as scaling raises its energy cost), developers push toward frontier scale even when the marginal megawatt-hour is fossil-based. In this regime, renewable expansion mainly relaxes scaling constraints rather than displacing fossil generation; clean capacity remains insufficient and fossil dependence persists. This yields an "adaptation trap": as climate damages rise, the value of AI-enabled adaptation increases, which strengthens incentives to enable frontier scaling while tolerating residual fossil use. When AI faces diminishing returns and lower scaling efficiency (so energy requirements outrun the market value of capability), energy costs discipline capability choices; renewable investment then both enables capability and decarbonizes marginal compute, generating an "adaptation pathway" in which climate stress spurs clean-capacity expansion toward a carbon-free equilibrium. A calibrated case study illustrates both mechanisms using observed magnitudes. The findings suggest that effective AI decarbonization requires policies that keep clean capacity binding at the margin as compute expands.

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