AI and renewable energy are increasingly framed as a "power couple" -- the idea that surging AI electricity demand will accelerate clean-energy investment -- yet concerns persist that AI will instead entrench fossil-fuel carbon lock-in. We reconcile these views by modeling the equilibrium interaction between AI growth and renewable investment. In a parsimonious game, a policymaker invests in renewable capacity available to AI and an AI developer chooses 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, developers push toward frontier scale even when the marginal megawatt-hour is fossil-based. In this regime, renewable expansion can primarily relax scaling constraints rather than displace fossil generation one-for-one, weakening incentives to build enough clean capacity and reinforcing fossil dependence. 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, energy costs discipline capability choices; renewable investment then both enables capability and decarbonizes marginal compute, generating an "adaptation pathway" in which climate stress strengthens incentives for clean-capacity expansion and can support a carbon-free equilibrium. A calibrated case study illustrates these mechanisms using observed magnitudes for investment, capability, and energy use. Decarbonizing AI is an equilibrium outcome: effective policy must keep clean capacity binding at the margin as compute expands.