Energy poverty persists even among households that are not income-poor, suggesting a deeper mechanism than mere budget constraints. We develop a model in which indoor thermal comfort is produced through a non-convex technology that couples energy input with dwelling efficiency. A critical efficiency threshold emerges below which the minimum comfort level is physically unattainable, regardless of how much energy is purchased. Households below this threshold suffer from structural energy poverty, which income transfers alone cannot cure. The model yields three sharp policy predictions: energy price shocks are strongly regressive, efficiency investments dominate income transfers and price subsidies in reducing energy poverty, and a cost-effective anti-poverty strategy must combine targeted retrofits with temporary income support. The results are illustrated with symbolic diagrams and formal proofs.