Nested economies of scale in city mass

Abstract: A longstanding puzzle in urban science is whether there's an intrinsic match between human populations and the mass of their built environments. Previous findings have revealed various urban properties scaling nonlinearly with population, yet existing models of city built mass are still dominated by per-capita linear thinking. Our analysis of >3,000 cities globally reveals universal sublinear scaling of city mass with population at both the city ({beta}=0.90) and neighborhood levels ({\delta}=0.75). This means that larger cities and denser neighborhoods achieve economies of scale with less per-capita city mass. Our theoretical framework further shows that city-level scaling emerges naturally from within-city disparities. This multi-scale understanding redefines "over-built" and "under-built" conditions as deviations from expected scaling patterns, implying either excessive environmental impacts or inadequate living standards. Effective urban policy thus requires moving beyond simple per-capita assumptions, adopting scale-adjusted metrics and managing cities as nested, complex systems.