Fractal clusters and urban scaling shape spatial inequality in U.S. patenting

Innovation is distributed unevenly across space, yet the mechanisms behind persistent patenting inequality are poorly understood. Analyzing over a century of georeferenced U.S. patent data across eight major technological domains, we identify three consistent spatial patterns: (1) Zipfian scaling in patent output, (2) fractal clustering of innovation hubs, and (3) a 20-km scale of maximal spatial disparity, which we term the “inequality horizon.” These general patterns suggests that the geography of innovation emerges not only from urbanization or policy but from underlying generative principles. A minimal spatial model combining local reinforcement and long-range diffusion replicates all observed patterns without sector-specific assumptions. Analytical results show that the inequality horizon arises at the spatial scale where cluster growth balances internal heterogeneity. Our multiscale analysis suggest that emergent inequality in U.S. patenting stems by the interaction of urban scaling (superlinear productivity linked to city size) and fractal clustering (hierarchical spatial organization beyond administrative boundaries), pointing to general mechanisms governing the spatial dynamics of innovation.