["The British Journal of Sociology, EarlyView. ", "\nABSTRACT\nScholars have long investigated intergenerational inequality, with attention to the associations among social origins, education, and socio‐economic destinations: the so‐called OED triangle. Meanwhile, recent research highlights distinct roles of education and skills in status attainment. Extending these studies, this article proposes a new framework—the OESD quadrangle—that incorporates origin, education, skills, and destination to better understand the structure of intergenerational inequality and its cross‐national patterns. As an application, causal mediation analysis based on the counterfactual framework is performed using data from the OECD Survey of Adult Skills (PIAAC) for 27,812 individuals in 23 countries, encompassing parental status, respondents' educational attainment, adult cognitive skills, and occupational outcomes. The findings corroborate prior research, demonstrating that education significantly mediates the origin‐destination association in all cases. Importantly, adult skills also exert a mediating power net of education, ranging from 4% (Sweden) to 25% (USA) of the total origin effect estimates in conservative models. These results suggest that fostering cognitive skills among the disadvantaged may help mitigate intergenerational inequality, although its magnitude varies across societies. By linking these cross‐national differences to fundamental concepts such as meritocracy and credentialism, the study concludes with a preliminary typology of societies from a comparative perspective.\n"]