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The claim says the hierarchy only runs one way: Harvard graduates can teach elsewhere, but graduates of "lower schools" cannot teach at Harvard. That is contradicted by Harvard's own faculty biographies.

For example, Emmanuel Akyeampong is listed by Harvard Kennedy School as an "Ellen Gurney Professor of History and Professor of African and African American Studies, Harvard University," and the same profile states that he "joined the History faculty at Harvard upon receiving his Ph.D. in African History from the University of Virginia in 1993."

Likewise, Harvard T.H. Chan School of Public Health lists Steve Gortmaker as "Professor of the Practice of Health Sociology, Emeritus" and states that "Dr. Gortmaker earned his PhD from the University of Wisconsin, Madison."

Harvard Chan also lists Rafael A. Irizarry as "Professor of Biostatistics" and states that he received his bachelor's degree from the University of Puerto Rico and his Ph.D. from the University of California, Berkeley.

These are direct counterexamples from Harvard's own official pages: people who did not graduate from Harvard do, in fact, teach at Harvard. That makes the blanket statement "but not vice versa" false.