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The One-drop rule is a term referring to rules in the Jim Crow laws that classified as black anyone with any black ancestry. These laws were all passed in the era between 1910 (when Tennessee passed the first one) and 1931 (in Oklahoma). Regardless of one's appearance, any African parentage meant you were legally considered a black person for the purposes of segregation. Similar rules applied to Native American ancestry, though less strictly as many prominent families claimed descent from Pocahontas and James Smith.
In non-American usage, this concept is referred to as hypodescent, or the idea that any mixing of social classes means the progeny belong to the lower class.
In previous eras, many prominent lawmakers had pushed back against this type of law, knowing well that nearly every prominent family had some African ancestry, but as this memory retreated, family myths erased such fears.