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I'm a little late to this exchange, but I'd like to add my two cents. I have found nothing on the parents of Rachael Pyron, only a claim she was born in 1788 in VA (or NC). Jesse Cate (1785-1832) I show as the son of JOHN PREACHER CATES and Mary Lawrence. JPC's parents were Robert Cates and Elizabeth Wyatt. Robert Cates was the brother of JOHN ROBERT EZRA CATES, their parents Robert Ezra Cates and Mary Ann Randall. JREC and Margery Lawrence were the parents of Margery Cates, who married John Pyron in Orange County, NC, around 1761. Their older son was also John Pyron, born in Orange Co. (later Caswell) in 1762.
In 1800 John's brother Joshua, born in 1770, ran off to Knox Co., TN (where John was living at the time), with a Caswell Co. neighbor's wife. Rachael may have been one of his legal daughters, joining him in TN, but that's just a guess.