Mr. Fox and another former DOGE employee, Nate Cavanaugh, had each testified in their depositions that they used ChatGPT to identify grants that ran afoul of President Trump’s executive order banning “radical and wasteful government D.E.I. programs.” The two men, who had previously worked in technology and finance, acknowledged they had no background in the humanities, but believed in DOGE’s broader mission of shrinking “useless small agencies,” as Mr. Cavanaugh put it.