Voters fill out ballots on primary election day at the Butte Civic Center in Butte, Mt., on Tuesday, June 2, 2026. The Trump administration is investigating nonprofits that it says may be recruiting noncitizens for voter registration, documents show, even though noncitizen voting in the United States is rare. (Will Warasila/The New York Times) By MADELEINE NGO, RICHARD FAUSSET and ALEXANDRA BERZON As the Trump administration promotes doubts about the integrity of the voting system ahead of…