The public face of New York’s diamond district is Diamond and Jewelry Way, a block of 47th Street between Fifth and Sixth Avenues lined with dazzlingly lit shops and exchanges and cluttered by hawkers, hustlers, cops and couriers. But beyond these street-level operations, in back rooms, upper floors and looming towers, toils an army of cutters, blockers, polishers, sorters, appraisers, graders, designers and dealers — most of whom the diamond-buying public never sees.