Along the ripped flanks of the Rubaya hills, in the northeast of the Democratic Republic of Congo, hundreds of workers dig through one of the world’s largest concentrations of coltan – a mineral that fuels both the 21st-century economy and one of its longest-running conflicts.

It does not matter to them, they say, who controls these mines. It could be the Congolese government, which has struggled for decades to assert its authority from a capital city some 1,600 miles away. It could be one of the militias tied to neighboring Rwanda, which took control of this swath of Congo last year.

Or it could be the United States, which this summer brokered a peace agreement between Congo and Rwanda. That could be a first step, U.S. officials have said, to an economic agreement that could help Americ

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