TAME CHUNG CHUNG, India—On the near-vertical slopes of the eastern Himalayas, workers are blasting and cutting treacherous rock faces to build a top-priority military asset: a 34-mile road to the country’s disputed border with China.

Far away in the capital New Delhi—a six-day journey by foot, road, rail and air—Prime Minister Narendra Modi’s office is monitoring their progress as he doubles down on a billion-dollar infrastructure plan in this contested region to strengthen India’s hand against its powerful neighbor.

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