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Determine experimentally whether round-trip gain exceeding total cavity loss, sustained exponential growth over many passes, and saturation with relative bandwidth $\\sim 10^{-7}$ can be obtained with existing high-repetition-rate linac beams, given crystal heat load and cavity alignment stability.","problem_ref":"2590722c932c601bc46523cec65d6f0d3e0892d5481d07c0791e0d31ff5e7ce1","references":"","settled_by":"Observation of exponential growth of stored x-ray pulse energy over many successive passes, followed by saturation.","status_note":"Spectral narrowing and proof-of-concept multi-pass amplification were reported at the European XFEL demonstrator (Li, Rauer, Huang, Deng, review arXiv:2608.15244, https://arxiv.org/abs/2608.15244, 2026); sustained growth to saturation is not reported.","title":"Exponential growth to saturation in a hard x-ray FEL 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