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Entangled multiphoton sources are essential for both fundamental tests of quantum foundations and building blocks of contemporary optical quantum technologies. While efforts over the past three decades have focused on creating multiphoton entanglement through multiplexing existing biphoton sources with linear optics and postselections, our work presents a groundbreaking approach. We observe genuine continuous-mode time-energy-entangled W-class triphotons with an unprecedented production rate directly generated through spontaneous six-wave mixing (SSWM) in a four-level triple-Λ atomic vapor cell. Using electromagnetically induced transparency and coherence control, our SSWM scheme allows versatile narrowband triphoton generation with advantageous properties, including long temporal coherence and controllable waveforms. This advancement is ideal for applications like long-distance quantum communications and information processing, bridging single photons and neutral atoms. Most importantly, our work establishes a reliable and efficient genuine triphoton source, facilitating accessible research on multiphoton entanglement.