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Hyperspectral imaging plays a critical role in numerous scientific and industrial fields. Conventional hyperspectral imaging systems often struggle with the trade-off between capture speed, spectral resolution, and bandwidth, particularly in dynamic environments. In this work, we present a novel event-based active hyperspectral imaging system designed for real-time capture with low bandwidth in dynamic scenes. By combining an event camera with a dynamic illumination strategy, our system achieves unprecedented temporal resolution while maintaining high spectral fidelity, all at a fraction of the bandwidth requirements of traditional systems. Unlike basis-based methods that sacrifice spectral resolution for efficiency, our approach enables continuous spectral sampling through an innovative ``sweeping rainbow" illumination pattern synchronized with a rotating mirror array. The key insight is leveraging the sparse, asynchronous nature of event cameras to encode spectral variations as temporal contrasts, effectively transforming the spectral reconstruction problem into a series of geometric constraints. Extensive evaluations on both synthetic and real data demonstrate that our system outperforms state-of-the-art methods in temporal resolution while maintaining competitive spectral reconstruction quality.