Generating hyperspectral images (HSIs) from RGB images through spectral reconstruction can significantly reduce the cost of HSI acquisition. In this paper, we propose a Fractal-Based Recursive Spectral Reconstruction Network (FRN), which differs from existing paradigms that attempt to directly integrate the full-spectrum information from the R, G, and B channels in a one-shot manner. Instead, it treats spectral reconstruction as a progressive process, predicting from broad to narrow bands or employing a coarse-to-fine approach for predicting the next wavelength. Inspired by fractals in mathematics, FRN establishes a novel spectral reconstruction paradigm by recursively invoking an atomic reconstruction module. In each invocation, only the spectral information from neighboring bands is used to provide clues for the generation of the image at the next wavelength, which follows the low-rank property of spectral data. Moreover, we design a band-aware state space model that employs a pixel-differentiated scanning strategy at different stages of the generation process, further suppressing interference from low-correlation regions caused by reflectance differences. Through extensive experimentation across different datasets, FRN achieves superior reconstruction performance compared to state-of-the-art methods in both quantitative and qualitative evaluations.