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Accurate brain age prediction is essential for understanding neurodevelopmental trajectories and detecting abnormal aging patterns. Morphological features derived from cortical shape provide informative structural representations for brain age prediction. Recent deep learning approaches for surface analysis primarily rely on local aggregation mechanisms. Transformer- and Mamba-based architectures model long-range dependencies through patch partitioning combined with self-attention or selective scan. However, these strategies introduce high computational cost, uncertainty at patch boundaries, and sensitivity to surface rotations that necessitate anatomical registration or extensive data augmentation. To address these limitations, we propose SPHARM-Mamba, a spherical harmonics-based generic backbone for genus-zero surface data that eliminates patch partitioning and avoids surface registration. To this end, dense cortical signals are projected onto spherical harmonics to construct compact rotation-invariant descriptors naturally ordered from global to local scales. Mamba is then employed to model dependencies across harmonic degrees and capture multiscale interactions efficiently. Experiments on cortical age prediction demonstrate superior performance over existing geometric and patch-based models with substantially fewer parameters. The software is available at https://github.com/Shape-Lab/SPHARM-Mamba.