Clifford Mamba: State Space Models with Geometric Product\\State Transitions for 3D Point Cloud Processing

We introduce \CM{}, a state space model (SSM) that replaces scalar element-wise state transitions with Clifford algebra geometric products in $\Cl(3,0)$. Each hidden state is an 8-dimensional multivector---encoding scalars, vectors, bivectors, and trivectors---and state updates are contractive rotors that simultaneously decay and rotate in the algebra. Combined with Morton Z-order curve linearization and bidirectional scanning, this yields a sequence model that natively processes 3D point clouds while preserving geometric structure. On N-body dynamics prediction, \CM{} achieves $7.3\times$ lower MSE than EGNN and $50\times$ lower than HC-Net (hybrid graph network) with $n{=}5000$ training samples. At matched parameter counts (${\sim}560$K), \CM{} still outperforms EGNN by $5.2\times$. The architecture maintains $\mathcal{O}(N)$ memory scaling---EGNN runs out of memory at $N{=}2000$ while \CM{} processes it in 16.8\,GB---and generalizes to larger systems ($N{=}50$) where it achieves $7.66 \times 10^{-6}$ MSE vs EGNN's $9.31 \times 10^{-6}$. Ablation confirms that the geometric product state transition contributes a consistent 13\% improvement over scalar SSM baselines.