A novel sequence architecture is introduced, Versor, which uses Conformal Geometric Algebra (CGA) in place of traditional linear operations to achieve structural generalization and significant performance improvements on a variety of tasks, while offering improved interpretability and efficiency. By embedding states in the $Cl_{4,1}$ manifold and evolving them via geometric transformations (rotors), Versor natively represents $SE(3)$-equivariant relationships without requiring explicit structural encoding. Versor is validated on chaotic N-body dynamics, topological reasoning, and standard multimodal benchmarks (CIFAR-10, WikiText-103), consistently outperforming Transformers, Graph Networks, and geometric baselines (GATr, EGNN). Key results include: orders-of-magnitude fewer parameters ($200\times$ vs. Transformers); interpretable attention decomposing into proximity and orientational components; zero-shot scale generalization (0.993 vs. 0.070 MCC for ViT); and featuring a Recursive Rotor Accumulator (RRA) for $O(L)$ linear temporal complexity in dynamical systems, and a Geometric Product Attention (GPA) mechanism for $O(L^{2})$ global relational modeling, allowing for task-specific architectural pruning or hybridization depending on the required scale. In out-of-distribution tests, Versor maintains stable predictions while Transformers fail catastrophically. Custom Clifford kernels achieve a cumulative over $100\times$ speedup via bit-masked contraction and specialized Matrix Isomorphism kernels, reducing per-step latency to 1.05 ms and outperforming highly-optimized Transformer baselines.