Deterministic Spatial Planning on Hypercomplex Manifolds: 1-Pass Maze Navigation with Zero Backpropagation and 53 KB Memory Footprint

Standard Reinforcement Learning (RL) and Frontier Foundation Models face severe structural bottlenecks in spatial planning. Deep RL algorithms (e.g., PPO, DreamerV3) suffer from extreme sample complexity (10^6–10^8 interactions) due to gradient dilution across delayed reward horizons, while Large Language Models (LLMs) display catastrophic looping beyond trivial topologies. We introduce the Hypercomplex Neuron (H-Neuron), a non-connectionist cognitive architecture that formulates spatial planning as closed-form geodesic traversal over an algebraic manifold M = R^d x {0, 1}^K. Operating over the non-commutative Lie group SO(3) via unit quaternions q in S^3, the H-Neuron achieves exact rotational invariance, single-pass trajectory synthesis without backpropagation (0 epochs, 0 FLOPs of gradient descent), and autonomous bit-level dead-branch pruning. On canonical Gym-MiniGrid benchmarks (DoorKey, MultiRoom-N6, KeyCorridor), the model synthesizes optimal paths in 2.80–4.07 ms in a single pass, delivering a 3,257x to 9,174x sample efficiency advantage over PPO. In continual multi-world benchmarks spanning 8 heterogeneous environments, the consolidated model retains all 8 worlds within a 53.14 KB binary archive with 0.00% catastrophic forgetting and 14,122 pruned cul-de-sac states.