Deterministic 4D Spacetime Geodesic Planning on Hypercomplex Toroidal Manifolds

Abstract

Dynamic obstacle avoidance and 4D spacetime pathfinding present severe computational bottlenecks in autonomous robotics. Classical time-expanded graph architectures (e.g., Space-Time A*) suffer from catastrophic state explosion O(|V| x T_horizon), exhausting memory over extended planning horizons. Local continuous optimizers (Timed Elastic Band, Velocity Obstacles) become trapped in non-convex barrier minima or fail when corridors mandate tactical waiting, while Deep Reinforcement Learning (PPO) requires millions of interactions and remains brittle to phase shifts. Building upon prior work, we introduce the 4D Spacetime Hypercomplex Neuron (4D-H-Neuron). Rather than treating time as an unbounded linear dimension, the architecture projects multi-periodic dynamics onto a compact torus M_4D = R^3 x T^m. By exploiting the Chinese Remainder Theorem, compound cyclic airlocks are resolved algebraically in O(1) without temporal branching. Furthermore, stationary holding is formulated as an active geodesic along the toroidal fiber, natively synthesizing phase-locked breaches through moving barriers. In empirical benchmarks on a 3-chamber, 3-airlock 4D dynamic labyrinth (World 9), the 4D-H-Neuron achieves 100% mission success with 0.00% collisions in 340.90 ms (4.16x faster than Space-Time A*), utilizing a 5.32 KB bitwise chunk latch (1,684x lighter than Space-Time A* and 9,022x lighter than PPO) with zero backpropagation and zero training epochs.