Autonomous Trajectory Redirection on Non-Linear Multi-Body Manifolds: A Moore-Penrose Gravitational Resolvent Approach to Orbital Keyhole Avoidance

Autonomous trajectory redirection and kinetic deflection of hazardous orbital bodies and near-Earth asteroids demands real-time terminal guidance capable of operating within non-linear, perturbed multi-body gravitational fields without high-latency ground loops. We formulate a geometric guidance framework embedding the relative phase space into a discrete symplectic transition manifold and computing the discrete Laplace-Beltrami Green's function via the Moore-Penrose pseudoinverse L†. Monotonic descent toward terminal capture without local saddle trapping is established via the discrete Maximum Principle, with continuous terminal proportional closing governed by a dimensionally homogeneous quadratic Lyapunov function. Validated against the April 13, 2029 Earth close encounter of (99942) Apophis under full J2 and lunar tidal perturbations, the guidance law navigates a continuous Pareto frontier from a fuel-optimal cruise (Δv = 28.4 m/s) to an emergency sprint mode (18.5 min closure). We further formulate the dual-use extension to extra-atmospheric orbital asset defense, demonstrating autonomous, trap-free Collision Avoidance Maneuvers (CAM) and Active Debris Removal (ADR) for space stations and orbital infrastructure in vacuum.