Empirical Validation of the 5D Nodal Restorative Law and 6D Parameter Space Integrity in 6-DOF Robotic Systems

Empirical Validation of the 5D Nodal Restorative Law and 6D Parameter Space Integrity in 6-DOF Robotic Systems Description / Abstract This dataset and preprint provide the definitive empirical validation of the "5D Nodal Restorative Law" within the configuration space of a 6-degree-of-freedom (DOF) industrial manipulator. While theoretical models in high-dimensional geometry predict a discrete stability manifold based on the $A_5$ root lattice, this work demonstrates the physical manifestation of these 109 equilibrium nodes. Using an ABB IRB 4600 manipulator within a high-fidelity Webots digital twin environment, we conducted 109 targeted simulations to verify the mathematical and kinematic integrity of the 6D parameter space. Key findings included in this record: Lattice Recovery: Evidence of the 109-node $A_5$ lattice structure recovered from robotic joint parameters $(q_1, q_2, q_3)$. Precision Metrics: Documentation of a mean orthonormality error of $4.42 \times 10^{-16}$, representing the convergence of physical robotic stability to the theoretical limits of machine epsilon. Statistical Validation: Spatial analysis using Ripley’s K-test ($< 1 \times 10^{-10}$) confirming the non-random, isomorphic nature of the settled nodal points. This work concludes the trilogy of research into high-dimensional lattice constraints in robotics, providing a rigorous framework for singularity-free path planning and restorative control laws. Keywords Robotics; A5 Lattice; Configuration Space; Webots; ABB IRB 4600; 6-DOF; Kinematic Integrity; Machine Epsilon; Nodal Restorative Law.