"Relational Metric Approach to Gravitational Energy: Compact System with Invariants and Symmetry Analysis"

System Description The proposed framework establishes a compact algebraic system of gravitational energy compensation, where the potential energy of a body is expressed as the exact balance between dual mass channels (inertial and gravitational) and a relational metric that encodes inter-object geometry. The system introduces a local constant kBk_B, derived from the ratio of the body’s gravitational field to a terrestrial reference, modulated by a universal invariant uu. This ensures that the formulation is transferable across different gravitational environments without loss of structural consistency. A key innovation lies in the substitution of the internal length scale ℓ\ell with a relational metric Λ(L)\Lambda(L), defined by the inter-object distance LL normalized to a reference length. This transformation shifts the system from a purely local description to a relational geometry, enabling direct comparison of energy differences between bodies such as Earth and Moon. The resulting expressions preserve dimensionality in SI units and yield clear, scalable measures of energy variation, gradients, and curvature. Although the system is not intended as a reformulation of General Relativity, it resonates with the principles of Special Relativity: by construction, the propagation of gravitational influence is treated as consistent with the invariant speed of light. In this sense, the framework aligns with the relativistic requirement that both light and gravity share the same limiting velocity, ensuring that the algebraic compensation remains compatible with relativistic kinematics. This subtle correspondence provides a didactic bridge between classical gravitational potential and relativistic mechanics, without invoking the full tensorial machinery of curved spacetime.