Fundamental Quantum Spacetime Equation: Geodesic Autonomy and Unified Field Structure in D ≥ 9

This work presents a generalized tensorial formulation aimed at describing the geometric structure of spacetime within an extended differential geometric framework. The proposed model builds upon the experimentally validated foundations of General Relativity and relativistic field theory, incorporating an effective metric tensor Gμν defined as an extension of the classical spacetime metric. The formulation is grounded in the well-established experimental confirmations of spacetime curvature, including gravitational time dilation, gravitational lensing, and the detection of gravitational waves by the LIGO Scientific Collaboration. These observations confirm that spacetime geometry governs the propagation of matter, radiation, and causal structure. Within this context, the generalized tensor modifies local geometric properties while preserving consistency with the differential geometric structure used in Einstein’s field equations. Geodesic trajectories are determined by the associated connection coefficients, and causal structure remains defined by the effective metric. The framework is expressed using tensor calculus and differential geometry, ensuring compatibility with the established mathematical structure of gravitational physics. This work proposes a mathematical extension intended for theoretical investigation while remaining consistent with experimentally verified principles of spacetime geometry.