High-Order Gradient Geometry of Spacetime: Distinguishing Classical and Quantum Gravitational Structures with Gravitational Wave Data

A new approach to the geometric modeling of spacetime is developed by explicitly distinguishing between third-order gradients (∇³) as descriptors of refined classical geometry and fourth-order gradients (∇⁴) as markers of quantum gravitational structure. This framework is implemented through a detailed analysis of LIGO gravitational wave data [2]. The results establish, for the first time, separate empirical constraints on the detailed classical and quantum geometries of spacetime, as encoded in high-order derivatives of observable signals.The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this workl