Energy-Geometry Coupling in Quantum Gravity: Framework and Physical Predictions

This paper develops a mathematical framework for energy-geometry coupling in quantum gravity. We propose two fundamental hypotheses: a functional coupling between metric geometry and energy-momentum distributions, and a spacetime strain formulation inspired by continuum mechanics. From these principles, we derive a master integral equation that generalizes Einstein's field equations while incorporating non-local and quantum gravitational effects. The framework systematically recovers general relativity in the appropriate limit and explores approaches to spacetime singularity regularization through non-local geometric effects. We address the cosmological constant problem through geometric compensation mechanisms and derive testable predictions for gravitational wave propagation. The work maintains mathematical consistency while identifying open challenges in quantization and renormalization.