Fractal Holographic Emergent String Theory (FHEST) Part D: A Unified Framework for Energy-Mass Relations and Field Equations in Fractal Theory

This paper proposes a unified framework for energy-mass relations and field equations based on fractal theory. In the context of fractal spacetime, the classical Einstein energy-mass equation and Einstein field equations require modifications to incorporate the effects of fractal dimensions and multi-scale geometry. By introducing fractal dimension correction terms and fractal curvature integral terms, we derive the fractal energy-mass formula and the fractal-modified Einstein field equations. These two formulas not only align formally with classical theories but also introduce critical corrections, providing a quantitative foundation for the experimental validation of fractal spacetime theories. This paper discusses the physical implications of these corrections and potential observable signals in detail, offering new theoretical tools for advancing research in quantum gravity, dark energy, and fractal superconductivity.