A Rigorous Framework for 3+1-Dimensional Fractal-Noncommutative Yang-Mills Model Coupled with Quantum Gravity

This paper proposes a quantum gravity coupling framework integrating 3+1dimensional fractal spacetime with noncommutative Yang-Mills fields. By introducing fractal-modified metric tensors and noncommutative coordinate algebra, we establish rigorous definitions for fractal connections and curvature, constructing fractal Einstein field equations coupled with quantum gravity. Addressing compatibility between fractal metrics and general relativity, we prove covariance of the fractal Ricci scalar and fractal manifold generalization of the equivalence principle via fractal Lie derivatives and fractional jet bundle theory. To resolve fractal parameter observability challenges, we propose a joint calibration method combining gravitational wave dispersion relations with lattice QCD. Numerical simulations reveal the glueball mass scaling law mG(DH) = mG(4) · (DH/4)1/3, predicting fractal-corrected gravitational wave dispersion parameters γ ∈ [10−5,10−3] (corresponding to DH ∈ (3.9,4)). This work provides new mathematical tools and testable predictions for unifying quantum gravity and high-energy phenomenology.