Construction of Exact Solutions for the 2+1-Dimensional Fractal-Noncommutative Yang-Mills Model

This paper rigorously constructs analytic solutions for the noncommutative Yang-Mills model on a 2+1-dimensional fractal manifold by integrating fractal geometry with noncommutative algebra. Addressing the challenges of fractal BPS equation validity and fractal lattice simulation feasibility, we propose the following advancements: 1. Strict Proof of Fractal Bogomolny Decomposition: Existence of energy lower bounds under fractional variational principles. 2. Mathematical Treatment of Fractal Topological Constraints: Instantonic suppression via fractal homology theory, proving constraints on instanton numbers by the fractal Euler characteristic. 3. Error Control in Fractal Lattice Simulations: A Hausdorff measurebased discretization scheme and regularization algorithm ensuring numerical stability in glueball mass calculations. Numerical simulations reveal that the glueball mass scaling law satisf ies mG(DH) = mG(3)·(DH/3)1/2, consistent with fractal energy gap theory. This work establishes a new paradigm for the experimental realization and mathematical rigor of fractal quantum field theories.