Discrete Topological Charges in Fractal Chern-Simons Theory: Rigorous Proof and Experimental Corrections

This paper generalizes Chern-Simons theory to fractal manifolds, rigorously proving the discreteness of topological charges in non-integer dimensions. By introducing fractal cohomology groups and the Strichartz framework of fractal differential geometry, we define fractal curvature forms and integration measures. The corrected theory shows that the fractal dimension modulates topological charges via a scaling enhancement effect, leading to experimentally observable Hall conductance: σxy = ν e2 h , ν = dH ·Z or ν = p q ·dH(p,q ∈ Z), where dH is the Hausdorff dimension. Numerical calculations and comparisons with fractal quantum Hall experiments validate the theoretical framework