A Rigorous Proof of the Fractal Leibniz Law,Correction of Fractal Derivatives and Analysis of Compatibility with Geometric Structures

This paper proposes a corrected fractal Leibniz law based on the geometricproperties of fractal manifolds, and provides a rigorous mathematical proof andnumerical verification. By combining fractional Caputo derivatives with the scal?ing laws of fractal measures, the fractal derivative operator suitable for non-integerdimensional manifolds is defined, correcting the local assumption in the classi?cal Leibniz law. Using the variational principle and fractal Hardy inequality, theuniqueness of the correction term is proven, and the self-consistency of the theo?retical results is verified on the Sierpi´nski manifold through discretization methods(with an error less than 10−4). This work provides a foundation for differentialoperations in field theory modeling on fractal spacetime, especially suitable for thephysical description of non-local interaction systems.