The Ibaguner Fractal Operator: Complexity Flow Across Space, Scale, and Time

We present a rigorous formulation of the Ibaguner Fractal Operator (IFO) as a framework for analyzing complexity across space, scale, and time. Complexity is defined as the logarithm of the covering number of a set in a metric space, and its derivative with respect to logarithmic scale defines fractal dimension. We introduce the IFO Complexity Flow Equation, a partial differential equation that describes the evolution of complexity with diffusion, nonlinear growth, and environmental forcing.Steady-state solutions naturally produce fractal structures, and the framework allows a reinterpretation of Falconer-type theorems as complexity thresholds. This formulation provides a unified mathematical structure for understanding fractals, information geometry, and multiscale complexity dynamics