Ibaguner Fractal Operator (IFO), A Complex-Operator Approach to Criticality Detection in Nonlinear Signals

We introduce a nonlinear complex-valued operator, termed the Ibaguner Fractal Operator (IFO), designed to capture coupled amplitudephase dynamics in complex systems. From this operator we derive a bounded scalar quantity, the Criticality Index (PCI), which quantifies proximity to critical regimes.We evaluate the method on representative signal classes, including stochastic noise, periodic signals, and chaotic dynamics. The results demonstrate consistent separation between these regimes. We further propose a conjectural connection between the PCI and geometric properties of operator-induced measures in the complex plane.