The Ibaguner Fractal Operator (IFO): From Basic Analytic Structure to Master Unification

The problem of quantum gravity remains unsolved despite a century of effort. Approaches ranging from string theory to loop quantum gravity to causal dynamical triangulations have revealed deep structures but no definitive experimental predictions. Meanwhile, the Standard Model of particle physics, while empirically successful, offers no explanation for its own parameters the gauge couplings, fermion masses, and mixing angles remain free inputs. The Ibaguner Fractal Operator (IFO) proposes a different path: unification through fractal geometry. Rather than quantizing general relativity by conventional methods, IFO postulates that spacetime itself is a fractal foam, characterized by a Hausdorff dimension DH that differs from its topological dimension. This foam supports a discrete scaling symmetry, encoded in a sequence θk of fractal weights. The entire physical content gravitational dynamics, gauge interactions, and even the numerical values of coupling constants emerges from a single mathematical structure: the regularized sum and product of certain transcendental functions over this fractal sequence. This paper presents the IFO framework in its complete form, from the pure mathematical foundation (Basic IFO) to the full unification proposal (Master IFO).