At (+∞ and −∞) Conjugate Asymptotic Symmetry, Fractal Reciprocity, and Dual-Boundary Cosmology in the Ibaguner Fractal Operator (IFO) Framework

We investigate the asymptotic and symmetry properties of the Ibaguner Fractal Operator (IFO), establishing a conjugate-reflection invariance that induces norm preservation under sign inversion. This symmetry gives rise to a principle of Fractal Reciprocity, wherein the asymptotic limits at +∞ and −∞ form a conjugate pair rather than independent endpoints. We formalize the resulting dynamical structure, introduce a minimal asymptotic model, and provide phase portrait evidence supporting a dual-boundary interpretation. Within the Final Ibaguner Conjecture (FIC), this leads to a cosmological formulation in which expansion and contraction
correspond to conjugate trajectories constrained by a shared invariant structure.