Fixed Point Self-Application Theory: Topological Realization of ψ0 = ψ0(ψ0)

We establish a rigorous mathematical framework for complete self-reference through topological fixed point theory. Building on the spectral fixed point established in Paper R4, we construct function spaces Hζ where mathematical objects can describe their own properties, achieving the topological realization of ψ0 = Λ(ψ0)(ψ0) without paradox. Using Scott domain theory and Banach fixed point techniques, we prove existence and uniqueness of self-referential fixed points, resolving classical paradoxes through continuous topology rather than discrete logic. The construction preserves the (2/3,1/3, 0) probability structure and golden ratio scaling throughout the self-application process, demonstrating that complete mathematical self-reference is not only possible but necessary within properly constructed topological frameworks.