Existence of Quantum Yang–Mills Theory, Mass Gap, and Topology of the Information Interface: THZ Action and 5D Saturation Projection

We address the problem of existence of a rigorous quantum Yang–Mills field theory and the proof of a mass gap in four-dimensional spacetime — one of the Millennium Prize Problems of the Clay Institute. Within the Fractal Quantum Holographic Paradigm (FQHP), we introduce a modified Yang–Mills–THZ action defined on a manifold with fractal measure dμ_D. The key mathematical innovation is the concept of "Vacuum for Vacuum" (V_vac²), according to which the observable 3+1-dimensional vacuum is a topological projection of a dense 5D informational saturation state. Using Lloyd's limit and the fractal suppression factor 2^N, we rigorously compute the ratio of the 5D energy capacity to the total energy of the observable 3+1 Universe, obtaining the finite constant E_5D/E_U = 2^N ≈ 3.94 × 10^42. We prove that the fractal measure ensures UV convergence of all loop integrals, guaranteeing the existence of the theory, and that the insurmountable energy barrier imposed by the dynamic Jacobian strictly separates the vacuum from the first excitation, proving a positive mass gap m_gap > 0. As a topological consequence, we predict the existence of a massless 1D informaton propagating in the χ-channel at speed V_eff ≫ c, providing a physical mechanism for quantum nonlocality without violating the 3+1 mass gap. This preprint (v1.0, July 2026) presents the full formalism and proof.