Topological Unification of the Hierarchy Problem and Fundamental Constants: Closed-Form Derivations of Gauge Couplings, Mass Ratios, and Newton's Constant from a Parameter-Free Hopf Fibration Invariant

**Problem---** The origin of fundamental physical constants, particularly the enormous hierarchy of approximately 10^17 orders of magnitude between gauge couplings and gravitational strength, constitutes one of the deepest puzzles in theoretical physics. The Standard Model cannot predict the absolute values of these constants, while candidate theories of quantum gravity have yet to provide an accurate spectrum of constants. **Method---** Based on the S^3 Hopf fibration and the holographic principle, this paper proposes a parameter-free bulk holographic topological invariant I_bulk = 4π^3 + π^2 + π = π(π(4π+1)+1), and demonstrates the algebraic uniqueness of this form under explicit constraints. **Basic Validation---** This single invariant simultaneously reproduces six independent Standard Model parameters with high precision: the fine-structure constant α⁻¹ (relative deviation 2.2×10⁻⁶), the weak mixing angle sin^2 θ_W (deviation 1.6×10⁻³), the strong coupling constant α_s (deviation 1.4×10⁻³), the Cabibbo angle sin θ_C (deviation 3.5×10⁻⁴), the proton-to-electron mass ratio m_p/m_e (deviation 1.9×10⁻⁵), and the muon-to-electron mass ratio m_μ/m_e (deviation 5.9×10⁻³), thereby validating the topological kernel. **Core Breakthrough---** Building upon this foundation, the framework naturally gives rise to dual benchmark energy scales for mass and gravity. We derive for the first time the complete topological closed-form expressions for the hierarchy factor and the gravitational constant: R = (5 I_bulk9.5) / (8(4π+1)), G = (64π(4π+1)^2) / (I_bulk²⁰), with theoretical values deviating from experimental observations by merely -0.78% and +0.01%, respectively, achieving a high-precision prediction of the gravitational constant without any free parameters. **Theoretical Extensions---** The framework further naturally yields closed-form expressions for the cosmological constant Λ and the heaviest neutrino mass m_ν3: Λ_planck = 8 / ((4π+1)^6 I_bulk⁵⁴), m_ν3/m_e = 1 / (2(4π+1) I_bulk^4), with deviations from observational values of -3.5% and the same order of magnitude, respectively, extending the constant network to cosmology and neutrino physics. **Significance---** This work demonstrates that the extreme weakness of gravity, the profound smallness of the cosmological constant, and the scale of neutrino masses are essentially geometric effects arising from the power-law discrepancy between the topological complexity of the spacetime bulk (bulk moduli space) and the boundary projection dimension. This framework unifies gravity, the cosmological constant, and quantum gauge interactions within a single topological algebraic network, achieving a high degree of phenomenological unification and self-consistency of fundamental constants without free parameters, and providing a set of precise "Rosetta Stone" boundary conditions that any quantum gravity theory must satisfy. **Keywords:** Hopf fibration, topological invariant, hierarchy problem, gravitational constant, cosmological constant, fine-structure constant, gauge coupling unification, holographic principle.