The Gravitational Coupling from Mirror Symmetry: Solving the Hierarchy Problem via Dual Asymptotic Geometry

The hierarchy problem — why gravity is 10⁴² times weaker than electromagnetism — has resisted explanation for decades. This paper demonstrates that the gravitational coupling constant α_G ≈ 1.75 × 10⁻⁴⁵ emerges from the same dual asymptotic geometry that produces the fine structure constant α ≈ 1/137. Where electromagnetic interactions couple to the ceiling of the velocity spectrum via α = sin²(θ_ceiling/4), gravitational interactions couple to the floor via α_G = cos²(θ_floor/4) × (βf/α)². The offset angles are complementary: θ_ceiling + θ_floor = 180°, revealing a mirror symmetry through the 45° point. The (βf/α)² ≈ 3 × 10⁻⁴⁵ factor encodes the timescale ratio between floor and ceiling regimes. Using three independently measured inputs — G from gravitational experiments, α from QED, and βf from cold-atom interferometry — the geometric prediction matches the measured gravitational coupling to 0.5%. The hierarchy is not a problem; it is the ratio of ceiling to floor timescales, built into the dual asymptotic structure.