Regular Simplex Hierarchical Gravity Part IV: Geometric Origin of the Fine-Structure Constant - Zero-Parameter Derivation from 600-Cell Projection and Discovery of the Ghost Edge Singularity

Key Highlights • Zero-Parameter Derivation: Demonstrates that the universal weight determining the fine-structure constant is strictly derived from the 600-cell topology, requiring absolute zero adjustable continuous parameters. • Integer Geometry over Numerology: The core derivation (W₁ = 12/120 5 = 0. 5) is not a numerical coincidence, but a direct product of topological invariants: the surface projection ratio (12/120) and the tetrahedral multiplicity (5). • Discovery of the "Ghost Edge": Reveals a 721st edge in the 600-cell projection caused by the deficit angle (7. 36^), providing a geometric origin for the arrow of time. Abstract The theoretical derivation of the fine-structure constant 1/137. 036 has remained unsolved since Sommerfeld's discovery in 1916. This paper demonstrates that the universal weight W₁=0. 5 of the Pan Framework, which determines, can be derived from the geometric structure of the 600-cell—a four-dimensional regular convex polytope—with zero adjustable parameters. The derivation relies on the product of two integer ratios: stereographic projection yields 12 surface vertices out of 120 total (12/120=0. 1), and multiplying by the topological fact that five regular tetrahedra meet at each edge gives W₁=0. 5. Sensitivity analysis confirms this result is a geometric necessity, not parameter adjustment. Additionally, introducing the deficit angle 7. 36^ produces 721 edges instead of the theoretical 720, revealing a "Ghost Edge" singularity that suggests a geometric origin for the arrow of time through symmetry breaking. This work completes the RSHG (Regular Simplex Hierarchical Gravity) trilogy by connecting discrete integer geometry to continuous wave description, unifying the derivation of gravitational constant G, cosmological constant, and fine-structure constant from a single geometric substrate. The RSHG Framework Series • Part I: Geometric derivation of G and • Part II: Projection caustics and mass spectrum • Part III: Hierarchical jamming and energy suppression • Part IV (This paper): Fine-structure constant and Ghost Edge