Discrete Angular Deficit Equals Gaussian Curvature: Gauss-Bonnet Unifies Geometry and Topology — E8 Intelligence Research

FINDING: Discrete angular deficit on polyhedra equals total Gaussian curvature; Gauss-Bonnet unifies local geometry with global topology across all dimensions. MATH: - Gauss-Bonnet (2D closed surface): ∮_M K dA = 2πχ(M), where χ = V − E + F (Euler characteristic) - Polyhedral defect angle: δ_v = 2π − Σθ_i (sum of face angles at vertex v) - Discrete Gauss-Bonnet: Σ_v δ_v = 2πχ(M) - Descartes' defect theorem (polyhedra): Σ_v δ_v = 4π for any convex polyhedron (χ = 2) - Generalization to 2n dimensions: ∫ Pf(Ω) = (2π)^n χ(M) — Pfaffian of curvature form - Odd dimensions: χ(M) = 0 for closed odd-dimensional manifolds (Chern-Gauss-Bonnet vanishes) - Higher-dim polyhedral analogue: Σ δ_σ = (2π)^n χ(M), where δ_σ is the solid-angle deficit at each (2n−2)-face CONNECTION: - Platonic solids: all five satisfy Σδ_v = 4π. Tetrahedron: δ = π (180°); Cube: δ = π/2; Octahedron: δ = 2π/3; Dodecahedron: δ = π/5; Icosahedron: δ = π/3. These deficits encode the 5-fold, 3-fold, 2-fold symmetries of the i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com