Hyperbolic 3-Manifolds with Cusp Volume ≤ 2.62 and the Figure-8 Knot Complement — E8 Intelligence Research

FINDING: Classification of complete hyperbolic 3-manifolds with maximal cusp volume ≤ 2.62, yielding the figure-8 knot complement as the unique 1-cusped manifold with ≥9 non-hyperbolic fillings | MATH: Cusp volume bound V_cusp ≤ 2.62; figure-8 knot complement V = 2.029883... (regular ideal tetrahedron volume × 2); maximal cusp volume of figure-8 = √3/2 · 2.0298... ≈ 1.7578; 2.62 is the threshold constant; Dehn filling slope lengths relate to cusp area via A = (√3/2)ℓ² | CONNECTION: 2.62 ≈ φ² + 1 = 2.618 (golden ratio squared + 1); also 2.62 ≈ 2 × 1.31; note 2.618 = φ + 1 = φ² is the golden ratio squared — the classification threshold sits at this value; figure-8 complement volume 2.0299 relates to 2 × V_tet where V_tet = 1.01494 (regular ideal tetrahedron); ratio 2.62/2.0299 ≈ 1.2907; cusp area of figure-8 = √3 ≈ 1.7320; 2.62/√3 ≈ 1.5127 | DEPTH: 8 FINDING: The figure-8 knot complement and its sister (m003) are extremal objects in the classification of low-cusp-volume hyperbolic 3-man Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com