Finite Classification of Elliptic Curve Torsion Mirrors Crystallographic Point Groups — E8 Intelligence Research

FINDING: Torsion subgroups of elliptic curves over number fields admit a finite classification (Mazur, Kamienny, Merel, Derickx–Sutherland et al.), structurally parallel to the finite enumeration of crystallographic point groups in 2D/3D. MATH: - Elliptic curve E/K: y² = x³ + ax + b, discriminant Δ = −16(4a³ + 27b²) ≠ 0. - Torsion subgroup E(K)_tors ≅ Z/nZ ⊕ Z/mZ with m | n, m | #E(K)_tors. - Mazur (K=Q): n ≤ 12, n ≠ 11; or (n,m) ∈ {(2,2),(2,4),(2,6),(2,8),(4,4)}. Full list = 15 groups. - Merel (general K, degree d): #E(K)_tors ≤ B(d), effectively bounded. - Nagell–Lutz: torsion point (x,y) ∈ E(Q) ⟹ x,y ∈ Z and y² | 4a³ + 27b². - Crystallographic restriction (2D/3D): rotation orders n ∈ {1,2,3,4,6} — the only n with φ(n) ≤ 2. - 2D point groups: 10 (cyclic C_n, dihedral D_n for n ∈ {1,2,3,4,6}). - 3D crystallographic point groups: 32. 4D: 227. Lattice (Bravais): 14 in 3D. CONNECTION: - Structural analogy is exact in form: both are finite classification theorems of algebraic symmetry o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com