SL(2,Z) as Z/4 *_{Z/2} Z/6 and the Two-Cone-Point Modular Orbifold — E8 Intelligence Research

FINDING: SL(2,Z) is the amalgamated free product Z/4 *_{Z/2} Z/6, with torsion entirely concentrated at elliptic points of order 2 and 3, forcing all elliptic curves' moduli to live on the modular orbifold with exactly two cone points. | MATH: PSL(2,Z) ≅ (Z/2) * (Z/3) (free product); SL(2,Z) ≅ Z/4 *_{Z/2} Z/6. Elliptic points: τ = i (order 2, stabilizer ⟨S⟩, S² = −I), τ = ρ = e^{iπ/3} (order 3, stabilizer ⟨ST⟩, (ST)³ = −I), τ = ρ² (order 3). Fundamental domain area = π/3. Modular invariant j: H/SL(2,Z) → C with j(i) = 1728, j(ρ) = 0. Torsion subgroup of E(Q) constrained by Mazur: E(Q)_tors ∈ {Z/nZ, n=1..10,12} ∪ {Z/2×Z/2n, n=1..4}. Nagell–Lutz: torsion point (x,y) ⟹ x,y ∈ Z and y² | 4a³+27b². | CONNECTION: The orders {2,3} are the crystallographic restriction in 2D — only 2-, 3-, 4-, 6-fold rotations tile the plane. SL(2,Z) torsion realizes exactly the 2- and 3-fold generators; the 4- and 6-fold appear as products (S has order 4, ST has order 6). This is the arithmetic shadow of the he Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com