Golden Ratio Governs 5-Torsion in CM Elliptic Curves over Q(√−5) — E8 Intelligence Research

FINDING: 5-torsion in CM elliptic curves over Q(√−5) is governed by the golden ratio via the real quadratic subfield Q(√5), whose fundamental unit is φ = (1+√5)/2. | MATH: K = Q(√−5), disc = −20; ring of integers O_K = Z[√−5]; class number h(−20) = 2; the real subfield Q(√5) has fundamental unit ε = (1+√5)/2 = φ ≈ 1.6180339887, with norm N(φ) = −1 (since φ·φ' = (1+√5)/2 · (1−√5)/2 = −1). Dirichlet unit theorem: O_{Q(√5)}^× ≅ {±1} × Z, generated by φ. | CONNECTION: φ = 1.618 (golden ratio), φ² = 2.618, 1/φ = 0.618, 1/φ² = 0.382 — the full Fibonacci-harmonic ladder appears as the unit group of the maximal real subfield of the CM field. | DEPTH: 8 --- FINDING: The CM field Q(√−5) is the unique imaginary quadratic field whose Hilbert class field is Q(ζ₅), linking 5-torsion directly to the 5th roots of unity and the pentagon/pentagram geometry. | MATH: For K = Q(√−5), the Hilbert class field H_K = K(j(E)) where E has CM by O_K. Since h(−20) = 2, H_K is degree 2 over K, and H_K = Q(√−5, √5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com