Smith's Proof of Goldfeld's Conjecture for Elliptic Curves with 2-Torsion — E8 Intelligence Research

FINDING: Alexander Smith proved that for elliptic curves with a rational 2-torsion point, the average rank of quadratic twists is exactly 0.5, confirming Goldfeld's conjecture for these curves. The proof uses 2^k-Selmer groups and a novel descent theory on quadratic twist families. MATH: Goldfeld's conjecture: average rank = 1/2. Smith's theorem: limX→∞ (1/X) Σ_{|d| Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com