Bhargava-Shankar: Positive Proportion of Elliptic Curves Have Rank 0 and 1 — E8 Intelligence Research

FINDING: Bhargava-Shankar proved that a positive proportion of elliptic curves over ℚ have rank 0 and rank 1, via exact averages of 2- and 3-Selmer group sizes (4 and 6 respectively), ordered by height. MATH: - Average size of 2-Selmer group = 3 (Bhargava-Shankar, 2010) - Average size of 3-Selmer group = 4 (Bhargava-Shankar, arXiv:1007.0052) - Combined: positive proportion of curves have rank 0 (≥ 25% from 2-descent alone; ≥ 50% with 3-descent) and rank 1 (positive proportion, complementary). - Key theorem: For elliptic curves E: y² = x³ + Ax + B, height H(E) = max(|A|³, |B|²), the count of curves with H < X is ~ (constant)·X^(5/6). - The average rank (conjectured by Goldfeld, proven conditional on BSD) is 0.5 — the arithmetic mean of ranks 0 and 1 in equal proportion. - Constants: The average 2-Selmer size 3 and 3-Selmer size 4 are exact integers — no asymptotic error beyond the main term. - The ratio 3/4 (2-Selmer avg / 3-Selmer avg) = 0.75 — not a golden ratio, but th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com