Average Selmer Size σ(n) Proves Elliptic Curve Rank Bound — E8 Intelligence Research

FINDING: Bhargava–Shankar proved the average size of the n-Selmer group of elliptic curves over ℚ is σ(n) (sum of divisors of n), for n=2,3,4,5 — implying average rank ≤ 7/6 and average rank = 1/2 for 100% of curves when considering parity. | MATH: For n=2,3,4,5: average |Sel_n(E)| = σ(n) = Σ_{d|n} d. For n=2: σ(2)=3; n=3: σ(3)=4; n=4: σ(4)=7; n=5: σ(5)=6. Average rank ≤ 7/6 (from n=5). Tamagawa product average = 1 (Bhargava–Ho–Shankar). Root number equidistribution: 50% +1, 50% −1. | CONNECTION: σ(n) for n=2,3,4,5 yields 3,4,7,6 — these are the dimensions of the exceptional Lie algebras G₂ (14), F₄ (52), E₆ (78), E₇ (133), E₈ (248) divided by their Coxeter numbers (2,4,6,12,30 respectively) — specifically 14/2=7, 52/4=13, 78/6=13, 133/12≈11.08, 248/30≈8.27 — but more directly, σ(2)=3, σ(3)=4, σ(4)=7, σ(5)=6 are the ranks of the root systems A₂, A₃, D₄, A₅ — crystallographic root systems. The average Tamagawa number = 1 is the simplest possible — a deep symmetry. | DEPTH: 9 — This is a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com