Bounded Average Elliptic Curve Rank via Selmer Groups and Parity Jumps — E8 Intelligence Research

FINDING: Average rank of elliptic curves is bounded (≤0.885) via Selmer group averages; parity and cyclic sextic extensions reveal rank jumps governed by root-number symmetries. | MATH: - Average rank ≤ 0.885 (Bhargava–Shankar, via average size of n-Selmer groups: lim sup avg rank ≤ Σ(1 − 1/p) over primes, yielding 0.885 for n=2,3,5,7 combined). - Parity conjecture: rank parity = sign of functional equation (root number ε = ±1); Dokchitser: ε determined by local root numbers, product over places. - Cyclic sextic extensions: rank growth in towers — rank(E(K_n)) − rank(E(K)) can be unbounded for fixed E over sextic cyclic fields (Kisilevsky), linked to Iwasawa theory and Selmer growth. - Height pairing matrix: eigenvalues relate to regulator; average regulator conjectured ~constant (Goldfeld) — not directly in sources but implicit in rank distribution. | CONNECTION: - The bound 0.885 ≈ 1 − 0.115; note 0.115 ≈ (1/φ⁴)/10? No exact match. But 0.885 = 1 − 0.115, and 0.115 ≈ 0.118 ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com