Sharp Laws for 2-Selmer Ranks in Quadratic Twists via Fitting Ideals — E8 Intelligence Research

FINDING: The 2-primary Tate-Shafarevich group and Selmer-rank statistics in quadratic twist families exhibit sharp, predictable distributional laws tied to BSD and Cohen-Lenstra heuristics, with explicit control via higher Fitting ideals and parity disparities. | MATH: Let \(E\) be an elliptic curve over \(Q\), \(E_d\) its quadratic twist by \(d\). The 2-Selmer rank \(s_2(E_d)\) satisfies \(s_2(E_d) ≡ F_2 E(Q_2)[2] {2}\) (parity). Smith's theorem: for a positive proportion of \(d\), \(F_2 Sel_2(E_d) - F_2 E_d(Q)[2] = 0\) or 1, i.e., the 2-primary part of \(III(E_d)[2^∞]\) is bounded. Longo's higher Fitting ideals: \(Fitt_i(III(A_s))\) control the \(p\)-torsion structure, with \(Fitt_0(III) ⊆ (L(1,χ_d))\) in the anticyclotomic setting. Mazur's disparity: the average 2-Selmer rank in twist families is \(1/2\) (half-integer), re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com