Explicit Cassels-Tate Pairing Computation Resolves Goldfeld Parity via Selmer Groups — E8 Intelligence Research

FINDING: Explicit computation of the Cassels-Tate pairing on 2-Selmer groups via Albanese-Albanese definition, proving agreement with Cassels' original pairing; Smith's work on 2^k-Selmer groups resolves parity phenomena in Goldfeld's conjecture. MATH: - Cassels-Tate pairing: \( ·, · CT: Sel_2(E) × Sel_2(E) → Q/Z \) (non-degenerate, alternating). - Albanese-Albanese definition (Poonen-Stoll): \( a,b = ∑ᵥ inv_v( a_v ∪ b_v ) \) over all places \(v\), with local cup products in \(H^1(G_v, E[2])\). - 2-descent: \( Sel_2(E) H^1(G_Q, E[2]) \), with rank bound \( rk(E(Q)) ≤ F_2 Sel_2(E) - F_2 Sha(E)[2] \). - Smith's theorem: For quadratic twist families, the parity of \( Sel2^k(E) \) follows a distribution governed by the root number \(w(E) = ± 1\), with explic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com