Parity Conjecture Proven for Elliptic Curves with Isomorphic 2-Torsion — E8 Intelligence Research

Abstract

FINDING: The 2-parity conjecture is proven for elliptic curves with isomorphic 2-torsion, linking analytic root numbers to algebraic rank parity. | MATH: Let \(E_1, E_2/\mathbb{Q}\) with \(E_1[2] \cong E_2[2]\) as Galois modules. The root number \(W(E) = \pm 1\) satisfies \(W(E) = (-1)^{\mathrm{rank}(E)}\) (BSD parity). The proof shows \(W(E_1)W(E_2) = (-1)^{\mathrm{rank}(E_1)+\mathrm{rank}(E_2)}\) under the 2-isogeny condition. Key invariants: conductor \(N_E\), discriminant \(\Delta_E\), Tamagawa product \(c_p\), and the local root number \(W_p(E) = \pm 1\) with \(W(E) = \prod_p W_p(E)\). The 2-torsion isomorphism forces \(W_p(E_1) = W_p(E_2)\) for all \(p \neq 2\), and the \(p=2\) correction is computed via the 2-adic unit \(u\) in \(\Delta_E\). | CONNECTION: The root number \(W(E) = (-1)^{\mathrm{rank}}\) is a binary symmetry — the same \(\pm 1\) structure as the golden ratio's continued fraction \([1;1,1,\dots]\) parity. The 2-torsion lattice \(E[2] \cong (\mathbb{Z}/2)^2\) is the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com