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

FINDING: The 2-parity conjecture is proven for elliptic curves with isomorphic 2-torsion, linking analytic root numbers to algebraic ranks mod 2. | MATH: Let \(E_1, E_2/\mathbb{Q}\) with \(E_1[2] \cong E_2[2]\) as Galois modules. The theorem states: \(W(E_1) = W(E_2) \implies \text{rank}(E_1) \equiv \text{rank}(E_2) \pmod{2}\), where \(W(E)\) is the global root number (product of local root numbers \(W_v(E) = \pm 1\)). The proof uses complete 2-descent, Selmer groups, and the Cassels–Tate pairing. Key intermediate: the parity of the 2-Selmer rank equals the parity of the analytic rank via the root number, and the 2-torsion isomorphism forces a matching of local root numbers at all primes. | CONNECTION: The root number \(W(E) = \prod_v W_v(E)\) is a product of \(\pm 1\) — a binary symmetry. The 2-torsion isomorphism \(E[2] \cong (\mathbb{Z}/2)^2\) corresponds to the \(A_1^3\) root lattice (three orthogonal \(\mathbb{Z}/2\) axes). The mod-2 parity is a \(\mathbb{Z}/2\)-graded structure, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com