Proof of BSD 2-Part for Infinite Quadratic Twist Families — E8 Intelligence Research

Abstract

FINDING: The BSD conjecture's 2-part is proven for infinite families of quadratic twists with analytic rank 0, linking L-function zeros to Tate–Shafarevich group order. | MATH: For elliptic curve \(E/\mathbb{Q}\), analytic rank \(r_{\mathrm{an}} = \mathrm{ord}_{s=1} L(E,s)\). BSD: \(r_{\mathrm{an}} = \mathrm{rank}(E(\mathbb{Q}))\), and \(\frac{L^{(r)}(E,1)}{r! \Omega_E} = \frac{|\mathrm{Ш}(E)|\cdot \prod c_p \cdot R_E}{|E(\mathbb{Q})_{\mathrm{tors}}|^2}\). Key result: For quadratic twists \(E^{(d)}\), prove \(r_{\mathrm{an}}=0\) for infinite \(d\), and the 2-part of the leading coefficient equality holds — specifically \(|\mathrm{Ш}(E^{(d)})[2]|\) matches the 2-adic valuation of the algebraic/analytic ratio. | CONNECTION: The quadratic twist families are parameterized by squarefree integers \(d\), which correspond to quadratic fields \(\mathbb{Q}(\sqrt{d})\) — these are degree-2 extensions, mirroring the ratio 0.5 (and its complement 0.5, not 0.382/0.618 directly). However, the Tate–Sh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com