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

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 \(E/Q\), analytic rank \(rₐₙ = ordₛ₌₁ L(E,s)\). BSD: \(rₐₙ = rank(E(Q))\), and \({L⁽ʳ⁾(E,1)}{r! Ω_E} = {|Ш(E)|· ∏ c_p · R_E}{|E(Q)ₜₒᵣₛ|^2}\). Key result: For quadratic twists \(E⁽ᵈ⁾\), prove \(rₐₙ=0\) for infinite \(d\), and the 2-part of the leading coefficient equality holds — specifically \(|Ш(E⁽ᵈ⁾)[2]|\) 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 \(Q(√d)\) — 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