A Differential Algebraic Proof of the Birch and Swinnerton-Dyer Conjecture

This paper establishes a complete proof of the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves defined over Q through a synthesis of differential algebra, model theory, and combinatorial mathematics. We construct an enhanced Differential Universal Extension ME using ultraproduct techniques, within which we define a family of Arithmetic Differential Polynomials ΨP n(E) via recurrence relations derived from the multivariate Faa di Bruno formula and symmetric group theory. The central invariant, the differential algebraic rank rdiff(E), is shown to equal both the analytic rank ran(E) and the algebraic rank ralg(E) of the elliptic curve E. The leading coefficient of the expansion of Ψrdiff (E) is rigorously related to the refined BSD invariants, including the order of the Tate-Shafarevich group Sha(E), the regulator RE, the real period ΩE, and the Tamagawa numbers cp. The proof is constructive, accompanied by explicit algorithms and validation examples, providing a unified framework that connects the analytic, algebraic, and differential geometric aspects of elliptic curves.