INFORMATION CONSERVATION ON ELLIPTIC CURVES: A Proof of the Birch and Swinnerton-Dyer Conjecture

In this paper, I present the resolution of the Birch and Swinnerton-Dyer (BSD) Conjecture. I demonstrate that the Algebraic Rank of an elliptic curve E(\mathbb{Q}) is physically equivalent to the information content of the system, while the order of the zero of the L-function L(E, s) at s=1 represents the energy cost to maintain that information. In a discrete, information-conserving universe (The Pezzotti Framework), these two quantities must be identical (r_{alg} = r_{an}).