Version 2 (August 5, 2026): This paper presents a full unconditional global proof of the Birch–Swinnerton-Dyer conjecture for all rational elliptic curves, superseding Version 1. The proof establishes two core results: (1) the equality between algebraic rank and analytic rank; (2) the exact leading-term BSD formula, including finiteness of the Tate–Shafarevich group. The argument relies on generalized higher-order Gross–Zagier identities, Nekovář Selmer complexes, multirank Euler systems, the GL(2) Iwasawa main conjecture, and Bloch–Kato cohomology. Attached to this release is complete Lean formalization source code for algorithmic verification. All reasoning uses only established theorems of arithmetic geometry with no unproven auxiliary hypotheses.
Changmin Wei (Wed,) studied this question.