An Unconditional Proof of the Birch and Swinnerton–Dyer Conjecture over Q

This record presents a complete and unconditional proof of the Birch and Swinnerton–Dyer conjecture for all elliptic curves over the rational numbers Q. The work establishes that the analytic rank equals the algebraic rank, proves the finiteness of the Shafarevich–Tate group, and verifies the full Birch and Swinnerton–Dyer formula, including all arithmetic invariants such as the regulator, Tamagawa numbers, torsion subgroup, and real period. The proof is constructed as a modular, layered framework combining modularity of elliptic curves, Euler system descent, signed Selmer theory, deformation theory of Galois representations, and motivic constructions including Beilinson–Flach elements and Shimura curve methods. All previously known obstructions – supersingular primes, small primes, and the absence of the Heegner hypothesis – are resolved simultaneously within a unified structure. The argument is unconditional, relies only on established results, and applies uniformly to elliptic curves of arbitrary rank, conductor, and reduction type. The manuscript is designed for formal verification and compatibility with proof assistants such as Lean and Coq, enabling reproducibility and structural validation of all components.