Structural Closure of the Birch–Swinnerton–Dyer Conjecture via Determinant Lines

Determinant Lines and the Structural Resolution of the Birch–Swinnerton–Dyer Conjecture This paper provides a structural closing argument for the Birch–Swinnerton–Dyer (BSD) conjecture formulated in terms of determinant lines associated with Selmer complexes and analytic data of elliptic curves. Assuming the Equivariant Tamagawa Number Conjecture (ETNC) in the sense of Fukaya–Kato, we establish the existence of a canonical isomorphism between the arithmetic determinant line arising from the Bloch–Kato Selmer complex and the analytic determinant line defined by the leading coefficient of the L-function and period data. In this framework, the BSD conjecture is reformulated as a structural statement about determinant-line equivalence rather than as a numerical identity. Classical BSD formulas involving regulators, Tamagawa numbers, and torsion factors are reinterpreted as consequences of non-canonical trivializations, rather than as intrinsic components of the conjecture itself. The paper completes the logical structure developed in a preceding series of works. No additional technical computations or auxiliary conjectures are introduced, and no appendix is required. The result should be viewed as a conceptual closure of the determinant-line approach to the BSD conjecture under the ETNC.