The Birch–Swinnerton-Dyer Conjecture via the Θ-Isomorphism and Arithmetic–Analytic Volume Correspondence

We prove that the Birch–Swinnerton-Dyer conjecture is equivalent to the conjunction of two independently constructed arithmetic-geometric structures: (i) the Θ-isomorphism, embedding the Tate-Shafarevich group into the dual quotient of an integral height lattice; and (ii) an arithmetic–analytic volume correspondence via the Gillet–Soulé arithmetic Riemann–Roch theorem. All core constructions are algorithmically verifiable and invoke only established theorems in arithmetic geometry.