The Spectral Birch–Swinnerton–Dyer Theorem in the CET Ω Framework

The Spectral Birch–Swinnerton–Dyer Theorem in the CET Ω Framework This article develops a complete spectral realization of the Birch–Swinnerton–Dyer (BSD) formula for elliptic curves over \mathbb{Q} inside the CET Ω framework. The central result is that all arithmetic invariants appearing in BSD arise as spectral invariants of two canonical operators: a global Dirac operator D_E, and a Selmer Laplacian \Delta_\Omega(E). The paper shows that the Hasse–Weil L-function of an elliptic curve is exactly the spectral zeta function of D_E, and that the leading coefficient at s=1 equals the Ray–Singer analytic torsion of the CET Ω Selmer Laplacian. Thus, BSD becomes a purely spectral identity.