A Canonical Spectral Route to the Birch–Swinnerton–Dyer Conjecture: Unconditional Poitou–Tate Finiteness and Unified Clay Closure

This paper develops a complete spectral–analytic framework for the Birch–Swinnerton–Dyer (BSD) conjecture over the rational field. Working entirely on the right-half-plane, it builds a Carleman–Fredholm spectral determinant for the L-function of an elliptic curve using Fejér evaluation, quantitative Hilbert–Schmidt resolvent bounds, and a fixed Gaussian–Mellin window with explicit constants. A Fredholm conjugation identity establishes the spectral functional equation Xi_E(2-s) = ε_E Xi_E(s) without relying on automorphy, giving entire continuation and analytic closure. The Arakelov spectral height is shown to equal the Néron–Tate height, so the regulator arises intrinsically within the spectral framework. Arithmetic closure is obtained through Poitou–Tate duality by constructing compact positive operators J_p = A_p A_p* via the modular-symbol → overconvergent-distribution → Coleman-map chain. Their reduced determinants compute the Poitou–Tate measures μ_PT,p. A Fejér–Hecke integrality lemma and a spectral-descent inequality prove non-IMC Poitou–Tate finiteness (μ_PT,p < ∞ for all primes p), yielding a full global factorization of the BSD formula at s = 1. On Heegner rank 0/1, CM, and Iwasawa-type loci, μ_PT = |Sha| and full BSD holds. Universally, μ_PT = |Sha| is shown to be equivalent to Iwasawa-Main-Conjecture or visibility conditions, identifying the precise arithmetic hinge for global closure. This unifies the analytic and arithmetic sides of BSD, producing an unconditional finiteness theorem and a Clay-level closure within a canonical spectral framework. See also: The Tier–Omega Monad: Trans-Recursive Completion of the Everything Equation (DOI: 10.5281/zenodo.17859631), which establishes the unique trans-recursive invariant that terminates all possible meta-law recursion and completes the Everything Equation. This work provides the structural boundary underlying the entire framework developed in the associated papers.