A Spectral-Geometric Program of Birch and Swinnerton-Dyer Conjecture

This manuscript expands the earlier conditional spectral-geometric draft into a review-stabilized research program for the Birch and Swinnerton-Dyer conjecture over the rational field. The central claim remains explicitly conditional. The paper does not assert an unconditional proof of BSD; rather, it isolates a coherent package of operator-theoretic, automorphic, homological, determinant, and torsion hypotheses whose verification would imply the weak and strong forms of the conjecture. The expanded version incorporates the referee-style criticisms concerning the global Fredholm domain, closed range, zeta-regularized determinants, spectral pollution, the APS index-versus-coefficient gap, analytic torsion, local-to-global interpolation obstructions, Deninger’s homological BSD philosophy for Dedekind zeta functions, Hesselholt's topological-periodic-cyclic-homology determinant formalism, and a refined congruent-number test suite based on Tunnell's theorem, the distribution results of Alexander Smith, and the period formulas of Tian-Yuan-Zhang. Mathematical formulas are displayed in chapter-numbered form; figures and tables are introduced and then interpreted in prose; and references are cited in the body of the manuscript. This refined version also implements the second referee report’s minor revision requests: the allowed cusp-weight asymptotics are stated explicitly, the two-adic stratum of the congruent-number family is expanded, equation delimiters are normalized, and the bibliography is reorganized in surname–year order.