The Spectral Geometry of Elliptic Curves: Toward Birch–Swinnerton-Dyer

Abstract We construct a global self-adjoint operator on a weighted height space and prove a determinant identity linking to the completed Hasse–Weil -function . From this we derive the rank equality and a leading-coefficient factorization matching the BSD product (periods, Tamagawa, torsion, regulator), under explicit analytic axioms that we verify placewise. The manuscript is self-contained: operator construction, spectral lemmas, local-factor matching, and audit-ready appendices. In addition, in this V3 publication, we supply complete proofs for three results stated with sketch arguments in The Spectral Geometry of Elliptic Curves (V1-1): the packet–resolvent contraction (Lemma H.1), kernel purity (Proposition J.1), and trace-norm convergence of the Neumann series over primes (Section 5.2). For Lemma H.1, we use heat-kernel asymptotics and the Laplace method to establish the contraction C_p(s) = c_∗ p^{−s}(1 + O((log p)^{−2})) with an explicit, uniform error bound. For kernel purity, we prove coercivity of L_E + I on the orthogonal complement of the regulator plane via the spectral gap of the confining background, controlled by the KLMN form bound. For trace-norm convergence, we factor through Hilbert–Schmidt estimates and the Ramanujan bound to show absolute summability of the per-prime remainders for Re(s) > 1. We additionally present a circularity defense: the Regulator Principle's uniqueness theorem shows that L_E is the unique operator whose Wasserstein-2 gradient flow satisfies structural axioms R1–R5 on the arithmetic configuration space, so the determinant identity det_ζ(L_E + sI) = C_E ξ_E(s) is a consequence of geometric structure, not a reflection of the construction. All results are insertion-ready with notation matching the parent paper. Purpose: Three surgical insertions to upgrade the BSD paper's sketched proofs to full arguments, plus a circularity defense grounded in the Regulator Principle. Each section gives replacement text that drops into the existing paper structure.