The Birch and Swinnerton-Dyer Endpoint by Exclusion of Analytic-Arithmetic Mismatch

Overview This record contains the AASC endpoint-closure manuscript for the Birch and Swinnerton-Dyer rank endpoint, together with the associated Lean 4 audit-facing project materials. The manuscript treats the BSD rank statement on the fixed elliptic-curve carrier. The proof route is not a conventional arithmetic-geometric construction by descent, Euler systems, Iwasawa theory, rational-point construction, Sha-finiteness proof, or analytic estimation. It is a kernel-first AASC endpoint proof. Main Claim Boundary The main closed endpoint is the BSD rank endpoint: analytic rank equals Mordell-Weil rank for every elliptic curve over (\mathbb{Q}). The refined leading-coefficient formula is treated separately and conditionally at the standing-fixed formula-factor layer. In particular, the manuscript does not claim to supply an independent arithmetic proof of finite-order standing for the Tate-Shafarevich group. The refined formula closure is located at the explicit factor-standing boundary. Proof Route The argument proceeds by fixed-carrier exclusion of analytic-arithmetic mismatch. Once the official BSD rank endpoint is used as a determinate theorem-bearing target, the fixed carrier already requires the kernel roles of: reference; standing; admissibility; irreversibility. Within that non-degenerate endpoint regime, analytic-arithmetic rank mismatch is routed through: standard BSD mismatch normal form; bridge-image exclusion; theorem-level endpoint-status governance; independent same-domain rank discrimination; kernel-forced no-second-classifier closure. The mismatch branch is thereby excluded, yielding pointwise equality between analytic rank and Mordell-Weil rank, and hence the official BSD rank endpoint correspondence. Lean 4 Audit Layer The accompanying Lean 4 material is an audit and reproducibility layer for the theorem-routing surface of the manuscript. It formalizes the proof-route architecture, including: endpoint-use structure; carrier adequacy predicates; rank-mismatch normal form; bridge-image exclusion; endpoint-governance routing; no-fifth-case closure; no-independent-discriminator closeout; official rank endpoint correspondence. The Lean 4 archive is not presented as a full formalization of elliptic curves, modularity, analytic continuation, Mordell-Weil theory, regulators, Tamagawa factors, or Sha from first principles. Its role is to audit the theorem-routing surface of the AASC endpoint proof. Associated Resources Lean 4 audit archive: https://github.com/somamaley-ux/AASC-BSD-Endpoint-Lean-Audit Zenodo record for the Lean 4 audit archive: https://doi.org/10.5281/zenodo.20619017