A Certificate-Based Terminal Assembly for the TEBAC Birch--Swinnerton--Dyer Program over \(Q\): Analytic Determinant Gates, Selmer--Kummer Bridge, Regulator Handoff, Finite Ledger, and Final Assembly

This preprint is a terminal certificate-based sequel to the author's TEBAC Birch--Swinnerton-Dyer master manuscript over \( Q\). It consolidates the post-master modules developed in the TEBAC BSD program into one submission-facing proof dossier: \({VII-D3}--{VII-D7} + {VIII-A} + {VIII-B} + {VIII-C} + IX\). The consolidated chain consists of the analytic determinant gate, the central Selmer--Kummer bridge, the integral lattice and regulator handoff, the residual Selmer/Cassels--Tate/Sha/Tamagawa/torsion finite ledger, and the final leading-coefficient assembly. The main result is the non-circular terminal implication \(ZBSD(E)=0 BSD(E/ Q)\), where \(ZBSD(E) = (Zₐₙ, ZSK, Zₗₐₜ, Zfin, Zₚₑᵣ, Zlead, Zforb)\). Thus, if the printed analytic, Selmer--Kummer, lattice/regulator, finite-ledger, period, leading-coefficient, and forbidden-import defect rows vanish, the BSD rank formula and leading-coefficient formula follow for elliptic curves over \( Q\): \(ordₛ₌₁L(E,s)=rankE( Q)\), and, for \(r=rankE( Q)\),\[{L⁽ʳ⁾(E,1)}{r!}={Ω_EReg(E)|(E)|∏_p c_p(E)}{|E( Q)ₜₒᵣₛ|^2}.\] The paper is deliberately claim-safe. It does not assert that the Birch--Swinnerton-Dyer conjecture is unconditionally resolved independently of the displayed terminal certificates. Instead, it presents a finite certificate system and a precise dependency graph. The remaining constructor-level tasks needed for a fully unconditional proof are explicitly separated: actual primitive trace state-space construction, actual prime-packet construction, endpoint estimates, residual Schur gap from the trace model, actual Selmer--Kummer realization, and finite Sha/Cassels--Tate closure. This record is a terminal certificate-based sequel to the author's earlier TEBAC BSD master manuscript and its associated Core modules. The earlier records are cited below by DOI as related identifiers.