A Certificate-Based Terminal Assembly for the TEBAC Birch--Swinnerton--Dyer Program over \(\mathbb{Q}\): 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 \(\mathbb Q\). It consolidates the post-master modules developed in the TEBAC BSD program into one submission-facing proof dossier: \(\mathrm{VII\text{-}D3}\text{--}\mathrm{VII\text{-}D7} + \mathrm{VIII\text{-}A} + \mathrm{VIII\text{-}B} + \mathrm{VIII\text{-}C} + \mathrm{IX}\).

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 \(Z_{\mathrm{BSD}}(E)=0 \Longrightarrow \mathrm{BSD}(E/\mathbb Q)\), where \(Z_{\mathrm{BSD}}(E) = \bigl(Z_{\mathrm{an}}, Z_{\mathrm{SK}}, Z_{\mathrm{lat}}, Z_{\mathrm{fin}}, Z_{\mathrm{per}}, Z_{\mathrm{lead}}, Z_{\mathrm{forb}}\bigr)\).

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 \(\mathbb Q\): \(\operatorname{ord}_{s=1}L(E,s)=\operatorname{rank}E(\mathbb Q)\), and, for \(r=\operatorname{rank}E(\mathbb Q)\),
\[
\frac{L^{(r)}(E,1)}{r!}
=
\frac{\Omega_E\operatorname{Reg}(E)|\Sha(E)|\prod_p c_p(E)}{|E(\mathbb Q)_{\mathrm{tors}}|^2}.
\]

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.