Actual Constructor Closure for the TEBAC Birch--Swinnerton-Dyer Program over \(\mathbb{Q}\): Full Core X Consolidation with Expanded Proof Chains

This preprint consolidates Core X of the TEBAC Birch--Swinnerton-Dyer program into a single expanded constructor-closure manuscript. The paper develops the determinant--Selmer proof chain for elliptic curves over \(\mathbb Q\) through actual primitive trace state spaces, row projectors, local packet constructors, endpoint finite-part legality, residual Schur gap execution, Selmer--Kummer and trace-to-cocycle comparison, finite residual/Sha/Cassels--Tate/Tamagawa/torsion ledger execution, and the terminal Core X no-certificate BSD implication. The resulting BSD leading-coefficient ledger is expressed by \(\frac{\Omega_E\operatorname{Reg}(E)|\Sha(E/\mathbb Q)|\prod_p c_p(E)}{|E(\mathbb Q)_{\mathrm{tors}}|^2}\), with \(r=\operatorname{rank}E(\mathbb Q)\). The manuscript should be read as a constructor-level hardening and consolidation of the TEBAC BSD program, not as an already externally refereed final proof of BSD. Its purpose is to replace earlier certificate-dependent modules by explicit execution criteria and expanded proof chains. The remaining proof burden is isolated into concrete execution rows, including the residual Schur gap, the central-to-etale compatibility obstruction, the residual Noetherian presentation, and the residual trace-energy bound.