Analytic Determinant Gate and Completed (L)-Function Ledger

This preprint is Core VII of the TEBAC BSD program. Version 2.0 supersedes the initial Core VII version by consolidating the analytic determinant gate into a single H1--H8 module. The paper isolates the analytic determinant identity layer in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(Q\). The central target is the completed determinant comparison \(D_Eᶜᵒᵐᵖ(s)=Λ(E,s)\), where \(D_Eᶜᵒᵐᵖ(s)\) is the completed determinant candidate on the spectral side and \(Λ(E,s)\) is the completed Hasse--Weil \(L\)-function of an elliptic curve \(E/Q\). The consolidated H1--H8 chain includes: Euler