Constructor-Level Audit, Gate Ledger, and Closure Protocol for the Analytic Determinant Gate

This preprint is Core VII-D of the TEBAC BSD program. It is an audit and closure-protocol module for the analytic determinant gate in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(\mathbb Q\). The purpose of this module is not to assert a standalone proof of the Birch--Swinnerton--Dyer conjecture, but to classify the analytic determinant layer according to its constructor-level status. It separates formal implication chains from primitive constructions, and records which parts of the Core VII analytic gate have been reduced to precise hard fronts. The audit covers the H1--H8 analytic gate chain, including Euler product log-derivative matching, prime-packet coefficient matching, gamma/conductor completion, quotient rigidity, central multiplicity transfer, Schur block factorization, residual Schur non-vanishing, and final analytic determinant gate assembly. It also reviews the C1--C5 constructor layer, including primitive row-projector closure, good/bad prime trace packets, archimedean and conductor heat-state constructors, endpoint finite-part estimates, and residual Schur non-vanishing from the trace package. The module identifies the remaining constructor-level obligations required before the analytic determinant identity \(D_E^{\mathrm{comp}}(s)=\Lambda(E,s)\) can be claimed unconditionally from first principles. In particular, it isolates the prime-packet theorem, completion matching, endpoint finite-part control, and residual Schur non-vanishing as the key hard fronts still requiring full primitive closure. This is a claim-safe audit preprint. It does not assert a standalone proof of BSD. Its role is to provide a transparent gate ledger and closure protocol for the analytic determinant identity layer of the TEBAC BSD program.