Primitive Trace Constructors and Row-Projector Closure for the Analytic Determinant Gate

This preprint is Core VII-C of the TEBAC BSD program. It is the constructor-level companion to Core VII v2.0, which consolidated the analytic determinant gate into H1--H8. The purpose of this module is to replace formal analytic gate assumptions by primitive trace-constructor data. It consolidates the C1--C5 constructor layer: \(\text{C1: primitive row-projector closure}\), \(\text{C2: good/bad prime trace packets}\), \(\text{C3: archimedean/conductor heat-state constructors}\), \(\text{C4: endpoint finite-part estimates}\), and \(\text{C5: residual Schur non-vanishing from the trace package}\). The module proves that, once the primitive trace space, row projectors, prime packets, completion packets, endpoint finite-part controls, and residual non-vanishing conditions are supplied with the stated compatibility properties, the analytic determinant gate of Core VII is activated. In particular, it supplies the constructor-level bridge from primitive trace rows to the H1--H8 determinant identity framework. This is a claim-safe preprint module. It does not assert a standalone proof of the Birch--Swinnerton--Dyer conjecture. Its role is to harden the analytic determinant identity layer by isolating the precise primitive trace-constructor requirements needed before a full unconditional BSD assembly can be claimed.