Canonical determinants and the Riemann ξ-function: E2 (heat-trace → Mellin → determinant), Trace–Prime package, and E3 wedge rigidity

This preprint develops a modular GL(1) “master paper” organized into the stages E2 + Trace–Prime + E3. E2 (determinant package). We construct the canonical GL(1) determinant and its s–reparametrization Dcomp(s), and prove well-definedness/holomorphy on the determinant domain (including (s)>1). In particular, the determinant normalization and cutoff subtractions are arranged to yield a stable, referee-facing determinant object. Trace–Prime (imported package). We assume/import the Trace–Prime package (as stated in the Appendix) which provides the prime-power / heat-trace identification and the resulting logarithmic-derivative identity on Ω₊. E3 (wedge rigidity). We prove a wedge/sector rigidity principle (Laplace transform + ray-independence + Phragmén–Lindelöf type control) that promotes the local identity to a global identity under the stated assumptions. Main conditional conclusion (interface theorem). Under the imported assumptions (CT package and Trace–Prime package), we obtain the determinant–ξ identity \[ Dcomp(s)≡ ξ(s) \] on the analytic continuation domain described in the paper. Status relative to RH. This paper does not prove the Riemann Hypothesis. It isolates a determinant–ξ identity intended to be coupled to a Hilbert–Pólya spectral model in the forthcoming E4 stage.