TEBAC HP III: The GL(1) Trace–Prime Package (TP), Fully Discharged

This module discharges the TP-specific assumptions from the TEBAC HP corpus (T1 arithmetic circle action/invariance, T3 end-translation model for prime correspondences, and T5 semigroupoid closure). It provides a justified trace--prime conversion on real $s>1$, producing an error term that extends holomorphically to the wedge \[ W_c \;=\; \{\,s∈ C : ((s-12)^2) > -c \}, \] with all Tonelli/Fubini interchanges proved at module level. A key analytic ingredient is a self-contained Gaussian heat-kernel upper bound for the explicit HP--II end operator \[ -∂_u^2 + α e²ᵘ + β e⁻²ᵘ + R(u), \] proved via truncation and Trotter domination and re-derived via a Brownian-bridge (Feynman--Kac) representation in the appendix. This preprint is part of the TEBAC Hilbert--Pólya program.