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 \[ \mathfrak W_c \;=\; \bigl\{\,s\in\mathbb C : \Re\bigl((s-\tfrac12)^2\bigr) > -c \bigr\}, \] 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 \[ -\partial_u^2 + \alpha e^{2u} + \beta e^{-2u} + R(u), \] 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.