Centered Heat Traces and Zeta-Regularized Determinants in a Balanced Hilbert–Pólya Model: The GS5 Proof Package (TEBAC Companion)

We isolate and close the GS5 analytic module in the TEBAC Hilbert--Pólya framework, focusing on the one-dimensional balanced $GL(1)$ channel\begin{equation*} L_{GL} = -\partial_u^2 + V(u), \quad V(u) = \alpha e^{2u} + \beta e^{-2u} + R(u), \quad \alpha, \beta > 0,\end{equation*}and the associated centered heat trace\begin{equation*} \Theta(t) := \Tr(e^{-tL_{GL}}) - P_{\mathrm{ref}}(t).\end{equation*}Working in the balanced exponential confinement regime, the paper assembles a referee-oriented, self-contained package for the centered heat trace, the centered spectral zeta function, and the zeta-regularized determinant, designed to be cited as the GS5 closure component in the broader TEBAC Hilbert--Pólya program. Under the explicit confining Schrödinger model on $\mathbb{R}$ and a trace-smoothing hypothesis on $R$, the paper proves the analytic backbone needed for the trace--determinant calculus and the prime-side exchanges, including: \item[(I)] Gaussian domination bounds for the heat kernel $K_{GL}(t; u, v)$ via the Feynman--Kac formula. \item[(II)] Compact resolvent and trace-class properties of $e^{-tL_{GL}}$ for all $t > 0$. \item[(III)] A small-time trace expansion with the characteristic logarithmic singularity \begin{equation*} \Tr(e^{-tL_{GL}}) = a_{-1} \, t^{-1/2} \log\left(\frac{1}{t}\right) + a_0 \, t^{-1/2} + a_1 + O(t^{1/2}), \quad a_{-1} = (4\pi)^{-1/2}, \end{equation*} together with a self-contained Duhamel remainder argument showing that trace-smoothing perturbations contribute only an $O(t^{1/2})$ correction after subtraction. \item[(IV)] Holomorphy at $w=0$ for the centered spectral zeta function \begin{equation*} \zeta_{\mathrm{cen}}(w; s) = \frac{1}{\Gamma(w)} \int_0^\infty t^{w-1} e^{-(s-1/2)^2 t} \, \Theta(t) \, dt, \end{equation*} and the resulting log-derivative identity for the centered determinant $D_{\mathrm{cen}}(s)$. \item[(V)] An exchange theorem on an explicit domain $U$ that justifies Tonelli/Fubini and differentiation under the integral sign for the prime test family, yielding the Euler--von Mangoldt block \begin{equation*} 2\left(s-\frac{1}{2}\right) \int_0^\infty e^{-(s-1/2)^2 t} \Theta(t) \, dt = -\sum_{p} \sum_{k \ge 1} (\log p) \, p^{-ks} = -\sum_{n \ge 1} \frac{\Lambda(n)}{n^s}, \quad \Re(s) > 1. \end{equation*} The paper carefully separates what is proved internally (heat-kernel bounds, trace-class control, small-time structure, zeta/determinant calculus, and exchange on $U$) from what is imported as structural geometric input (the prime/Hecke correspondences $T_p$ and the trace--prime decomposition of $\Theta(t)$). This makes the GS5 component a checkable, standalone analytic module and reduces the remaining identification targets in the full TEBAC Hilbert--Pólya closure chain to explicit, auditable geometric/microlocal deliverables in the main preprint.