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{equation*} LGL = -∂_u^2 + V(u), V(u) = α e²ᵘ + β e⁻²ᵘ + R(u), α, β > 0,{equation*}and the associated centered heat trace{equation*} Θ(t) := (e^{-tLGL}) - Pref(t).{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 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: [(I)] Gaussian domination bounds for the heat kernel KGL(t; u, v) via the Feynman--Kac formula. [(II)] Compact resolvent and trace-class properties of e^-tLGL for all $t > 0$. [(III)] A small-time trace expansion with the characteristic logarithmic singularity {equation*} (e^{-tLGL}) = a₋₁ \, t-1/2 log(1/t) + a_0 \, t-1/2 + a_1 + O(t1/2), a₋₁ = (4π)-1/2, {equation*} together with a self-contained Duhamel remainder argument showing that trace-smoothing perturbations contribute only an O(t1/2) correction after subtraction. [(IV)] Holomorphy at $w=0$ for the centered spectral zeta function {equation*} ζcen(w; s) = 1/Γ(w) ∫_0^∞ tʷ⁻¹ e-(s-1/2)^2 t \, Θ(t) \, dt, {equation*} and the resulting log-derivative identity for the centered determinant Dcen(s). [(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 {equation*} 2(s-1/2) ∫_0^∞ e-(s-1/2)^2 t Θ(t) \, dt = -∑ₚ ∑k ≥ 1 (log p) \, p⁻ᵏˢ = -∑n ≥ 1 Λ(n)/n^s, (s) > 1. {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 Tₚ and the trace--prime decomposition of Θ(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.