The Centered GL(1) Trace, Laplace Transforms, and an Analytic Trace–Prime Package for the Completed Riemann Zeta Function

In the scalar Hilbert--P\'{o}lya framework developed in Karadzhov (2025) and refined in Karadzhov (2025b), a key role is played by a $\mathrm{GL}(1)$ trace and by two analytic hypotheses: a trace--prime property (TP) and a $\mathrm{GL}(1)$ wedge analyticity hypothesis. In the geometric formulation, these relate a Dirac-type operator on a compact spin manifold to the completed Riemann zeta function $\xi(s)$ through a centered spectral determinant. The purpose of this note is to isolate the $\mathrm{GL}(1)$ trace and the hypotheses (TP) and Wedge($\mathrm{GL}(1)$) in a purely analytic form that involves only the classical function\[F(z) = -\frac{\xi'}{\xi}(1/2 + z),\]its Laplace transform and the explicit formula. First, a centered $\mathrm{GL}(1)$ trace $T_{\xi,\mathrm{cen}}$ is defined as a tempered distribution on $\mathbb{R}$ characterized by the Laplace representation\[F(z) = \int_{\mathbb{R}} T_{\xi,\mathrm{cen}}(t) e^{-zt} \, dt \quad (\Re z > 1),\]and $T_{\xi,\mathrm{cen}}$ is explicitly identified in terms of the zeros of $\xi(s)$ and of its Euler product and $\Gamma$-factor. Weil's explicit formula is then shown to be equivalent to the equality of distributions\[T_{\xi,\mathrm{cen}} = T_{\mathrm{zeros}} - T_{\mathrm{arch}} = T_{\mathrm{primes}},\]where the three pieces correspond, respectively, to the non-trivial zeros of $\xi$, the Archimedean factor $\Phi_{\infty}(s) = \pi^{-s/2} \Gamma(s/2)$, and the von Mangoldt weights $\Lambda(n)$. Second, analytic versions (TP$_{\mathrm{an}}$) and Wedge($\mathrm{GL}(1)$)$_{\mathrm{an}}$ of the trace--prime and wedge hypotheses are formulated as statements about the meromorphic function $F(z)$ in the center variable $z = s - 1/2$. These conditions are satisfied for the classical $\xi(s)$ by standard results on the analytic continuation and growth of $\zeta(s)$ and its logarithmic derivative. Under (TP$_{\mathrm{an}}$) and Wedge($\mathrm{GL}(1)$)$_{\mathrm{an}}$ we prove a prime-side rigidity theorem: any even tempered distribution whose Laplace transform has the same prime-side Dirichlet expansion and wedge-type growth as $-\xi'/\xi(1/2 + z)$ necessarily agrees with $T_{\xi,\mathrm{cen}}$ up to a finite linear combination of derivatives of the Dirac mass at $t = 0$. At the level of centered determinants, this implies that any completed $L$-function with the classical von Mangoldt prime-side and Archimedean factor must be of the form\[e^{a + b(s-1/2)} \xi(s)\]for suitable constants $a, b \in \mathbb{C}$. Third, the analytic $\mathrm{GL}(1)$ package is in agreement with the scalar Dirac framework of Karadzhov (2025, 2025b). Any geometric $\mathrm{GL}(1)$ trace $T_{\mathrm{GL}(1),\mathrm{cen}}$ whose Laplace transform satisfies the analogs of (TP$_{\mathrm{an}}$) and Wedge($\mathrm{GL}(1)$)$_{\mathrm{an}}$ is shown to coincide with $T_{\xi,\mathrm{cen}}$ up to a distribution supported at $t = 0$, and this residual freedom corresponds exactly to the linear ambiguity in the logarithm of the central determinant. Combined with the order-one growth condition (O1) and the Archimedean alignment (AA) from Karadzhov (2025b), this yields the analytical side of the conditional identification of the twisted determinant with $\xi(s)$ and clarifies the rigidity constraints on scalar Dirac models. As a byproduct, we obtain a $\mathrm{GL}(1)$ trace formulation of Weil's positivity criterion and a trace-based simplicity criterion for the zeros of $\xi(s)$, phrased entirely in terms of $T_{\xi,\mathrm{cen}}$ and a natural Paley--Wiener class of test functions.