Exact Schwarzian Metric Factor and Holographic Wilson-Loop Screening

We evaluate exactly the radial metric factor $h(\zeta)$ generated by Schwarzian averaging in the AdS$_2$ throat of an extremal Reissner--Nordstr\"om AdS$_5$ black brane. The result is a Gaussian integral against $\coth(\pi y)$, valid at all radial depths, which Mordell's identity turns into an exact Appell--Lerch $q/q^\ast$-series representation. The dual series identifies the nonperturbative scale $e^{-\pi^2 C/\zeta}$ missed by any finite near-boundary truncation. The third parametric derivative required by the evaluation generates the quasimodular Eisenstein series $E_2$, absent from the classical Mordell identity. From the integral representation we prove that $\mathcal{G}_0(\zeta):=h(\zeta)/\zeta^2$ is completely monotone and hence has no interior minimum, so any confining minimum produced by a finite near-boundary truncation is an artifact. We also compute the exact relative variance of the Schwarzian kernel, which makes the averaged-metric approximation error quantitative and shows that the absence of a confining minimum is robust across moment-based effective geometries. Applied to the temporal rectangular Wilson loop, the exact throat gives algebraic screening, $E(L)\sim -\kappa_{\rm IR}/L^2$, the Wilson-loop diagnostic of the semi-local quantum-liquid IR of the extremal RN brane. A numerical check in a simple matched geometry confirms that the screened saddle is the dominant string configuration, and an exact-versus-truncated force comparison shows that the apparent constant-force regime of the fourth-order truncation is not a feature of the exact geometry.