Exact Schwarzian metric factor and holographic Wilson-loop screening

A bstract We evaluate exactly the radial metric factor h ( ζ ) generated by Schwarzian averaging in the AdS 2 throat of an extremal Reissner-Nordström AdS 5 black brane. The result is a Gaussian integral against coth( πy ), valid at all radial depths, which Mordell’s identity turns into an exact Appell-Lerch q / q * -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 𝒢 0 ( ζ ) := h ( ζ )/ ζ 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 ) ~ − κ 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.