Linear Spectral Statistics for Entrywise-Transformed Spiked Wigner Matrices under Shifted $L^4$ Profile Admissibility

Abstract

We prove a derivative-free analytic linear spectral statistics theorem for entrywise-transformed rank-one spiked Wigner matrices with a general microscopic noise law. The transform is required to be centered, variance-normalized, and admissible under the small translations generated by the spike: its shifted mean and second moment have first- and second-order profiles, while its centered shifted fourth cumulants and fourth tails are stable. At every microscopic shift we construct an explicit uniformly bounded three-point variable matching the first four centered moments of the target entry exactly. A generalized-Wigner LSS theorem applies to the resulting bounded triangular array, and a global Fourier--Duhamel derivative estimate transfers the analytic statistic back to the rough transform without a common truncation, coefficient-stability assumption, or local law. The order-one mean consists of the homogeneous Wigner bias, a rank-one Woodbury response, a zero-diagonal correction, and a quadratic variance-profile response. The centered covariance is the standard zero-diagonal real-Wigner covariance with fourth-cumulant parameter $\kappa_4^{f,\nu}$. We distinguish the bulk contour statistic from the full trace in the supercritical regime and show that the separated outlier adds exactly $\varphi(\theta+\theta^{-1})$ to the full-trace centering. As a self-contained consequence, Gaussian noise with $f\in L^{4+\epsilon}(\gamma)$ satisfies the theorem without differentiability of $f$; the resulting bulk and full-trace corollary has explicit Hermite coefficients and includes every centered, variance-normalized polynomial-growth transform. Concrete likelihood-ratio, smooth-transform, bounded rough-transform, and atomic criteria are provided, together with obstructions showing that bare $L^4(\nu)$ is insufficient.