Gaussian-Smoothed Explicit Formulas, Spectral Rigidity, and Arithmetic Reconstruction

We develop a unified analytic and statistical framework for reconstructing the von Mangoldt function \Lambda(n) from the nontrivial zeros of the Riemann zeta function. Introducing a Gaussian-smoothed inverse Mellin operator S_T(x), we prove that its distributional stability is equivalent to the Generalized Riemann Hypothesis (GRH). Interpreting S_T(x) as a linear statistic of the zero point process, we show that the variance of the reconstruction is governed by the structure factor S(k) of the underlying spectrum. Poisson spectra yield linear variance growth, while GUE-type spectra exhibit logarithmic growth, demonstrating that spectral rigidity suppresses low-frequency noise. Numerical experiments with up to N=10,000 zeros validate the theoretical predictions.