Hidden Dyson Universality in Inverse-Spectral Geometry

Abstract: Dyson universality typically manifests itself in local eigenvalue statistics. Here we show that its signature survives a nonlinear inverse-spectral reconstruction and reappears in the matrix geometry of the reconstructed operator. Using a dressing transformation, we map each unfolded spectrum to a deformation $f(x)$ of a fixed harmonic oscillator and represent it in the common oscillator basis by $F_{mn}=\bra m|f|n\ket$. We resolve the matrix-element weight into shells of fixed distance $d=|m-n|$, corresponding to the energy-transfer channels of the reference oscillator, and characterize the resulting distribution by distance-shell moments. Independently calibrated on Gaussian $β$-ensembles, these moments vary smoothly with $β$ and distinguish the GOE, GUE, and GSE. With this calibration fixed, applying the same diagnostic to the nontrivial zeros of the Riemann zeta function places the reconstructed operators in the GUE sector. Thus, the GUE character of the zeros is recovered not through direct statistics of the input levels, but from the distance-resolved geometry of the reconstructed operator.