Researchers have developed an analytical framework that relates spread complexity to local spectral statistics in quantum systems. This advancement provides a better understanding of the microscopic origin of the finite-time complexity peak, a key feature in the study of quantum chaos. The new model is based on an energy-space representation of the Krylov kernel, showing that this kernel is approximately banded and that its diagonal expansion converges rapidly, dominated by nearby energy levels in the ordered spectrum.

The work proposes an approximate universality hypothesis for the Krylov kernel: once unfolded, this kernel is well approximated by that of a uniform lattice. By combining this universal kernel with local spectral statistics, a simple analytical expression for spread complexity is obtained. This expression is formulated in terms of the Fourier transforms of the k-th nearest-neighbor spacing distributions. In particular, at leading order, the finite-time complexity peak is controlled by the Fourier transform of the nearest-neighbor spacing distribution.

This unified framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, providing an explanation for the spectral origin of the complexity peak and its late-time behavior. Furthermore, it establishes a direct connection between Krylov dynamics and spectral statistics, offering a powerful tool for studying thermalization and quantum information. This breakthrough is crucial for understanding how information propagates in complex quantum systems and how chaos manifests at the spectral level.