Emergent Wigner-Dyson statistics and self-attention-inspired prediction in driven Bose-Hubbard chains

We propose an algorithm based on modulable hidden variables and adaptive step lengths, inspired by statistical physics and renormalization group principles, to study the emergent Wigner-Dyson level statistics in a driven many-body system. We apply this method to the boundary-driven Bose-Hubbard chain to illustrate the competition between coherent driving, hopping, and on-site interactions. Unlike random matrix models where chaos is induced through quenched disorder, here ergodicity emerges intrinsically from many-body configuration hybridization induced by the interplay between the driving field $F$ and the nonlinearity $U$. The resulting system follows statistics approaching the Gaussian orthogonal ensemble (GOE) limit. Using a self-attention-inspired Gaussian relevance map, the algorithm organizes the discrete site-occupation features and iteratively redistributes a coarse-grained spectral measure. This reduces the computational overhead relative to calculating the full eigenspectrum