Analytic Spread Complexity from Level Statistics: From Chaos to Integrability

What the math gives to ML

The paper gives an explicit spectral-statistics formula for a normalized spread-complexity curve, in which the contribution of each k-th neighbor spacing is weighted by the universal-kernel coefficient $(4k^2-1)^{-1}$. Its transferable asset is a cheap, interpretable probe of local eigenvalue geometry: Fourier transforms of unfolded spacings distinguish clustered or degenerate spectra from repulsive, well-spread spectra. A practical neural-network adaptation is to apply this probe to feature-covariance or attention-score spectra and train against a target spectral profile, providing a differentiable anti-collapse regularizer rather than relying only on trace or condition-number penalties.