The Information Content of Krylov Observables: A Machine Learning Approach

Abstract

We employ machine learning to quantify the information carried by three Krylov-space observables: the spread complexity $\mathcal{C}(t)$, the discrete Wigner negativity $N(t)$, and the normalized negativity $\chi(t)=N(t)/|S(t)|$, with $S(t)$ the survival amplitude, recently proposed as a second-moment infall probe (arXiv:2607.04065). Small residual networks (16-32 neurons) and boosted trees are trained on half of $\sim 57{,}000$ labeled evolutions spanning the GUE, GOE and Poisson ensembles, the integrable $SL(2,\mathbb{R})$/CFT sector, and the chaos interpolation $H(\varepsilon)=H_{SL(2,\mathbb{R})}+\varepsilon R_0 W_{GUE}$. Either moment determines the thermofield temperature at $R^2\simeq 0.999$. Neither reconstructs the fine spectral form factor ($R^2\simeq 0.18$ in every ensemble); smoothing the target does not repair this, and windows wide enough to help erase the dip-ramp physics itself: the SFF strictly refines both moments. The coarse $e^S$ plateau is nevertheless recovered at $R^2=0.861$, mostly from the first 20% of $\mathcal{C}(t)$. A single curve identifies the symmetry class at up to 98% accuracy. In the integrable sector the observables are informationally equivalent, as exact negative-binomial slaving demands, while the negativity best resolves the $(h,\alpha)$ degeneracy ($N\to h$: 0.999). Along the interpolation the asymmetry gap of $\chi$ over $\mathcal{C}$ switches on with chaos, growing from +0.33 to +0.77 as the level statistics cross to GUE, while the raw-$N$ gap decays to zero. The second-moment informational surplus is therefore a signature of chaos, carried specifically by the normalized negativity, and we derive an analytical mechanism and a quantitative bound for it.