From phase space to Krylov space, one shell at a time

What the math gives to ML

The paper provides a constructive Krylov-space description of classical dynamics: phase-space Poisson evolution is represented in an orthonormal Lanczos chain, and the spreading of an initial observable along that chain defines Krylov complexity. Its most transferable mechanisms are microcanonical, energy-shell-resolved complexity and the quantitative early-time growth scale controlled by Lanczos coefficients. A neural-network implementation can apply the same construction to hidden-state or parameter-space vector fields to detect shell-dependent instability and adapt learning rates, recurrent step sizes, or inference horizons before conventional loss divergence appears.