Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

What the math gives to ML

The paper provides an exact spectral-basis update for polynomially prepared initial states: replacing x_0 by Q(H)x_0 reweights the reference spectral measure by |Q|^2, and the new orthogonal-polynomial basis is connected to the old one through a finite-width transformation. The important transferable asset is computational reuse: one Lanczos factorization can support many learned polynomial seeds without rerunning Lanczos in the ambient state dimension. This can become a shared spectral propagation module for graph neural networks or linear state-space layers, where several seed responses are combined by a learned gate.