arXiv:2606.02938v2 Announce Type: replace-cross Abstract: This paper develops a data-driven realization of the generalized Koopman operator (GeKo) [1], in which states and inputs are lifted independently and the dynamics are expressed as a tensor bilinear system. The first contribution is a time-sequenced multi-step Khatri-Rao kernel regression formulation that exposes the operator to evolved snapshots along trajectories rather than only single one-step pairs, which reduces compounded prediction error. Secondly, we develop a kernel- and input-agnostic structured SVD reduction that compresses the lifted state and input spaces while preserving the Khatri-Rao realization. We instantiate the framework with random Fourier features and describe a complete predictive-control pipeline, including a multi-step roll-out diagnostic that guides the choice of MPC horizon. The framework is validated on the chaotic Lorenz system, where the learned reduced-order GeKo model reduces the long-horizon prediction error by up to a factor of five relative to one-step and lifted linear baselines fitted on the same data, and stabilizes an unstable equilibrium.