arXiv:2603.16851v2 Announce Type: replace Abstract: Nonlinear hereditary systems combine nonlinear state dependence with memory. Koopman lifting provides linear predictors for nonlinear dynamics, but standard formulations are Markovian, while unrestricted lag based extensions require progressively more fitted coefficients as the retained memory horizon increases. This work proposes a structured Koopman finite memory model that represents history through a shared normalized Gunwald Letnikov temporal profile, while learning how the history couples the lifted coordinates directly from data. This structure retains explicit history dependence while making the number of fitted coefficients independent of the retained memory horizon. For fixed memory parameters, identification remains a linear regression problem with closed-form least squares and ridge solutions. We derive deterministic bounds that distinguish structured memory approximation, omitted history, regressor conditioning, and regularization effects, and construct an exact augmented Markovian realization for recursive prediction and stability analysis. Numerical studies on a nonlinear viscoelastic robotic system demonstrate the complementary roles of nonlinear lifting and explicit memory, while the structured representation requires substantially fewer fitted coefficients than unrestricted lag based Koopman models.