Online Taylor Residual World Model

1. Integration point: attach this module to a pretrained neural dynamics model used by a Dyna agent, MPC planner, or trajectory optimizer. Initially freeze the neural parameters. Maintain an operating point $(x_0,u_0)$, construct monomials through degree $r=1$, $2$, or $3$, and maintain one RLS covariance and coefficient matrix per output dimension. 2. Pseudocode: observe $(x_k,u_k,x_{k+1})$; compute $z_k=[x_k-x_0,u_k-u_0]$ and $\phi_k=\phi_r(z_k)$; calculate residual target $y_k=x_{k+1}-\hat{F}_{\theta}(x_k,u_k)$; apply the displayed RLS update; predict with $\hat{F}_{\theta}+W_k\phi_r$; calculate $A_k$ and $B_k$ by automatic differentiation; freeze them for the next $H$ planning steps; recenter $(x_0,u_0)$ when $\|z_k\|$ exceeds a threshold. 3. Computed directly: monomials, RLS gains, covariance, and Jacobians. Estimated empirically: the forgetting schedule, covariance initialization, recentering threshold, and useful Taylor order. 4. First cheap experiment: train a small MLP world model on CartPole or Pendulum, then inject an abrupt mass or actuator-scale change. Compare neural-only, order-1, order-2, and order-3 residual models inside an MPC controller with horizon $H=10$. Measure one-step error, ten-step error, rollout divergence, and recovery time. The quantitative prediction is that effective adaptation memory is approximately $N_{\mathrm{eff}}\approx1/(1-\lambda)$: recovery time should scale with this quantity, while stationary variance should increase as $\lambda$ decreases. Degree two should reduce local multi-step error over degree one only within a neighborhood where its quadratic term exceeds measurement noise; degree three should help only in a smaller, sufficiently excited neighborhood.