Taylor-Structured Online Adapter

Use this as a lightweight adaptation head on a pretrained neural state-space or world model. Freeze the base network $h_\theta(z)$ and predict $x_{k+1}=h_\theta(z_k)+R_k^T\varphi_D^{\mathrm{FE}}(z_k)$, where $R_k$ is updated online by RLS; alternatively let the polynomial head replace the final transition layer. Build the monomial list once, remove even-total-degree monomials only when the environment has verified joint-odd symmetry, and apply forward-Euler pruning only when the corresponding interaction is physically absent. Pseudocode: initialize $R=0$, $P=\alpha I$; at each transition compute $\varphi_k$, residual $e_k=x_{k+1}-h_\theta(z_k)-R^T\varphi_k$, gain $K_k=P\varphi_k/(\lambda+\varphi_k^TP\varphi_k)$, update $R\leftarrow R+e_kK_k^T$, and update $P\leftarrow\lambda^{-1}(P-K_k\varphi_k^TP)$. The paper provides the Taylor-informed reduced dictionary and online RLS; the neural residual-head construction is the transfer. Compute covariance from $P$ and damp or reject updates when $\varphi_k^TP\varphi_k$ is excessively large. The first cheap experiment is a regime-shifted Lorenz-63 or nonlinear-pendulum dataset: pretrain on one parameter range, shift the parameter online, and compare full fine-tuning, unstructured polynomial RLS, and the symmetry/pruned adapter. The prediction is that, when symmetry is correct, even-degree coefficient estimates remain statistically indistinguishable from zero and the reduced adapter reaches a target one-step error with fewer samples. After a regime shift, adaptation error should decrease approximately geometrically at a rate controlled by the forgetting factor $\lambda$ until reaching a persistent-excitation noise floor.