Peel-and-pass polynomial latent dynamics

Integrate this at the recurrent transition of a latent ODE, world model, or continuous-time SSM, specifically replacing the usual sequence of K ordinary Euler or RK hidden updates inside a chunk with a coefficient-prediction module. Use normalized local time s in [-1,1], choose shifted Legendre polynomials p_j(s)=P_j(s), and choose K between 3 and 8. For each chunk, inputs are the incoming state h_n, the chunk input tokens or observations x_{n,0:L-1}, and the chunk length Δt. A small transformer or MLP coefficient head predicts A_n=[a_{n,0},...,a_{n,K}], with each a_{n,j} in R^d. Evaluate the trajectory at C fixed Gauss-Legendre nodes s_l using H_l=Σ_j a_{n,j}p_j(s_l). If a neural vector field f_theta is available, add λ_dyn/C Σ_l ||r_n(s_l)||² to the task loss, where r_n is the displayed collocation residual and p'_j is precomputed once. The recurrent update is exactly h_{n+1}=Σ_j a_{n,j}; discard all H_l after the chunk, so activation memory is O(dK) rather than O(dL). A simpler first MVP can omit the residual and train the coefficient head end-to-end against next-chunk prediction, while retaining an endpoint consistency penalty between the coefficient sum and a teacher-forced final hidden state. Compute polynomial values and derivatives analytically or precompute their C×(K+1) matrices; no numerical estimation of mathematical constants is needed. Start with a 2-layer GRU or small MLP world model on sine-wave system identification and Penn Treebank character prediction, comparing against an equally sized GRU unrolled for L steps and a standard neural ODE solver. Measure peak activation memory, wall-clock time, long-horizon rollout error, and loss at equal parameter count. The expected signal is at least 2x lower activation memory for L much larger than K, with no degradation in one-chunk prediction and slower growth of rollout error when K is increased from 3 to 6. A falsification criterion is failure to match the baseline at equal FLOPs or instability when Δt is doubled.