Bernstein-Certified Scheduled Recurrent Core

1. Exact integration point: implement a continuous-time latent state-space block inside a sequence model, with dot{x} = A_theta(rho)x + B_theta(rho)u and y = C_theta x. Start with A_theta(rho) = A_0 + sum_s rho_s A_s, where rho is a normalized context such as timestep, operating condition, or a low-dimensional encoder output. Use matrix exponentials or a stable ODE discretization for rollout. 2. Pseudocode: create a rectangular grid over rho; introduce symmetric Bernstein coefficients P for every cell; impose P_coeff - epsilon*I >> 0; for every cell and every rate-box vertex v, symbolically assemble the Bernstein coefficients of v dot grad(P) + P A + A^T P + 2 alpha P and impose each coefficient << 0; impose equality of Bernstein coefficients on shared faces; solve the SDP for P and, optionally, A when A enters affinely. During neural training, either freeze A and periodically recertify P, or alternate several gradient steps on the predictor with an SDP feasibility solve and reject updates that destroy the certificate. 3. Computed versus estimated: Bernstein coefficient matrices, derivatives, face equalities, and rate vertices are computed exactly from polynomial coefficients; neural approximation error, discretization error, and whether rho remains in the declared box are estimated empirically. 4. First cheap experiment: use a 2- or 4-dimensional latent linear SSM on Copy Memory and sequential MNIST, comparing unconstrained, spectral-normalized, and certified models. Sweep maximum context-rate bound r and polynomial degree m. The quantitative prediction is a feasibility transition: the SDP becomes infeasible near the largest admissible rate r_star, while increasing m or grid-cell count should weakly increase feasible r_star and reduce long-horizon hidden-state error. Validate by dense random sampling and require every sampled largest eigenvalue of F to be nonpositive, with the observed boundary agreeing with the SDP prediction within 20%.