Lifting-free PSD certificate layer

Integrate this as a training-time certificate solver for a neural dynamics or energy model. Let the network f_theta(x) predict dynamics, and let V_c(x) be a polynomial Lyapunov candidate with coefficient vector z; optionally let a second network produce coefficients entering the equality constraints. Sample collocation points x_i and construct coefficient or value-matching equations Az+Bs=b for conditions such as V_c(x_i)>=epsilon||x_i||^2 and -nabla V_c(x_i)^T f_theta(x_i)>=epsilon||x_i||^2. Use a PSD Gram matrix for each SOS constraint, but do not introduce Schur-complement lifting variables. For each minibatch, run K dual iterations: initialize y=0; compute z=-Q^{-1}(w+A^T y); compute X=-B^T y; eigendecompose each symmetric X and set Pi_K(X)=U max(diag(lambda),0) U^T; set s=Pi_K(X)/mu; compute r=b+Az+Bs; update y with accelerated ascent using the displayed t recurrence and step size 1/L. Backpropagate through the unrolled iterations, or stop-gradient through the solver and alternate certificate and network updates. Estimate L with 5-10 power iterations on AQ^{-1}A^T+BB^T/mu, and reduce mu geometrically only after ||r|| falls below a threshold. The cheap first experiment is a 2-layer MLP learning a stable 2D or 4D nonlinear system, comparing this solver against a CVXPY SDP baseline and a penalty-only certificate loss. Measure peak memory, wall-clock time per certificate update, equality residual, minimum sampled Lyapunov margin, and closed-loop rollout stability. Success means at least 2x lower peak memory or a materially larger certificate degree at equal memory, while preserving the baseline certificate margin and reducing the constraint residual below 1e-4.