arXiv:2603.24503v2 Announce Type: replace Abstract: Repeated online optimization makes nonlinear model predictive control (NMPC) computationally demanding for real-time applications. Neural approximations reduce this cost but often require large datasets of expert NMPC solutions. We introduce Sequential Approximate MPC (Seq-AMPC), a label-free recurrent framework that generates complete control sequences along predicted trajectories using parameters shared across the prediction horizon. The policy is trained directly through known dynamics with losses for control cost, constraint satisfaction, and a cost-decrease condition aligned with a feasible-backup safety filter. Building on this filter, we introduce a stage-cost-dependent acceptance tolerance that preserves recursive feasibility, constraint satisfaction, and asymptotic convergence under nominal MPC assumptions, independently of neural approximation error. We further derive Hoeffding-based finite-sample bounds for the expected fraction of applied proposals and closed-loop cost discrepancy relative to NMPC. We evaluate Seq-AMPC on a quadcopter, chain of masses, obstacle-avoidance vehicle, and robotic manipulator, comparing recurrent and feedforward policies under label-free and imitation training. Across four benchmarks, training takes 0.2-1.7 GPU hours per policy without expert control labels. The filtered controller applies 97.2-99.1% of neural proposals and achieves mean closed-loop cost ratios within 0.3% of NMPC on jointly completed episodes. Tests with suboptimal feasible initial plans further demonstrate the benefit of neural replanning.