Path-Complete Robust Critic

1. Integration point: use this as the critic and actor structure in a robust model-based or model-free RL loop. Define M known or learned transition modes f_i; examples include different simulators, actuator models, corruption operators, or adversarial disturbance choices. Build a small path-complete graph G and a reachability graph H offline. Instantiate one scalar critic V_alpha(x) per graph node alpha and one policy head pi_A(x) per reachability node A, sharing a trunk but using separate output heads. At action time compute q_A=max_{alpha in A} V_alpha(x), choose A*=argmin_A q_A, and execute pi_A*(x). 2. Pseudocode: sample x; for every A and every graph edge (alpha,beta,i) with alpha in A, set r=c(x,pi_A(x))+V_beta_target(f_i(x,pi_A(x))); violation=softplus((r-V_alpha(x))/tau); minimize the sum of violation terms plus ordinary actor/environment return loss. Use stop-gradient target critics, target-network Polyak averaging, and a multiplier or hinge schedule that gradually increases the inequality penalty. For real transitions, sample i uniformly or from a worst-case mode distribution; for model-based training evaluate all M modes. 3. The graph construction and reachability relation are computed exactly from the discrete graph. Neural values, transition outputs, and Bellman residuals are estimated empirically. To test certification, bound each residual on a box using interval bound propagation or dense state gridding, and report the largest upper-bound violation rather than claiming formal validity from samples. 4. First experiment: train a 2-layer MLP actor/critic on a toy 2D environment with four arbitrary switching modes and compare SAC, a single robust critic using max-over-modes, and this 4-node path-complete critic at equal environment steps and model evaluations. Measure worst-mode return, ordinary return, mode-switch frequency, Bellman residual, and catastrophic-state rate. Success is a higher minimum-over-switch-sequences return and lower worst-case residual without reducing average return by more than 5 percent.