Certified Learning and Equilibrium Implementation under Opaque Partial Commitment

As an extension of existing Bayesian persuasion framework with inadequate message mechanism, we study direct recommendation when a sender is bound by an installed information policy only with probability $\rho$, the realization of binding is hidden, and the receiver does not observe the persistent structural environment. The receiver first sees a payoff-neutral, nonmanipulable calibration sample and then faces a fresh, non-certified deployment interaction. In common, the calibration law identifies only the receiver-facing reduced form, not the latent binding and discretionary kernels. We characterize type-wise $\rho$-implementability, construct the receiver's posterior over the full deployment node, and prove a static direct-following implementation theorem. After every calibration history that passes a posterior-predictive obedience test, the deployment assessment is an exact perfect Bayesian equilibrium: Bayes consistency, receiver sequential rationality, sender sequential rationality, and off-path completion are all verified. Under finite-type separation, common recommendation support, and a positive obedience margin, the test activates such an equilibrium with high probability. Our results keep statistical failure probability distinct from equilibrium approximation. Finally, we embed the original robust value frontier, support-wise linear-programming algorithm, and binary-action fractional-knapsack specialization into this implementation framework