Post-quantum cryptography (PQC) must secure large-scale communication systems against quantum adversaries where classical hardness alone is insufficient and purely quantum schemes remain impractical. Lattice-based key encapsulation mechanisms (KEMs) such as CRYSTALS-Kyber provide efficient quantum-resistant primitives but rely solely on computational hardness assumptions that are susceptible to hybrid classical-quantum attacks. To overcome this limitation, we introduce the first Clauser-Horne-Shimony-Holt (CHSH)-certified Kyber protocol, which embeds quantum non-locality verification directly within the key exchange phase. The proposed design integrates CHSH entanglement tests using Einstein-Podolsky-Rosen (EPR) pairs to yield measurable quantum advantage values exceeding classical correlation limits, thereby coupling information--theoretic quantum guarantees with lattice-based computational security. Formal reductions demonstrate that any polynomial-time adversary breaking the proposed KEM must either solve the Module Learning With Errors (Module-LWE) problem or a Quantum Merlin-Arthur (QMA)-complete instance of the 2-local Hamiltonian problem, under the standard complexity assumption QMA $\subset$ NP. The construction remains fully compatible with the Fujisaki-Okamoto (FO) transform, preserving chosen-ciphertext attack (CCA) security and Kyber's efficiency profile. The resulting CHSH-augmented Kyber scheme therefore establishes a mathematically rigorous, hybrid post-quantum framework that unifies lattice cryptography and quantum non-locality to achieve verifiable, composable, and forward-secure key agreement.