No finite level of the NPA hierarchy is exact for the doubly-tilted CHSH functional near the critical tilt

This paper addresses a fundamental question regarding the convergence of the Navascues-Pironio-Acin (NPA) hierarchy, a standard tool for bounding quantum correlations. Specifically, it investigates whether any finite level of the NPA hierarchy can exactly recover the quantum maximum of the doubly-tilted CHSH functional as the parameters approach the critical line where the quantum advantage vanishes. Previous numerical observations suggested that the level required for exactness grows without bound, and this paper provides a definitive negative answer.

The author employs a primal construction to prove that for every NPA level $k \geq 2$, the NPA value strictly exceeds the quantum value in a neighborhood of the critical point. The proof is computer-assisted, utilizing exact-integer arithmetic to verify structural laws of the NPA hierarchy at the critical face. The argument relies on a level-independent signed witness—a closed-form class function—that forces a quadratic overshoot in the non-quantum part of the NPA tangent cone. The entire proof chain, including the witness identity and regime lemmas, has been independently verified through clean-room implementations.

The study proves that for every finite level $k$ of the NPA hierarchy, there exists an explicit interval $(0, \epsilon_k]$ where the NPA value $c_k(s)$ is strictly greater than the quantum value $c_Q(s)$. This confirms that no finite level is exact on any neighborhood of the critical point. The overshoot is shown to be at least quadratic in the parameter $s$. The author demonstrates that this failure is not a limitation of the numerical implementation but a structural feature: finite NPA levels admit a signed tangent direction that is strictly forbidden by quantum mechanics.