No Finite NPA Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

The Navascués–Pironio–Acín (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether \emph{any} finite level can already equal the quantum set has remained open even in the bipartite scenario with two binary measurements per party. We demonstrate that \emph{no finite level} is exact. For the symmetric doubly tilted CHSH functional $h_{\alpha }=A_0{B}_0+A_0{B}_1+A_1{B}_0-A_1{B}_1+\alpha (A_0+B_0)$, set $T=1-\alpha$. Its quantum maximum satisfies $[{\omega }_{\mathrm{Q}}(1-T)-(4-2T)]\mathrm{/}{T}^3\to 4\mathrm{/}3$, whereas every fixed NPA level satisfies $[{\omega }_L(1-T)-{\omega }_{\mathrm{Q}}(1-T)]\mathrm{/}{T}^3\to +\mathrm{\infty }$. Under the corresponding boundary rescaling, an explicit expectation of the positive operator ${\omega }_{\mathrm{Q}}(1-t^2)I-H_t$ converges to the Motzkin polynomial. A bounded fixed-level error would therefore make the Motzkin polynomial plus a nonnegative constant a sum of squares, which is impossible. Consequently, every standard NPA relaxation based on a fixed finite list of words in the measurement projectors strictly contains the complete quantum set, and its nonquantum behaviors accumulate at a local deterministic behavior. Thus, the finite-level exactness of CHSH and all one-sided tilted CHSH maxima does not extend to an exact finite-level description of the complete quantum set in the minimal scenario.