 Solution

For the spin singlet state, measurements along the same axis are perfectly anticorrelated. By measuring either spin component of particle $B$, one can therefore predict with certainty the corresponding result for distant particle $A$. The Einstein–Podolsky–Rosen criterion of reality then says that every such spin component of $A$ is an element of physical reality, because the choice at $B$ can be made without disturbing $A$. Since the quantum state assigns no simultaneous sharp values to noncommuting spin components, the EPR argument concludes that the wavefunction is not a complete description, provided this locality premise is accepted.

Let $\lambda$ denote a proposed complete hidden state, and let $a,b$ be the chosen axes.

For two axes on each side, every $\lambda$ obeysMeasurement independence permits averaging the same distribution of $\lambda$ for all four setting pairs, producing the CHSH inequality

The singlet prediction isChoose coplanar unit vectorsThenand thereforeThus the predictions of quantum theory violate the conjunction of outcome determinism, parameter independence, and measurement independence. This is Bell theorem; the calculation alone does not select which premise a deeper theory must abandon.