= Solution
An <eigenvalue> of an <observable> in a known corresponding <eigenstate> is a simple example satisfying the <Einstein–Podolsky–Rosen criterion of reality>: its value is predictable with certainty. For example, a <spin angular momentum> component in a prepared spin-up eigenstate has a definite predicted value. For a distant system, <Albert Einstein>, <Boris Podolsky> and <Nathan Rosen> invoked locality: a measurement performed on a sufficiently separated, noninteracting partner should not disturb that system. In the relativistic version here, the measurement events have <spacelike separation>.
The <Pauli matrices> obey $[\sigma_z,\sigma_x]=2i\sigma_y$, so
$$
\boxed{[S_z,S_x]=i\hbar S_y\ne0.}
$$
Their nondegenerate <eigenstates> are incompatible: an $S_z$ eigenstate has equal probabilities for the two $S_x$ outcomes, and conversely. Thus no single <quantum state> has both components sharply determined. The <Robertson uncertainty relation> gives $\Delta S_z\Delta S_x\ge\hbar|\langle S_y\rangle|/2$, but its right-hand side may be zero; that alone is not a proof that simultaneous zero uncertainties exist. The absence of a common eigenstate supplies the stronger conclusion for this spin-half pair.
For the specified <Bell state>, Bob obtains $+\hbar/2$ or $-\hbar/2$ with probability $1/2$ when he measures $S_z$. Conditional on his result, Alice obtains the same sign with certainty if she measures $S_z$. Under the EPR locality premise, this certainty supplies an element of reality for her $S_z$; it is a conditional prediction after Bob learns his own random result, not a premeasurement prediction of its sign.
The normalized $S_x$ eigenstates are $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ and $|-\rangle=(|0\rangle-|1\rangle)/\sqrt2$. Expanding their <tensor products> gives
$$
|++\rangle+|--\rangle=|00\rangle+|11\rangle.
$$
Consequently the same <Bell state> has equal, perfectly correlated $S_x$ outcomes. Bob's two outcomes again each have probability $1/2$, and the same locality-and-certainty argument applies separately to Alice's $S_x$.
The disputed step is to combine predictions conditional on two incompatible choices into two simultaneous, measurement-independent pre-existing values. Quantum theory provides the two conditional ensembles, not a joint sharp assignment of $S_x$ and $S_z$. This is <quantum steering>. Bob cannot measure both incompatible observables sharply on the same member of the pair. Nor does conditional steering allow signalling: ignoring his result leaves Alice's <reduced density matrix> equal to $I/2$ for either measurement choice, by the <no-communication theorem>.
To test the stronger claim of a <local hidden-variable theory>, Alice needs several correlation settings, not just perfect agreement along the two common axes. For the <CHSH inequality>, choose observables with outcomes $\pm1$:
$$
A_0=\sigma_z,\quad A_1=\sigma_x,\qquad
B_0=\frac{\sigma_z+\sigma_x}{\sqrt2},\quad
B_1=\frac{\sigma_z-\sigma_x}{\sqrt2}.
$$
In the given <Bell state>, $\langle\sigma_z\otimes\sigma_z\rangle=\langle\sigma_x\otimes\sigma_x\rangle=1$ and the crossed correlations vanish. Hence
$$
\boxed{\mathcal S=\langle A_0B_0\rangle+\langle A_0B_1\rangle+\langle A_1B_0\rangle-\langle A_1B_1\rangle=2\sqrt2,}
$$
exceeding the local bound $|\mathcal S|\le2$. The bound assumes <Bell local causality> and <measurement independence>, with the appropriate treatment of sampling and settings.
Experimental <Bell tests> support violations rather than that local assignment. The <Aspect switching experiment> changed analyzer settings during separated photon flights, while leaving sampling and switching limitations. By the time of this examination, the <Delft electron-spin Bell test> had used electron spins at stations $1.3\,\mathrm{km}$ apart, efficient spin readout and rapid random basis choices. The combination addressed both detection and locality loopholes, providing evidence against the tested local model class.<Experimental loophole-free violation of a Bell inequality by Bas Hensen and others>[2015 abstract: separated electron spins, efficient readout and fast random settings]{ref}
This does \b[not] prove that every conceivable notion of reality is false, or that noncommutation alone rules out every hidden-variable theory. A nonlocal theory can evade the local premise. What Alice can challenge is Bob's claim that locality, measurement-independent simultaneous values and all observed correlations can be retained together. An ordinary measurement disturbance at Bob's lab is not by itself a local explanation of Alice's conditional ensemble.
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