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Write $|0⟩=|\uparrow ⟩$, $|1⟩=|\downarrow ⟩$, and use $X_j,Y_j$ for the Pauli matrices on particle $j$. Their actions on the computational basis areSet $P_1=X_1{Y}_2{Y}_3$, $P_2=Y_1{X}_2{Y}_3$, $P_3=Y_1{Y}_2{X}_3$, and $P_4=X_1{X}_2{X}_3$. Any two distinct operators in this list differ by an $X$ versus $Y$ at exactly two sites. At each such site $XY=-YX$, while factors on different particles commute. Swapping the two full products therefore introduces two minus signs, givingThis is the Pauli-string commutation parity rule. Each product is Hermitian and squares to the identity, so its only eigenvalues are $\pm 1$.

For the negative-phase Greenberger–Horne–Zeilinger state, put $|G_{-}⟩=(|000⟩-|111⟩)/\sqrt{2}$. The two $Y$ factors in each of $P_1,P_2,P_3$ contribute $i^2=-1$ on $|000⟩$ and $(-i{)}^2=-1$ on $|111⟩$. Thus these operators take $|000⟩$ to $-|111⟩$ and $|111⟩$ to $-|000⟩$. By contrast, $P_4$ flips these two basis states without a phase. ConsequentlyThe GHZ state is therefore a simultaneous eigenstate of the four commuting products. As a sign check, direct multiplication givesThis is compatible with the displayed quantum eigenvalues.

Now apply the EPR criterion of reality together with locality for the spacelike separated particles. In the context with local settings $X_1,Y_2,Y_3$, the product of the three outcomes is certainly $+1$. Therefore the two remote $Y$ outcomes predict the first particle's $X$ outcome with certainty, without disturbing it under the locality assumption. The other perfect-correlation contexts similarly predict each particle's $X$ or $Y$ outcome from measurements performed only on the other particles. The EPR criterion of reality then assigns predetermined local values $x_j,y_j\in \{+1,-1\}$ to these six observables. Locality makes the value assigned at one site independent of which measurements were chosen at the other sites.

To reproduce the three certain mixed-setting quantum predictions, every such assignment must satisfyMultiplying these scalar equations cancels each squared $y_j$ and gives $x_1{x}_2{x}_3=+1$. But the all-$X$ quantum prediction, from the $P_4$ eigenvalue, is $x_1{x}_2{x}_3=-1$ with certainty. HenceThe perfect GHZ correlations contradict the simultaneous local elements of reality inferred by the EPR argument. No statistical Bell inequality is needed. The locality assumption is essential to that inference; the certainty criterion without the premise that distant choices cannot disturb the local element does not by itself imply a local hidden-variable model. Nor does the argument require simultaneous measurement of $X$ and $Y$ on a single particle: different experimental contexts establish the perfect predictions, and it is the EPR/locality interpretation that attempts to assign all their values at once.