= Solution
Write $|0\rangle=|\uparrow\rangle$, $|1\rangle=|\downarrow\rangle$, and use $X_j,Y_j$ for the <Pauli matrices> on particle $j$. Their actions on the <computational basis> are
$$
X|0\rangle=|1\rangle,\quad X|1\rangle=|0\rangle,\quad Y|0\rangle=i|1\rangle,\quad Y|1\rangle=-i|0\rangle.
$$
Set $P_1=X_1Y_2Y_3$, $P_2=Y_1X_2Y_3$, $P_3=Y_1Y_2X_3$, and $P_4=X_1X_2X_3$. 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, giving
$$
\boxed{[P_i,P_j]=0\quad\text{for every }i,j.}
$$
This is the <Pauli-string commutation parity> rule. Each product is Hermitian and squares to the identity, so its only <eigenvalues> are $\pm1$.
For the negative-phase <Greenberger–Horne–Zeilinger state>, put $|G_-\rangle=(|000\rangle-|111\rangle)/\sqrt2$. The two $Y$ factors in each of $P_1,P_2,P_3$ contribute $i^2=-1$ on $|000\rangle$ and $(-i)^2=-1$ on $|111\rangle$. Thus these operators take $|000\rangle$ to $-|111\rangle$ and $|111\rangle$ to $-|000\rangle$. By contrast, $P_4$ flips these two basis states without a phase. Consequently
$$
\boxed{P_1|G_-\rangle=P_2|G_-\rangle=P_3|G_-\rangle=|G_-\rangle,\qquad P_4|G_-\rangle=-|G_-\rangle.}
$$
The <GHZ state> is therefore a simultaneous eigenstate of the four commuting products. As a sign check, direct multiplication gives
$$
P_1P_2P_3=(XYY)_1(YXY)_2(YYX)_3=-X_1X_2X_3=-P_4.
$$
This 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 satisfy
$$
x_1y_2y_3=1,\qquad y_1x_2y_3=1,\qquad y_1y_2x_3=1.
$$
Multiplying these scalar equations cancels each squared $y_j$ and gives $x_1x_2x_3=+1$. But the all-$X$ quantum prediction, from the $P_4$ eigenvalue, is $x_1x_2x_3=-1$ with certainty. Hence
$$
\boxed{\text{local EPR assignments predict }+1,\qquad\text{quantum mechanics predicts }-1.}
$$
\b[The 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.
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