= GHZ contradiction for N congruent to 3 modulo 4
{c}
{title2=$N\equiv3\pmod4$}
For $|G_N^-\rangle=(|0\rangle^{\otimes N}-|1\rangle^{\otimes N})/\sqrt2$, let $O_j$ have one <Pauli X gate> at site $j$ and <Pauli Y gates> elsewhere, and let $X_N=X^{\otimes N}$. The relations $Y|0\rangle=i|1\rangle$, $Y|1\rangle=-i|0\rangle$ show that $O_j|G_N^-\rangle$ is proportional to $|G_N^-\rangle$ exactly when $N$ is odd. In that case its <eigenvalue> is $\lambda_N=(-1)^{(N+1)/2}$, while $X_N$ always has <eigenvalue> $-1$.
Under the <EPR criterion of reality> and locality, each site's $X$ and $Y$ outcomes have predetermined values $x_j,y_j\in\{\pm1\}$. Multiplying the $N$ certain equations $x_j\prod_{k\ne j}y_k=\lambda_N$ gives $\prod_jx_j=\lambda_N$ when $N$ is odd, since every $y_k$ appears an even number of times. This contradicts the certain quantum value $-1$ precisely when $N\equiv3\pmod4$. Thus the displayed <GHZ state> gives an infinite family of perfect-correlation contradictions with <local hidden-variable theory>, without an inequality.
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