otation implicitly contains the assumption that the identity of Alice’s measurements as random variables does not depend on Bob’s settings, and vice versa. It is well known [@fine_hidden_1982] that under the no-signaling conditions the existence of the jpd is equivalent to the CHSH inequalities being satisfied. Applying the NP approach, the minimal L1 norm of the probability distribution is given by [@oas_exploring_2014] $$\Gamma_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} ,\label{eq:M-inequality-AliceBob}$$ where $$\begin{array}{r}
S_{CHSH}=\raisebox{0pt}[0pt][0pt]{\ensuremath{{\displaystyle \max_{\#^{-}=1,3}}}}\{\pm\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle \pm\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle \phantom{\mbox{\}}.}\\
\pm\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle \pm\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \mbox{\}}.
\end{array}$$ Here $\Gamma_{\min}=0$ corresponds to the CHSH inequalities, and $\Gamma_{\min}>0$ to contextuality.

Turning now to the CbD approach, we have four pairs of random variables, $$\left(\mathbf{A}_{1,1},\mathbf{B}_{1,1}\right),\left(\mathbf{A}_{1,2},\mathbf{B}_{1,2}\right),\left(\mathbf{A}_{2,1},\mathbf{B}_{2,1}\right),\left(\mathbf{A}_{2,2},\mathbf{B}_{2,2}\right).\label{eq:AB pairs}$$ Here, $\mathbf{A}_{i,j}$ denotes Alice’s measurement under her setting $i=1,2$ when Bob’s setting is $j=1,2$, and analogously for $\mathbf{B}_{i,j}$. We seek a jpd with the smallest value $\Delta_{\min}$ of $$\begin{array}{r}
\Pr\left[\mathbf{A}_{1,1}\neq\mathbf{A}_{1,2}\right]+\Pr\left[\mathbf{A}_{2,1}\neq\mathbf{A}_{2,2}\right]+\Pr\left[\mathbf{B}_{1,1}\neq\mathbf{B}_{2,1}\right]+\Pr\left[\mathbf{B}_{1,2}\neq\mathbf{B}_{2,2}\right].\end{array}$$ No contextuality means $\Delta_{\min}=0$. A computer assisted Fourier-Motzkin elimination algorithm yields (this is a special case of the result in Ref. [@dzhafarov_generalizing_2014; @DKL2014; @KDL2014]) $$\Delta_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} .\label{eq: