\{5,6\}, \{7,8\} $ form a partition $\Pi_{12}$ of $\Omega$. Thanks to experiment $AB$, the four parts of $\Pi_{12}$ are coobservable. Define partitions $\Pi_{23}$ and $\Pi_{31}$ similarly. The four parts of $\Pi_{23}$ are coobservable thanks to the experiment $BC$, and the four parts of $\Pi_{13}$ are coobservable thanks to the experiment $AC$.

The four parts of any partition $\Pi_{ij}$ generate a Boolean algebra $E_{ij}$ of subsets of $\Omega$. Define $${\ensuremath{\mathrm{CO}}}= \big\{ E: E\subseteq E_{ij}\text{ for some }i,j\big\},$$ so that each $E_{ij}$ is an ensemble.

[Probability distribution.]{} Since the experimental results support quantum mechanics, the experiments $AB$, $BC$ and $AC$ produce results approximating the following three tables.

$$\begin{aligned}
\label{AB}
{\mathcal P}(++\pm) &= {\mathcal P}\{1,2\} = \frac12 \cos^2(\pi/4) = 1/4,\\
{\mathcal P}(--\pm) &= {\mathcal P}\{7,8\} = \frac12 \cos^2(\pi/4) = 1/4,\tag{AB}\\
{\mathcal P}(+-\pm) &= {\mathcal P}\{3,4\} = \frac12 \sin^2(\pi/4) = 1/4,\\
{\mathcal P}(-+\pm) &= {\mathcal P}\{5,6\} = \frac12 \sin^2(\pi/4) = 1/4.\end{aligned}$$

$$\begin{aligned}
\label{BC}
{\mathcal P}(\pm++) &= {\mathcal P}\{1,5\} = \frac12 \cos^2(\frac\pi8) = \frac18(2+\sqrt2),\\
{\mathcal P}(\pm--) &= {\mathcal P}\{4,8\} = \frac12 \cos^2(\frac\pi8) = \frac18(2+\sqrt2),
             \tag{BC}\\
{\mathcal P}(\pm+-) &= {\mathcal P}\{2,6\} = \frac12 \sin^2(\frac\pi8) = \frac18(2-\sqrt2),\\
{\mathcal P}(\pm-+) &= {\mathcal P}\{3,7\} = \frac12 \sin^2(\frac\pi8) = \frac18(2-\sqrt2).\end{aligned}$$

$$\begin{aligned}
\label{AC}
{\mathcal P}(+\pm+) &= {\mathcal P}\{1,3\} = \frac12 \cos^2(3\pi/8) 