= Chained modular Bell inequality
{title2=$I_{N,d}\geq d-1$}
For outcomes in $\{0,\ldots,d-1\}$, let $[x]_d$ be their nonnegative modular representative. A <deterministic local hidden-variable model> satisfies
$$
I_{N,d}=\sum_{j=1}^N\mathbb E\big([A_j-B_j]_d\big)+\sum_{j=1}^{N-1}\mathbb E\big([B_j-A_{j+1}]_d\big)+\mathbb E\big([B_N-A_1-1]_d\big)\geq d-1.
$$
The unreduced chain telescopes to $-1$, so its reduced nonnegative sum is at least $d-1$ for each assignment. For binary outcomes and two settings, it is a form of the <CHSH inequality>: $I_{2,2}=2-(E_{11}+E_{21}+E_{22}-E_{12})/2$.
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