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For outcomes in $\{0,\dots ,d-1\}$, let $[x{]}_d$ be their nonnegative modular representative. A deterministic local hidden-variable model satisfiesThe 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: $I_{2,2}=2-(E_{11}+E_{21}+E_{22}-E_{12})/2$.

Combining the EPR criterion of reality with perfect predictive correlations and locality motivates pre-existing values for the locally selectable observables. The general local hidden-variable theory additionally assumes a complete shared variable $\lambda$ and conditional factorizationThe second condition is measurement independence. The reality criterion alone does not prove this factorization for arbitrary imperfect correlations: it motivates the local completion whose consequences are being tested.

A deterministic local hidden-variable model may be used without loss of generality by putting local random seeds into $\lambda$. For each $\lambda$, write its setting responses as $A,A^{\prime },B,B^{\prime }\in \{-1,1\}$. Integrating with the same setting-independent distribution and using the triangle inequality givesExactly one of $B-B^{\prime }$ and $B+B^{\prime }$ vanishes and the other has magnitude two. Hence the requested CHSH bound is two. No assumption about the spacelike quantum state enters this local-model derivation.

The EPR criterion of reality says that a physical quantity has an element of reality if its value can be predicted with certainty without physically disturbing the system. The proposed criterion distinguishes what the system possesses from what one happens to measure.

A local hidden-variable theory supplements the preparation by a variable $\lambda$, drawn from a distribution $\mu (\lambda )$ independent of the later measurement settings. Its locality assumption is conditional factorization:Alice's local response does not depend on Bob's setting $y$, and Bob's does not depend on Alice's $x$. Correlations can arise from the shared past variable $\lambda$. In a deterministic local hidden-variable model, the response probabilities are point masses at functions $a=A_x(\lambda )$ and $b=B_y(\lambda )$. In particular, all possible local settings have definite values for a fixed $\lambda$, even when only one setting is used in a trial.

The setting-independence assumption is needed for comparing these predetermined values across different experiments. Conditional factorization is stronger than quantum no-signalling: no-signalling constrains observed marginal probabilities, whereas local hidden-variable theory constrains their decomposition at fixed $\lambda$. This distinction is what Bell theorem tests.

Fix a hidden variable in the deterministic local hidden-variable model, so that every $A_j,B_j$ is an integer. Let $[x{]}_d$ be the representative of $x$ modulo $d$ in $\{0,\dots ,d-1\}$. The terms in the chained modular Bell inequality alternate between the two parties. Before reduction their sum telescopes:After reduction, the sum $T$ is a nonnegative integer congruent to $-1$ modulo $d$. For $d\ge 2$, the smallest possible such integer is $d-1$. Thus $T\ge d-1$ for each hidden variable separately. Averaging over the setting-independent distribution givesHere each expectation is the expectation value of the reduced random variable, not the residue of its expectation. Each term can be measured using one setting at each site. The proof uses their common deterministic assignments rather than any joint quantum measurement of incompatible local settings. Stochastic local hidden-variable theories satisfy the same bound: include their local random seeds in $\lambda$ and average the resulting deterministic assignments.

The printed final coefficient in the definition of the average is typographically incomplete. The expectation used here is the usual $\mathbb{E}X={\sum }_{x=0}^{d-1}xP(X=x)$, with final coefficient $d-1$.