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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: $I_{2,2}=2-(E_{11}+E_{21}+E_{22}-E_{12})/2$.

For local outcomes $A_0,A_1,B_0,B_1\in \{-1,1\}$, the Clauser–Horne–Shimony–Holt inequality isAn arbitrary no-signalling correlation can reach the algebraic maximum $4$, while quantum correlations are bounded by $2\sqrt{2}$.

For a short proof of the local bound, fix a deterministic local hidden-variable model assignment. The combination is $A_0(B_0+B_1)+A_1(B_0-B_1)$. Exactly one bracket vanishes and the other is $\pm 2$, so every assignment contributes $\pm 2$. Averaging with a single setting-independent probability distribution preserves the interval $[-2,2]$. Mixtures also cover stochastic local hidden-variable theories. Rejecting a different sample at each setting would invalidate precisely that common-average step.

The quantum bound follows just as economically. For Hermitian operators with $A_i^2=B_j^2=I$, the CHSH inequality operator $C=A_0\otimes (B_0+B_1)+A_1\otimes (B_0-B_1)$ obeys $C^2=4I-[A_0,A_1]\otimes [B_0,B_1]$. Each commutator has operator norm at most two, giving $\Vert C\Vert \le 2\sqrt{2}$. The Polarization CHSH count model attains this bound without allowing either station to control the distant marginal probability distribution.

A two-setting nonlocality argument in which several zero-probability events forbid a further coincidence in any deterministic local hidden-variable model, but quantum theory assigns that coincidence positive probability. For binary indicators, suppose $C_{+}^R{C}_{-}^R=0$, $D_{+}^I=1\Rightarrow C_{-}^R=1$ and $D_{-}^I=1\Rightarrow C_{+}^R=1$. Any assignment with $D_{+}^I{D}_{-}^I=1$ would violate the first relation. A quantum implementation can nevertheless give such a coincidence. Real imperfect experiments require an inequality that accounts for their nonzero error probabilities. Lucien Hardy gave a two-particle inequality-free argument in 1992.^[ref]

For each hidden variable $\lambda$, a deterministic local hidden-variable model assigns values $a_i(\lambda ),b_j(\lambda )\in \{0,\dots ,d-1\}$ to all settings. The responses depend only on the local setting; their averaging distribution is setting independent, as required by measurement independence.

Let $T(\lambda )$ be the sum of the nonnegative modular representatives in the chained modular Bell inequality. Before reduction modulo $d$, the chain telescopes:Reduction changes this integer only by multiples of $d$. Since all reduced terms are nonnegative,The smallest possible nonnegative value in this residue class is $d-1$. Averaging over the hidden variable provesAdding local random seeds to $\lambda$ gives the same result for stochastic local hidden-variable theories.

For the coplanar qubit realization of the chained Bell inequality, shareAt angle $\theta$ in the $xz$ plane, use the projective measurement of the observableHere ${\sigma }_x,{\sigma }_z$ are Pauli matrices, and the classical bit $a$ records eigenvalue $(-1{)}^a$. Since $⟨{\sigma }_z\otimes {\sigma }_z⟩=⟨{\sigma }_x\otimes {\sigma }_x⟩=1$ and both crossed expected values vanish in this Bell state,The one-party expected values vanish, so the complete joint probabilities areFor binary outcomes, $[a-b]$ is one exactly when $a\ne b$, while $[b-a-1]$ is one exactly when $a=b$. Consequently

For $N\ge 2$, set $\delta =\pi /(2N)$ and choose Alice's angle ${\theta }_i=2(i-1)\delta$ and Bob's angle ${\varphi }_i=(2i-1)\delta$. Every adjacent pair $A_i,B_i$ or $B_i,A_{i+1}$ differs by $\delta$, giving expectation ${\sin }^2(\delta /2)$. The closing pair has angle difference $(2N-1)\delta =\pi -\delta$; its agreement probability is also ${\sin }^2(\delta /2)$. Thus every one of the $2N$ terms contributes equally:Indeed $\sin x<x$ for $x>0$, so this value is below ${\pi }^2/(8N)\le {\pi }^2/16<1$. For an explicit two-setting instance, Alice measures ${\sigma }_z,{\sigma }_x$ and Bob measures $({\sigma }_z+{\sigma }_x)/\sqrt{2},(-{\sigma }_z+{\sigma }_x)/\sqrt{2}$. The four relevant expected values sum toviolating the local bound one. At $N=1$ there is no violation; the two terms always sum to one. As $N\to \infty$, the displayed quantum value is asymptotic to ${\pi }^2/(8N)$ and approaches the algebraic lower bound zero.

The violation rules out a local hidden-variable theory with setting-independent preassigned local responses reproducing these quantum correlations. It does not permit faster-than-light communication: the joint formula gives each local outcome probability $1/2$, independent of the remote setting, as required by the no-signalling theorem. The contradiction concerns the joint correlations and the Bell assumptions, not the ability to send a signal.

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 $d-1$.

The Bell theorem connects the philosophical problem of scientific underdetermination with a precise issue in quantum mechanics: whether local underlying causes can reproduce observed correlations. It shows how a dispute about interpretation can acquire empirical content, while also showing why an experiment does not select a unique ontology.

A shared cause can produce strong correlations without a signal between distant measurements. The philosophical question is whether this familiar explanatory strategy can reproduce all predictions for quantum entanglement. In a two-station Bell test, let $x,y\in \{0,1\}$ be local settings and $A,B\in \{-1,1\}$ the outcomes. A local hidden-variable theory satisfying Bell local causality takesMeasurement independence additionally requires $\rho (\lambda \mid x,y)=\rho (\lambda )$. The same ensemble of underlying states must therefore apply across the four setting pairs.

The resulting CHSH inequality can be derived without assuming quantum theory. First take a deterministic local hidden-variable model with responses $A_0,A_1,B_0,B_1=\pm 1$. For each $\lambda$,because exactly one bracket vanishes. Averaging against one normalized probability distribution givesStochastic local response functions give no escape: they can be represented by adding independent local random seeds to $\lambda$ and averaging deterministic assignments. The essential restriction is the common setting-independent ensemble and local dependence of responses, not a separate insistence that all nature be deterministic.

For two spin-one-half systems in a spin singlet state, ideal quantum measurements along unit directions $\mathbf{a},\mathbf{b}$ give $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$. ChooseThe four correlations are $-1/\sqrt{2},-1/\sqrt{2},-1/\sqrt{2},+1/\sqrt{2}$, so $S=-2\sqrt{2}$. Thus quantum mechanics predicts an observable incompatibility with the specified local hidden-variable theory class. This is stronger than saying that two interpretations happen to fit the observations collected so far.

Actual Bell tests require more than the ideal calculation. Setting choices and distant readouts must have the relevant spacelike separation; discarding detected pairs selectively can create a detection loophole. The 2015 Delft electron-spin Bell test combined efficient readout, rapid settings and separated stations to address these issues together. The reported statistic was $S=2.42\pm 0.20$, with a test of the local-model hypothesis accounting for device memory.^[ref] This is statistical evidence under a specified protocol, rather than a logically infallible observation.

The philosophical conclusion must be equally precise. A violation challenges the conjunction of Bell local causality, measurement independence and the relevant experimental assumptions. It does not disprove every possible hidden-variable account, or establish that the observer's mind creates the outcome. Different theories can reject different premises while agreeing on the observable correlations. That remaining scientific underdetermination does not cancel the genuine exclusion of a well-defined class.

Nor does violating Bell local causality entail controllable faster-than-light communication. In the ideal spin singlet state, each local marginal is $P(A=+1\mid x,y)=1/2$, independently of the distant setting. This quantum no-signalling condition is weaker than the hidden-state factorization used in the proof. Conflating them turns a precise result into an unwarranted conclusion about signalling.

The example therefore illustrates both the power and the limit of empirical testing in philosophy of science. Mathematical clarification can transform an apparently metaphysical disagreement into rival observational constraints. Evidence can discriminate those constraints without determining every question about the reality represented by a successful theory.

In a deterministic local hidden-variable model, write $C^{+}(R,\lambda )$ or $D^{+}(I,\lambda )$ without an argument for the remote splitter setting. That omission is the parameter independence assumption: the local response for a fixed complete hidden state does not change when the other wing changes its apparatus. Measurement independence also requires the hidden-state ensemble to be independent of the freely selected settings, allowing the same $\lambda$ assignments to be compared across configurations.

The zero probabilities derived above then constrain those assignments, apart from sets of hidden states of zero measure:

For a trial with $D^{+}(I,\lambda )D^{-}(I,\lambda )=1$, the last two implications force $C^{+}(R,\lambda )=C^{-}(R,\lambda )=1$, contradicting the first condition. Thus these deterministic local assignments predict zero probability for the double-$D$ event, whereas quantum theory predicts $1/16$. This is Hardy's paradox, an inequality-free ideal contradiction. With imperfect apparatus the zero-event conditions become small nonzero rates and require a statistical inequality rather than declaring a contradiction from one imperfect detection.^[ref]

The logical contradiction combines counterfactual hidden-variable assignments; it does not assign simultaneously observed path properties to a quantum state. Nor does it allow a message to be sent by changing a distant splitter: the observable local marginal obeys quantum no-signalling. For example, fixing the plus-side splitter to $I$ gives $P(D^{+})=1/8$ for either minus-side setting in the results above.