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Outcome independence requires $P(A,B\mid a,b,\lambda )=P(A\mid a,b,\lambda )P(B\mid a,b,\lambda )$. It says that, once settings and the proposed complete state are fixed, learning the distant outcome does not change the local outcome probability. Combining it with parameter independence gives Bell factorization. Entangled quantum states generally fail outcome independence even though their observable marginal distributions obey quantum no-signalling.

For a spin singlet state, quantum spin measurements along unit vectors $\mathbf{a},\mathbf{b}$ predict $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$. Take ${\mathbf{a}}_0=\mathbf{z}$, ${\mathbf{a}}_1=\mathbf{x}$, ${\mathbf{b}}_0=(\mathbf{z}+\mathbf{x})/\sqrt{2}$ and ${\mathbf{b}}_1=(\mathbf{z}-\mathbf{x})/\sqrt{2}$. These giveThe violation therefore excludes the whole conjunction of the sufficient assumptions, for any model reproducing these correlations. It does not identify which premise fails without further commitments about causal explanation, state completeness and measurement setting generation.

A defensible verdict is to retain ordinary independent setting preparation and reject Bell local causality, specifically its classical screening-off factorization. In standard quantum theory, taking the pure entangled state as the complete state preserves parameter independence: each local marginal is $1/2$, regardless of the distant setting. Yet outcome independence fails. For equal spin axes, $P(A=+,B=-)=1/2$, whereas $P(A=+)P(B=-)=1/4$. The distant outcome changes the conditional prediction even after the quantum state is specified. This failure is compatible with quantum no-signalling, since conditioning on that outcome requires learning a result which cannot itself be controlled.

Adding further past variables does not restore the entire Bell conjunction while retaining the violating correlations; the CHSH inequality already allowed arbitrary such $\lambda$. Alternatively, one may reject measurement independence, permit retrocausal setting dependence in the relevant past state, or adopt another non-Bell-local description. Those are logically possible responses, but each must explain its replacement probability and causal model rather than treating the inequality as a failed algebraic theorem. A deterministic theory is not exempt from this requirement.

The assumption of measurement independence is supported by the purpose and design of independently varied settings, but cannot be proved for every conceivable hidden variable merely by inspecting observed setting frequencies. Conversely, a rejection of Bell factorization does not alone establish a controllable superluminal signal or select a unique ontology. The reasoned verdict is against Bell's screening-off locality, conditional on the other stated premises; the unconditional conclusion concerns their conjunction.

Consider two spacelike-separated measurement regions $R_A$ and $R_B$, with locally chosen settings $a,b\in \{0,1\}$ and recorded outcomes $A,B\in \{-1,1\}$. A relevant past region supplies a sufficient specification $\lambda$ of the common preparation and other hidden physical information needed to screen the outcomes. This is not necessarily just a quantum state written at the source, nor a conditioning on arbitrary records which already contain the future choices. The spacetime shielding and sufficiency requirements are substantive causal assumptions.

Use Bell local causality: once the relevant past is sufficiently specified, learning facts in the remote measurement region changes no local outcome probability. In particular, requirewhere conditional probabilities are defined. The probability chain rule givesThis factorization combines parameter independence with outcome independence. Neither probability is required to be zero or one: the argument includes stochastic local hidden-variable theories.

Also assume measurement independence, $\rho (\lambda \mid a,b)=\rho (\lambda )$, so all four setting pairs average over one normalized nonnegative past-variable distribution. This is a statistical condition on the setting procedure and relevant hidden information, not a proof of a metaphysical doctrine of free will. Finally the recorded outcomes must represent the same ensemble of trials. One can include nondetections in a specified bounded-outcome rule; if instead one discards events depending on setting or outcome, a further sampling assumption is required. Setting-dependent postselection is not covered silently by the argument.

Let ${\alpha }_a(\lambda )={\sum }_{A}AP(A\mid a,\lambda )$ and ${\beta }_b(\lambda )={\sum }_{B}BP(B\mid b,\lambda )$. They lie in $[-1,1]$, and the factorization givesFor each $\lambda$,Averaging with the same $\rho$ proves the CHSH inequalityNo separate assumption of simultaneous predetermined values for incompatible quantum observables was needed. The common probability model and its screening relations did the work.

A Reichenbach common cause principle can motivate the same calculation: a complete classical common cause in the past screens the two outcomes, while each setting affects only its own wing. Together with setting-independent sampling this gives the displayed factorization. It is not enough to provide a different screening variable for each pair of settings: those constructions need not form one compatible common-cause model for the four correlations.

A stochastic Einstein locality formulation instead constrains objective chances by facts in the appropriate causal past. To use it as a sufficient premise here, specify both local dependence on settings and a joint screening condition for the two outcomes. Merely requiring local marginal chances not to depend on a remote setting gives parameter independence, which is weaker than factorization. Thus the precise stochastic locality version matters; one should not infer a Bell inequality from every condition loosely described as no superluminal influence.

For the spin singlet state, measurements along the same axis are perfectly anticorrelated. By measuring either spin component of particle $B$, one can therefore predict with certainty the corresponding result for distant particle $A$. The Einstein–Podolsky–Rosen criterion of reality then says that every such spin component of $A$ is an element of physical reality, because the choice at $B$ can be made without disturbing $A$. Since the quantum state assigns no simultaneous sharp values to noncommuting spin components, the EPR argument concludes that the wavefunction is not a complete description, provided this locality premise is accepted.

Let $\lambda$ denote a proposed complete hidden state, and let $a,b$ be the chosen axes.

For two axes on each side, every $\lambda$ obeysMeasurement independence permits averaging the same distribution of $\lambda$ for all four setting pairs, producing the CHSH inequality

The singlet prediction isChoose coplanar unit vectorsThenand thereforeThus the predictions of quantum theory violate the conjunction of outcome determinism, parameter independence, and measurement independence. This is Bell theorem; the calculation alone does not select which premise a deeper theory must abandon.