= Solution
For local dichotomic outcomes define $A(a,\lambda),B(b,\lambda)=\pm1$ and a setting-independent normalized distribution $\rho(\lambda)$. At each $\lambda$,
$$
A(a)[B(b)+B(b')]+A(a')[B(b)-B(b')]=\pm2,
$$
because exactly one bracket vanishes and the other is $\pm2$. Integrating and using the <triangle inequality> yields
$$
\boxed{|E(a,b)+E(a,b')+E(a',b)-E(a',b')|\leq2.}
$$
For stochastic local response functions the same bound follows by replacing outcomes with <Conditional mean>[conditional means] in $[-1,1]$, or by including local randomness in $\lambda$. The important locality condition is factorization of joint conditional outcome probabilities, together with independence of the setting distribution and $\lambda$.
The <Spin singlet state>[spin singlet] instead predicts $E(a,b)=-a\cdot b$. Two orthogonal directions at one wing and directions halfway between them at the other give $|S|=2\sqrt2$. Therefore reproducing the quantum predictions requires giving up at least one of the stated premises. In the perfect-anticorrelation version, $B(a,\lambda)=-A(a,\lambda)$ gives Bell's three-setting form $|E(a,b)-E(a,c)|\leq1+E(b,c)$ by the same algebra.
This does not establish that all deterministic or hidden-variable theories are impossible: nonlocal guidance is a counterexample to that reading. Nor does it equate Bell locality with no-signalling; averaging quantum outcomes can preserve setting-independent local marginals despite nonfactorizing correlations. The theorem is significant because it turns a local-completeness dispute into incompatible quantitative correlation bounds, not because it refutes every possible meaning of physical reality.<On the Einstein Podolsky Rosen paradox by John Stewart Bell>[1964, assumptions and three-setting inequality]{ref}
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