= Solution
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, require
$$
P(A\mid a,b,B,\lambda)=P(A\mid a,\lambda),\qquad P(B\mid a,b,\lambda)=P(B\mid b,\lambda),
$$
where conditional probabilities are defined. The probability chain rule gives
$$
P(A,B\mid a,b,\lambda)=P(A\mid a,\lambda)P(B\mid b,\lambda).
$$
This 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 gives
$$
E_{ab}=\int\rho(\lambda)\alpha_a(\lambda)\beta_b(\lambda)\,d\lambda.
$$
For each $\lambda$,
$$
|\alpha_0(\beta_0+\beta_1)+\alpha_1(\beta_0-\beta_1)|\le|\beta_0+\beta_1|+|\beta_0-\beta_1|=2\max(|\beta_0|,|\beta_1|)\le2.
$$
Averaging with the same $\rho$ proves the <CHSH inequality>
$$
\boxed{|E_{00}+E_{01}+E_{10}-E_{11}|\le2.}
$$
No 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.
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