The CHSH bound for every binary correlation model admitting a local-response representation

Let $\mu$ be a binary correlation model on the setting sets $I=J=\{0,1\}$ with outcome labels $\mathrm{\Sigma }=\{-1,+1\}$, with correlations $E_{ij}(\mu )$ and CHSH functional $S(\mu )$ as in Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional clauses (d), and suppose that $\mu$ admits a local-response representation with response laws $({\alpha }_i{)}_{i\in I}$, $({\beta }_j{)}_{j\in J}$ on a probability space $(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }},\mathbb{P})$ (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws). Then $\mid S(\mu )\mid \le 2.$ The statement is a theorem about the tables and their representation; it asserts no measurement, no dynamics, no spatial or causal structure, and it contains no physics. No choice principle is used.

• Deterministic special case. If the representation has response functions $a_i,b_j:\mathrm{\Lambda }\to \mathrm{\Sigma }$ in the sense of clause (b) of Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws, then the four compositions $a_i{b}_j$ are $\pm 1$-valued random variables on one probability space and the present bound reduces to the classical CHSH inequality for such variables, which is also proved directly in The CHSH bound and the finite-sampling law for independent trials (a). The probabilistic form proved here is the general case: it does not assume determinism of the responses.
• Sharpness. The bound $2$ is not improvable for the class: the model $I=J=\{0,1\}$ with ${\mu }_{ij}$ the point mass at $(\epsilon ,\delta )$ with $\epsilon =\delta =+1$ for all $(i,j)$ admits the deterministic representation with $a_i\equiv +1$, $b_j\equiv +1$ and has $S(\mu )=2$; reversing the sign of every $b_j$ gives $S(\mu )=-2$.