Volume 12, Issue 2, pages 210-247

Parker Emmerson (Yaohushuason)

Bell–CHSH is often read as a universal prohibition on \emph{local hidden variables}. Mathematically, however, the CHSH bound $S\le 2$ is a theorem about a particular hypothesis class: a \emph{single} Kolmogorov probability space carrying all counterfactual outcomes, together with measurement independence, Bell locality (factorized response functions), and bounded outputs. On that class the proof is pointwise and unconditional, so no algebraic reformulation of a parameter (including a “phenomenological velocity” (PV)) can yield Tsirelson-scale singlet correlations $E(a,b)=-\cos(a-b)$ on a Tsirelson quartet. We then make precise two distinct ways in which PV can remain meaningful as a \emph{local} and in that sense “hidden” parameter while coexisting with Tsirelson-scale correlations, without contradicting Bell’s theorem. First, we formalize “exceptional-locus/defined-only” conventions as setting-dependent acceptance (post-selection): reported correlators are conditional expectations under a setting-indexed family of accepted laws $\{\nu_{ab}\}$. Within this rung-1 semantics we give an explicit Bell-local deterministic base model whose accepted-sample joint law exactly matches the singlet law (unbiased marginals and correlator $-\cos(a-b)$) for all $a,b$, via an explicit acceptance rule $\gamma(a,b,\lambda)\in[0,1]$. Second, we give a PV-indexed microcausal (operator-algebraic) realization: PV parameterizes a canonical $SU(2)$ unitary that conjugates local observables inside commuting Alice/Bob subalgebras, and in the singlet state the unconditional correlator equals $-\cos(a-b)$ and attains Tsirelson’s value $2\sqrt2$. The results clarify a semantic and structural trichotomy: any “PV-to-Tsirelson” account must (i) change the effective ensemble by selection/context, or (ii) leave the Kolmogorov (commutative) model class (e.g.\ via noncommutativity or quasi-probability), or (iii) relax a Bell premise such as measurement independence or Bell locality. In this sense, Bell–CHSH constrains a specific formalization of hidden variables rather than ownership of the terms “local”, “real”, or “hidden” in physics discourse.

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