Local hidden-variable model of singlet-state correlations discussed in M. Czachor, Arithmetic loophole in Bell's Theorem: Overlooked threat to entangled-state quantum cryptography, Acta Phys. Polon. A 139, 70-83 (2021), is shown to be a particular case of an infinite hierarchy of local hidden-variable models based on an infinite hierarchy of calculi. Violation of Bell-type inequalities is shown to be a `confusion of languages' problem, a result of mixing different but neighboring levels of the hierarchy. Mixing of non-neighboring levels results in violations beyond the Tsirelson bounds.
The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value 22, known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than 22. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.
Bell’s theorem is widely understood as demonstrating that quantum correlations cannot be explained by any theory of local hidden variables. The experimental violation of Bell inequalities has therefore often been interpreted as evidence for nonlocality or “spooky action at a distance.” This paper proposes a different geometric interpretation. It accepts Bell’s theorem as a decisive refutation of separable preassigned vector realism, but argues that entangled systems need not be modeled as pairs of particles carrying hidden local vector outcomes. Instead, the entangled state is interpreted as a nonseparable bivector coherence relation whose correlated vector outcomes are disclosed only through measurement. On this view, measurement is a basis-conditioned reduction from bivector structure to local vector registration. The standard singlet correlation E(a,b)=-a⋅b, including the Tsirelson bound S=2√2, is then interpreted as the scalar trace of a shared bivector relation rather than as the result of superluminal influence. The framework does not deny Bell’s theorem, Bell experiments, or quantum statistics. It instead reframes their ontological meaning: Bell violation marks the failure of separable vector completion, not the necessity of action at a distance. Keywords: Bell’s theorem; entanglement; bivectors; geometric algebra; spin; nonlocality; measurement; quantum foundations; Tsirelson bound; separable realism; relational ontology; UCCF. Explainer video Watch on YouTube (Roots of Reality): https://youtu.be/wfpY4QZ45ME The video, its thumbnail and description are included in this record.