Geometric Substrate Models and Bell–CHSH Correlations: A Structural Analysis of Assumption Relaxation

This paper presents a structural analysis of Bell–CHSH correlations using a hierarchy of geometric hidden-variable models defined on the unit circle. Bell’s theorem constrains hidden-variable models under the joint assumptions of locality, deterministic response functions, and measurement independence. The paper examines how Bell–CHSH behavior changes when these assumptions are modified one at a time. A minimal local model with setting-independent preparation obeys the classical Bell–CHSH bound |S| <= 2. A setting-dependent update can produce Bell violation but introduces operational signalling. Direct imposition of the cosine correlation is used as a structural reference, reproducing Tsirelson’s bound while preserving no-signalling marginals. The paper then shows that Tsirelson-level correlations can also arise from a geometric, context-dependent sampling rule without inserting the cosine function directly into the preparation law. In that construction, local deterministic response functions and operational no-signalling are preserved, while measurement independence is relaxed. The models are presented as structural illustrations rather than physical hidden-variable proposals. Their purpose is to make the assumption-dependence of Bell correlations explicit in a transparent geometric framework. Internal reference: CGI-RSR-000021.