Geometric Substrate Models and Bell–CHSH Correlations: A Structural Analysis of Assumption Relaxation
Internal ID: CGI-RSR-000021
Author(s): Barry L. Petersen
Document Type: Research Paper
Publication Date: May 2026
Original Creation Date: Feb 28, 2025
Revised Document Date: N/A
Status: Public
Domains: Physics
Sub-Domain: Quantum Foundations
Research Topics: Bell theorem, Bell-CHSH, Tsirelson bound
Abstract
Bell’s theorem constrains hidden-variable models under the joint assumptions of locality, deterministic response functions, and measurement independence. The Bell–CHSH inequality therefore restricts correlations only when these structural conditions are simultaneously satisfied. We construct a hierarchy of geometric hidden-variable models defined on the unit circle, modifying one assumption at a time. A minimal local model with setting-independent preparation obeys the classical bound $|S|\le 2$. Introducing a setting-dependent update produces Bell violation accompanied by operational signalling. As a structural reference, direct imposition of the cosine correlation reproduces Tsirelson’s bound while preserving no-signalling marginals. We then show that Tsirelson-level correlations also arise from a purely geometric, context-dependent sampling rule without inserting the cosine function at the level of the preparation law. In this construction, local deterministic response functions and operational no-signalling are preserved, while measurement independence is relaxed. These models are presented as structural illustrations rather than physical proposals. They provide a transparent geometric framework in which the dependence of Bell correlations on underlying assumptions can be examined explicitly.
Available Document
DOI: 10.5281/zenodo.20258246
Citation:
Petersen, B. L. (2026). Geometric Substrate Models and Bell–CHSH Correlations: A Structural Analysis of Assumption Relaxation. Zenodo. https://doi.org/10.5281/zenodo.20258246
Source Code and Supporting Materials
None.
Summary and Notes
Document role:
This paper presents a structural analysis of Bell–CHSH correlations using a hierarchy of geometric hidden-variable models defined on the unit circle. The purpose is not to propose a complete physical hidden-variable theory, but to make the assumption-dependence of Bell-type correlations explicit in a transparent geometric setting.
The paper examines how Bell–CHSH behavior changes when standard assumptions are modified one at a time, including locality, deterministic response functions, and measurement independence.
Core result:
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 reproduces 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 this construction, local deterministic response functions and operational no-signalling are preserved, while measurement independence is relaxed.
Scope:
The models are presented as structural illustrations rather than physical proposals. Their purpose is to clarify how Bell-type correlations depend on underlying assumptions and to provide a geometric framework in which those assumptions can be inspected explicitly.
Framework context:
This paper belongs to the Quantum Foundations area of the Coherence Geometry research corpus. It is related to later and companion CG work on deterministic geometric preparation, Born-type interference, phase projection, and coherence-based interpretations of quantum structure. The paper is largely self-contained and does not require the full Coherence Geometry framework in order to follow the Bell–CHSH assumption analysis.
Related Work
Petersen, B. L. (2026). Coherence Geometry Foundations, Part I: Orientation,
Closure, and Algebraic Foundations (Version 0.1). Zenodo.
https://doi.org/10.5281/zenodo.20156532
Petersen, B. L. (2026). Coherence Geometry Foundations, Part II: Physical
Projections (Version 0.1). Zenodo.
https://doi.org/10.5281/zenodo.20156997