{"id":3301,"date":"2026-05-17T07:19:35","date_gmt":"2026-05-17T14:19:35","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=3301"},"modified":"2026-05-17T09:06:30","modified_gmt":"2026-05-17T16:06:30","slug":"geometric-substrate-models-and-bell-chsh-correlations-a-structural-analysis-of-assumption-relaxation","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/17\/geometric-substrate-models-and-bell-chsh-correlations-a-structural-analysis-of-assumption-relaxation\/","title":{"rendered":"Geometric Substrate Models and Bell\u2013CHSH Correlations: A Structural Analysis of Assumption Relaxation"},"content":{"rendered":"\n

Internal ID:<\/strong> CGI-RSR-000021
Author(s):<\/strong> Barry L. Petersen
Document Type:<\/strong> Research Paper
Publication Date:<\/strong> May 2026
Original Creation Date:<\/strong> Feb 28, 2025
Revised Document Date:<\/strong> N\/A
Status:<\/strong> Public
Domains:<\/strong> Physics
Sub-Domain:<\/strong> Quantum Foundations
Research Topics:<\/strong> Bell theorem, Bell-CHSH, Tsirelson bound<\/p>\n\n\n\n

Abstract<\/h3>\n\n\n\n


Bell\u2019s 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\u2019s 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.<\/p>\n\n\n\n

Available Document<\/h3>\n\n\n\n

DOI:<\/strong> 10.5281\/zenodo.20258246<\/code><\/p>\n\n\n\n

Citation:<\/strong>
Petersen, B. L. (2026). Geometric Substrate Models and Bell\u2013CHSH Correlations: A Structural Analysis of Assumption Relaxation. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.20258246<\/a><\/p>\n\n\n\n

Source Code and Supporting Materials<\/h3>\n\n\n\n

None.<\/p>\n\n\n\n

Summary and Notes<\/h3>\n\n\n\n

Document role:<\/em>
This paper presents a structural analysis of Bell\u2013CHSH 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.<\/p>\n\n\n\n

The paper examines how Bell\u2013CHSH behavior changes when standard assumptions are modified one at a time, including locality, deterministic response functions, and measurement independence.<\/p>\n\n\n\n

Core result:<\/em>
A minimal local model with setting-independent preparation obeys the classical Bell\u2013CHSH bound |S| <= 2. A setting-dependent update can produce Bell violation but introduces operational signalling. Direct imposition of the cosine correlation reproduces Tsirelson\u2019s bound while preserving no-signalling marginals.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

Scope:<\/em>
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.<\/p>\n\n\n\n

Framework context:<\/em>
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\u2013CHSH assumption analysis.<\/p>\n\n\n\n

Related Work<\/h3>\n\n\n\n

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<\/a><\/p>\n\n\n\n

Petersen, B. L. (2026). Coherence Geometry Foundations, Part II: Physical
Projections (Version 0.1). Zenodo.
https:\/\/doi.org\/10.5281\/zenodo.20156997<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

CGI-RSR-000021 | This paper presents a structural analysis of Bell\u2013CHSH 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\u2013CHSH behavior changes when standard assumptions are modified one at a time, including locality, deterministic response functions, and measurement independence.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","footnotes":""},"categories":[31,68,56],"tags":[],"class_list":["post-3301","post","type-post","status-publish","format-standard","hentry","category-physics","category-quantum-foundations","category-research-papers"],"_links":{"self":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/3301","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/comments?post=3301"}],"version-history":[{"count":7,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/3301\/revisions"}],"predecessor-version":[{"id":3331,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/3301\/revisions\/3331"}],"wp:attachment":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/media?parent=3301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/categories?post=3301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/tags?post=3301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}