{"id":3313,"date":"2026-05-17T08:52:59","date_gmt":"2026-05-17T15:52:59","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=3313"},"modified":"2026-05-17T09:06:00","modified_gmt":"2026-05-17T16:06:00","slug":"deterministic-geometric-preparation-and-the-emergence-of-the-born-rule-in-two-path-interference","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/17\/deterministic-geometric-preparation-and-the-emergence-of-the-born-rule-in-two-path-interference\/","title":{"rendered":"Deterministic Geometric Preparation and the Emergence of the Born Rule in Two-Path Interference"},"content":{"rendered":"\n

Internal ID:<\/strong> CGI-RSR-000022
Author(s):<\/strong> Barry L. Petersen
Document Type:<\/strong> Research Paper
Publication Date:<\/strong> May 2026
Original Creation Date:<\/strong> March 3, 2026
Revised Document Date:<\/strong> May 17, 2025
Status:<\/strong> Public
Domains:<\/strong> Physics
Sub-Domain:<\/strong> Quantum Foundations
Research Topics:<\/strong> Planck’s Constant, Coherence Quantization, Phase Geometery<\/p>\n\n\n\n

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


The two-path interference law
<\/p>\n\n\n\n

P<\/mi>(<\/mo>\u0394<\/mi><\/mrow>)<\/mo>=<\/mo>cos<\/mi>2<\/mn><\/msup>\u2061<\/mo><\/mspace><\/mspace>(<\/mo>\u0394<\/mi><\/mrow>2<\/mn><\/mfrac>)<\/mo><\/mrow><\/mrow>P(\\Delta) = \\cos^2\\!\\left(\\frac{\\Delta}{2}\\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
<\/div>\n\n\n\n

is conventionally derived from complex amplitude superposition together with the Born rule. We present an alternative structural derivation based on a deterministic geometric preparation model defined on the unit circle. A hidden directional variable is uniformly distributed prior to preparation and subsequently biased in proportion to geometric alignment with a preparation axis. Measurement is modeled deterministically by the sign of a directional projection onto a measurement axis. We show analytically that this preparation-weighted geometric ensemble yields the correlation<\/p>\n\n\n\n

E<\/mi>(<\/mo>\u0394<\/mi><\/mrow>)<\/mo>=<\/mo>cos<\/mi>\u2061<\/mo><\/mspace><\/mrow>\u0394<\/mi><\/mrow>,<\/mo><\/mrow>E(\\Delta) = \\cos \\Delta,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
<\/div>\n\n\n\n

and therefore reproduces the standard two-path interference probability exactly. The harmonic phase dependence arises as an overlap identity on \\(S^1\\) rather than from probabilistic collapse or intrinsic complex amplitude addition. The construction is minimal and does not attempt to reproduce the full Hilbert space structure of quantum theory. It demonstrates only that the Born interference law for two coherent paths is consistent with deterministic geometric preparation and projection on a continuous hidden-direction space.<\/p>\n\n\n\n

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

DOI:<\/strong> 10.5281\/zenodo.20259054<\/code><\/p>\n\n\n\n

Citation:<\/strong>
Petersen, B. L. (2026). Deterministic Geometric Preparation and the Emergence of the Born Rule in Two-Path Interference. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.20259054<\/a><\/p>\n\n\n\n

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

Code availability:
A Jupyter notebook, Born_From_Geometric_Preparation.ipynb<\/em>, is included to reproduce the analytic\/Monte Carlo comparison figure in the paper. The notebook is provided as a lightweight research artifact for inspection and figure generation, not as maintained software. No technical support is implied.<\/p>\n\n\n\n

The Jupyter notebook generates the figure comparing the analytic law P(+ | Delta) = cos^2(Delta\/2) with Monte Carlo estimates obtained by sampling the geometric preparation and deterministic measurement rules. Agreement is within sampling uncertainty.<\/p>\n\n\n\n

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

Scope:<\/em>
The construction is intentionally minimal. It does not attempt to reconstruct the full Hilbert space structure of quantum theory or provide a complete physical hidden-variable theory. It shows only that the Born interference law for two coherent paths is consistent with deterministic geometric preparation and projection on a continuous hidden-direction space.<\/p>\n\n\n\n

Framework context:<\/em>
This paper belongs to the Quantum Foundations area of the Coherence Geometry research corpus. It is structurally paired with CGI-RSR-000021, “Geometric Substrate Models and Bell\u2013CHSH Correlations: A Structural Analysis of Assumption Relaxation,” which studies related geometric preparation and projection mechanisms in Bell-type correlation settings.<\/p>\n\n\n\n

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

Petersen, B. L. (2026). Geometric Substrate Models and Bell\u2013CHSH Correlations:
A Structural Analysis of Assumption Relaxation (Version 1.0). Zenodo.
https:\/\/doi.org\/10.5281\/zenodo.20258246<\/p>\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<\/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<\/p>\n\n\n\n","protected":false},"excerpt":{"rendered":"

CGI-RSR-000022 | The two-path interference law is conventionally derived from complex amplitude superposition together with the Born rule. We present an alternative structural derivation based on a deterministic geometric preparation model defined on the unit circle.<\/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-3313","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\/3313","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=3313"}],"version-history":[{"count":12,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/3313\/revisions"}],"predecessor-version":[{"id":3329,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/3313\/revisions\/3329"}],"wp:attachment":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/media?parent=3313"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/categories?post=3313"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/tags?post=3313"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}