{"id":2887,"date":"2026-05-11T12:53:09","date_gmt":"2026-05-11T12:53:09","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=2887"},"modified":"2026-05-11T13:17:15","modified_gmt":"2026-05-11T13:17:15","slug":"a-multi-phase-extension-of-complex-numbers-and-the-global-coherence-theorem","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/11\/a-multi-phase-extension-of-complex-numbers-and-the-global-coherence-theorem\/","title":{"rendered":"A Multi-Phase Extension of Complex Numbers and the Global Coherence Theorem"},"content":{"rendered":"\n


Internal ID:<\/strong> CGI-RSR-000009
Document Type:<\/strong> Research Paper
Publication Date:<\/strong> May 2026
Status:<\/strong> Public
Domains:<\/strong> Foundations, Mathematics
Research Topics:<\/strong> Coherence Geometry Canon<\/p>\n\n\n\n

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


We introduce \u00b5-numbers<\/em> (short for Multi-Phase Numbers<\/em>), a generalization of complex numbers that extends traditional amplitude-phase representations to multi-phase systems under a single amplitude constraint. This framework unifies real, complex, and multi-phase constructs while preserving local coherence without requiring global orthonormality constraints. A central result of this work is the Global Coherence Theorem (GCT)<\/em>, which guarantees that systems of \u00b5<\/em>-numbers remain structurally stable and interference-free under local bounded interactions, even as they grow arbitrarily large. We rigorously prove that multi-phase coherence remains intact while ensuring bounded energy evolution, making \u00b5<\/em>-numbers naturally suited for infinite-dimensional settings. Key contributions include the formal definition of \u00b5<\/em>-numbers, the introduction of a local synergy function governing amplitude-phase interactions, rigorous bounds on amplitude evolution, an illustrative example of amplitude stabilization, and a proof of coherence for infinite sets. This work establishes a mathematical foundation for structured stability in high-dimensional multi-phase systems.<\/p>\n\n\n\n

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

DOI:<\/strong> 10.5281\/zenodo.<\/a>20116654<\/a><\/code><\/p>\n\n\n\n

Citation:<\/strong>
Petersen, B. L. (2026). A Multi-Phase Extension of Complex Numbers and the Global Coherence Theorem. Zenodo.\u00a0https:\/\/doi.org\/10.5281\/zenodo.20116654<\/a><\/p>\n\n\n\n

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

N\/A<\/p>\n\n\n\n

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

This paper is listed in the Foundation Papers section because it serves as a source document for CDR-00.<\/p>\n\n\n\n

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

N\/A

<\/p>\n\n\n\n","protected":false},"excerpt":{"rendered":"

CGI-RSR-000009 | This is the foundational document that introduces multi-phase numbers and the Global Coherence Theorem (GCT).<\/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":[56,58,30],"tags":[],"class_list":["post-2887","post","type-post","status-publish","format-standard","hentry","category-research-papers","category-foundation-papers","category-mathematics"],"_links":{"self":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2887","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=2887"}],"version-history":[{"count":5,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2887\/revisions"}],"predecessor-version":[{"id":2894,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2887\/revisions\/2894"}],"wp:attachment":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/media?parent=2887"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/categories?post=2887"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/tags?post=2887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}