{"id":2920,"date":"2026-05-11T13:52:09","date_gmt":"2026-05-11T13:52:09","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=2920"},"modified":"2026-05-12T08:52:31","modified_gmt":"2026-05-12T08:52:31","slug":"the-unified-coherence-functional-a-closed-generative-basis-for-mathematics","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/11\/the-unified-coherence-functional-a-closed-generative-basis-for-mathematics\/","title":{"rendered":"The Unified Coherence Functional: A Closed Generative Basis for Mathematics"},"content":{"rendered":"\n

Internal ID:<\/strong> CGI-RSR-000012
Document Type:<\/strong> Research Paper
Status:<\/strong> Public
Domains:<\/strong> Mathematics, Foundations
Research Topics:<\/strong> Functional, UCF<\/p>\n\n\n\n

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


We introduce the Unified Coherence Functional <\/em>(UCF), a closed variational framework in which broad classes of mathematical structures arise as stationary projections<\/em>. A coherence field \\(X=(A,\\Theta)\\) over the multi\u2013phase algebra $\\mathbb{M}$ is evaluated by a scalar functional
$$
\\mathcal{U}[X] \\;=\\; \\int_{\\mathcal{D}} \\mathcal{C}\\!\\big(A,\\nabla A,\\{\\nabla\\theta_k\\},\\kappa(\\Theta)\\big)\\,d\\mu,
$$
where \\(\\mathcal{C}\\) balances alignment and curvature across phase channels and \\(\\kappa(\\Theta)\\) denotes intrinsic phase curvature. Axioms (U1\u2013U8) encode closure, invariance, and \\emph{projection consistency}. A Projection\u2013Reconstruction Bridge<\/em> establishes a commutation principle (\u201cproject\u2013then\u2013vary\u201d versus \u201cvary\u2013then\u2013project\u201d) under admissibility\/regularity hypotheses, and the associated coherence gradient flow <\/em>yields existence and stability of equilibria. We develop domain projections\u2014algebraic, geometric, analytic, topological, probabilistic, and logical\/computational\u2014and derive representative induced functionals and Euler\u2013Lagrange systems in standard silo variables. Highlights include: (i) an analytic rigidity principle at critical scaling, expressed as a strict bilinear gap<\/em> \\(\\|\\mathcal{B}(f,g)\\|_{\\mathrm{crit}}\\le \\alpha\\|f\\|_{\\mathrm{crit}}\\|g\\|_{\\mathrm{crit}}\\) with \\(\\alpha<1\\) under angular deficit (data) or null\u2013form (symbol) structure; (ii) a geometric variation producing Einstein-type balance from metric variation; (iii) a topological sector where homotopy classes are stationary and selected indices are quantized; (iv) a probabilistic sector in which entropy and Fisher information arise as induced coherence terms and the amplitude map realizes Fisher\u2013Rao geometry; and (v) a discrete logical sector where fixed\u2013point consistency and semi\u2013decidability correspond to discrete Euler\u2013Lagrange conditions and nonattainment phenomena. Thus, classical structures appear as constrained stationary points of a single closed functional, providing a coherence-based foundation that organizes cross-domain invariants and Noether-type correspondences within one generative variational language.<\/p>\n\n\n\n

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

DOI:<\/strong> 10.5281\/zenodo.20120296<\/code><\/p>\n\n\n\n

Citation:<\/strong>
Petersen, B. L. (2026). The Unified Coherence Functional: A Closed Generative Basis for Mathematics. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.20120296<\/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

This paper is listed in the Foundation Papers section because it serves as a source document for CDR-02.<\/p>\n\n\n\n

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

CGI-RSR-000012 : Unified Coherence Geometry: A Common Action for Physical Fields<\/p>\n\n\n\n","protected":false},"excerpt":{"rendered":"

CGI-RSR-000012 | We introduce the Unified Coherence Functional (UCF), a closed variational framework in which broad classes of mathematical structures arise as stationary projections.<\/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":[30,58,56],"tags":[],"class_list":["post-2920","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-foundation-papers","category-research-papers"],"_links":{"self":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2920","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=2920"}],"version-history":[{"count":7,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2920\/revisions"}],"predecessor-version":[{"id":2939,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2920\/revisions\/2939"}],"wp:attachment":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/media?parent=2920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/categories?post=2920"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/tags?post=2920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}