{"id":2920,"date":"2026-05-11T13:52:09","date_gmt":"2026-05-11T13:52:09","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=2920"},"modified":"2026-05-17T09:12:16","modified_gmt":"2026-05-17T16:12:16","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
Author(s):<\/strong> Barry L. Petersen
Document Type:<\/strong> Research Paper
Publication Date:<\/strong> May 2026
Creation Date:<\/strong> December 19, 2025
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. A coherence field \\(X=(A,\\Theta)\\) over the multi\u2013phase algebra \\(\\mathbb{M}\\) is evaluated by a scalar functional<\/p>\n\n\n\n

<\/div>\n\n\n\n
\ud835\udcb0<\/mi>[<\/mo>X<\/mi>]<\/mo><\/mspace>=<\/mo><\/mspace>\u222b<\/mo>\ud835\udc9f<\/mi><\/msub>\ud835\udc9e<\/mi><\/mspace><\/mspace>(<\/mo>A<\/mi>,<\/mo>\u2207<\/mo>A<\/mi>,<\/mo>{<\/mo>\u2207<\/mo>\u03b8<\/mi>k<\/mi><\/msub>}<\/mo>,<\/mo>\u03ba<\/mi>(<\/mo>\u0398<\/mi><\/mrow>)<\/mo>)<\/mo><\/mspace>d<\/mi>\u03bc<\/mi>,<\/mo><\/mrow>\\mathcal{U}[X] \\;=\\; \\int_{\\mathcal{D}} \\mathcal{C}\\,\\!\\big(A,\\nabla A,\\{\\nabla\\theta_k\\},\\kappa(\\Theta)\\big)\\,d\\mu,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
<\/div>\n\n\n\n

where \\(\\mathcal{C}\\) balances alignment and curvature across phase channels and \\(\\kappa(\\Theta)\\) denotes intrinsic phase curvature. Axioms (U1\u2013U8) encode closure, invariance, and projection consistency. A Projection\u2013Reconstruction Bridge 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<\/p>\n\n\n\n

\u2016<\/mi>\u212c<\/mi>(<\/mo>f<\/mi>,<\/mo>g<\/mi>)<\/mo>\u2016<\/mi><\/mtext>crit<\/mi><\/mrow><\/msub>\u2264<\/mo>\u03b1<\/mi>\u2016<\/mi>f<\/mi>\u2016<\/mi><\/mtext>crit<\/mi><\/mrow><\/msub>\u2016<\/mi>g<\/mi>\u2016<\/mi><\/mtext>crit<\/mi><\/mrow><\/msub>,<\/mo><\/mrow>\\|\\mathcal{B}(f,g)\\|_{\\mathrm{crit}}\\le \\alpha\\|f\\|_{\\mathrm{crit}}\\|g\\|_{\\mathrm{crit}},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
<\/div>\n\n\n\n

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

Unified Coherence Functional Equation<\/h3>\n\n\n\n
<\/div>\n\n\n\n
\ud835\udcb0<\/mi>[<\/mo>X<\/mi>]<\/mo><\/mspace>=<\/mo><\/mspace>\u222b<\/mo>\ud835\udc9f<\/mi><\/msub>\ud835\udc9e<\/mi><\/mspace><\/mspace>(<\/mo>A<\/mi>,<\/mo>\u2207<\/mo>A<\/mi>,<\/mo>{<\/mo>\u2207<\/mo>\u03b8<\/mi>k<\/mi><\/msub>}<\/mo>,<\/mo>\u03ba<\/mi>(<\/mo>\u0398<\/mi><\/mrow>)<\/mo>)<\/mo><\/mrow><\/mspace>d<\/mi>\u03bc<\/mi>,<\/mo><\/mrow>\\mathcal{U}[X] \n\\;=\\; \n\\int_{\\mathcal{D}} \n\\mathcal{C}\\,\\!\\left(A,\\nabla A,\\{\\nabla\\theta_k\\},\\kappa(\\Theta)\\right)\n\\,d\\mu,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
<\/div>\n\n\n\n

Duality with the Coherence Action<\/strong> (embedding correspondence)<\/p>\n\n\n\n

In embeddings where a time parameter and a compatible action formulation are defined,
the physical Coherence Action<\/a> \\(\\mathcal{S}\\) and the mathematical Coherence Functional \\(\\mathcal{U}\\)
are related by a Legendre-type correspondence:<\/p>\n\n\n\n

\ud835\udcae<\/mi>=<\/mo>\u222b<\/mo>(<\/mo>p<\/mi>\u22c5<\/mo>q<\/mi>\u02d9<\/mo><\/mover>\u2212<\/mo>\ud835\udcb0<\/mi>)<\/mo><\/mspace>d<\/mi>t<\/mi>,<\/mo><\/mrow>\\mathcal{S} = \\int (p\\cdot \\dot{q} – \\mathcal{U})\\,dt,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
<\/div>\n\n\n\n

with \\((q,p)\\) and \\(t\\) specified by the chosen physics embedding\/projection.

This expresses that the same generative coherence principle can be realized in both
\u201cfunctional\u201d (mathematics) and \u201caction\u201d (physics) form when the correspondence is available.<\/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":"

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