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The Unified Coherence Functional: A Closed Generative Basis for Mathematics

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Internal ID: CGI-RSR-000012
Author(s): Barry L. Petersen
Document Type: Research Paper
Publication Date: May 2026
Creation Date: December 19, 2025
Status: Public
Domains: Mathematics, Foundations
Research Topics: Functional, UCF

Abstract


We introduce the Unified Coherence Functional (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–phase algebra \(\mathbb{M}\) is evaluated by a scalar functional

𝒰[X]=∫𝒟𝒞(A,∇A,{∇θk},κ(Θ))dμ,\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–U8) encode closure, invariance, and projection consistency. A Projection–Reconstruction Bridge establishes a commutation principle (“project–then–vary” versus “vary–then–project”) under admissibility/regularity hypotheses, and the associated coherence gradient flow yields existence and stability of equilibria. We develop domain projections—algebraic, geometric, analytic, topological, probabilistic, and logical/computational—and derive representative induced functionals and Euler–Lagrange systems in standard silo variables. Highlights include: (i) an analytic rigidity principle at critical scaling, expressed as a strict bilinear gap

‖ℬ(f,g)‖crit≤α‖f‖crit‖g‖crit,\|\mathcal{B}(f,g)\|_{\mathrm{crit}}\le \alpha\|f\|_{\mathrm{crit}}\|g\|_{\mathrm{crit}},

with \(\alpha<1\) under angular deficit (data) or null–form (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–Rao geometry; and (v) a discrete logical sector where fixed–point consistency and semi–decidability correspond to discrete Euler–Lagrange 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.

Unified Coherence Functional Equation

𝒰[X]=∫𝒟𝒞(A,∇A,{∇θk},κ(Θ))dμ,\mathcal{U}[X] \;=\; \int_{\mathcal{D}} \mathcal{C}\,\!\left(A,\nabla A,\{\nabla\theta_k\},\kappa(\Theta)\right) \,d\mu,

Duality with the Coherence Action (embedding correspondence)

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

𝒮=∫(p⋅q˙−𝒰)dt,\mathcal{S} = \int (p\cdot \dot{q} – \mathcal{U})\,dt,

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
“functional” (mathematics) and “action” (physics) form when the correspondence is available.

Available Document

DOI: 10.5281/zenodo.20120296

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

Source Code and Supporting Materials

None.

Summary and Notes

This paper is listed in the Foundation Papers section because it serves as a source document for CDR-02.

Related Work

CGI-RSR-000012 : Unified Coherence Geometry: A Common Action for Physical Fields

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