The Unified Coherence Functional: A Closed Generative Basis for Mathematics
Internal ID: CGI-RSR-000012
Document Type: Research Paper
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
$$
\mathcal{U}[X] \;=\; \int_{\mathcal{D}} \mathcal{C}\!\big(A,\nabla A,\{\nabla\theta_k\},\kappa(\Theta)\big)\,d\mu,
$$
where \(\mathcal{C}\) balances alignment and curvature across phase channels and \(\kappa(\Theta)\) denotes intrinsic phase curvature. Axioms (U1–U8) encode closure, invariance, and \emph{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 \(\|\mathcal{B}(f,g)\|_{\mathrm{crit}}\le \alpha\|f\|_{\mathrm{crit}}\|g\|_{\mathrm{crit}}\) 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.
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