The Unified Coherence Functional: A Closed Generative Basis for Mathematics

This paper introduces the Unified Coherence Functional (UCF), a closed variational framework in which broad classes of mathematical structures arise as stationary projections of a common coherence-geometric functional. The manuscript defines a coherence field over the multi-phase algebra and evaluates it through a scalar functional balancing amplitude, alignment, curvature, and phase-channel structure. It introduces axioms for closure, invariance, and projection consistency, together with a Projection–Reconstruction Bridge relating “project–then–vary” and “vary–then–project” under admissibility and regularity hypotheses. The associated coherence gradient flow provides a framework for equilibrium, existence, and stability. The paper develops representative projections into algebraic, geometric, analytic, topological, probabilistic, and logical/computational settings. These include induced Euler–Lagrange systems, analytic rigidity through bilinear gap structure, metric variation leading to Einstein-type balance, stationary homotopy sectors and quantized indices, entropy and Fisher information as induced coherence terms, and discrete fixed-point or semi-decidability phenomena as variational consistency structures. This record contains the original hash-committed foundation paper associated with Coherence Geometry Canon CDR-02. The PDF is released in the same form referenced by the CDR-02 provenance record so that the public file remains consistent with the recorded SHA-256 hash. Later documents may expand individual derivations, equivalent formulations, or domain-specific consequences, while this record preserves the original foundation statement.