A Variational–Relaxation Framework for Coherence-Driven Structure Formation

This paper develops a variational–relaxation framework for coherence-driven structure formation. It places phase-coherent self-organization within a geometric and variational setting compatible with broader physical and mathematical formulations of constrained dynamics. Building on the Global Coherence Theorem and the Structured Coherence Refinement Corollary, the manuscript introduces an intrinsic structure functional whose extrema define coherent configurations. Coherence formation is treated not as a conservative Hamiltonian process, but as dissipative relaxation toward minima of this functional. In this formulation, coherence refinement is understood as constraint satisfaction and landscape extremization rather than conservative energy preservation. The paper describes phase evolution through gradient-like flow on a coherence potential induced by shared amplitude constraints, with optional inertial extensions for transient dynamics. It provides a foundation for later Coherence Geometry work on emergence, basin structure, relaxation dynamics, self-organization, wave systems, physical processes, and coherence-based computational models. This record contains the original hash-committed foundation paper associated with Coherence Geometry Canon CDR-03. The PDF is released in the same form referenced by the CDR-03 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.