Pattern Formation via Curvature-Driven Amplitude Relaxation in Coherence Geometry

This paper develops a coherence-geometric formulation of pattern formation based on curvature-driven amplitude relaxation. Instead of treating diffusion and reaction as separate processes acting on multiple scalar fields, the framework represents spatial organization through the evolution of a constrained amplitude-phase field governed by geometric coherence couplings. In the minimal single-phase case, the system relaxes from noise into stable spatial patterns including isotropic nodal aggregates, labyrinthine networks, segmented textures, mosaic structures, and multi-scale morphologies, without imposing multi-species chemistry, feed-kill kinetics, or externally prescribed reaction terms. Multi-phase extensions enrich the pattern space through competing internal coherence fluxes, producing aligned ridge-like, dune-like, and textured morphologies while preserving the same underlying relaxation principle. The paper provides numerical evidence of robust convergence from random initial conditions and stable morphology across parameter ranges. It positions classical diffusion-reaction behavior as a projected or limiting description of a more structured coherence-geometric substrate, with relevance for pattern formation in physical, biological, and computational contexts. This record contains the original hash-committed foundation paper associated with Coherence Geometry Canon CDR-05. The PDF is released in the same form referenced by the CDR-05 provenance record so that the public file remains consistent with the recorded SHA-256 hash. Supplementary notebooks, code, or numerical materials are not included in this initial release and may be added later as a separate versioned record. Later documents may also expand individual derivations, equivalent formulations, or domain-specific consequences, while this record preserves the original foundation statement.