Pattern Formation via Curvature-Driven Amplitude Relaxation in Coherence Geometry
Internal ID: CGI-RSR-000015
Document Type: Research Paper
Status: Public
Domains: Pattern Formation, Chemistry, Physics
Research Topics: Diffusion-Reaction, coherence-driven amplitude relaxation (CDAR)
Abstract
We present a coherence–geometric formulation of diffusion–reaction–like pattern formation based on curvature-driven amplitude relaxation (CDAR). Rather than modeling diffusion and reaction as distinct processes acting on multiple scalar fields, the proposed approach represents spatial structure as the evolution of a constrained amplitude–phase field governed by geometric coherence couplings. In its minimal single-phase instantiation (N = 1), the system relaxes from noise into stable spatial patterns including isotropic nodal aggregates, labyrinthine networks, and multi-scale textured morphologies, without invoking multi-species chemistry, feed–kill kinetics, or externally prescribed reaction terms. Depending on operating regime and projection, the same dynamics admit both sharply segmented, Dalmatian-like textures and richer mosaic or tapestry-like structures arising from continued coherence-governed relaxation. Multi-phase extensions (N ≥ 2) enrich the accessible pattern space by introducing competing internal coherence fluxes, yielding aligned ridge- and dune-like morphologies while preserving the same underlying relaxation principle. Numerical experiments demonstrate robust convergence from random initial conditions and stable morphology across wide parameter ranges. These results suggest that classical diffusion–reaction behavior can be understood as a projected or limiting description of a more structured coherence–geometric substrate, providing a compact and extensible foundation for pattern formation across physical, biological, and computational contexts.
Available Document
DOI: 10.5281/zenodo.20121185
Citation:
Petersen, B. L. (2026). Pattern Formation via Curvature-Driven Amplitude Relaxation in Coherence Geometry. Zenodo. https://doi.org/10.5281/zenodo.20121185
Source Code and Supporting Materials
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Summary and Notes
This paper is listed in the Foundation Papers section because it serves as a source document for CDR-05.
Related Work
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