{"id":2945,"date":"2026-05-11T14:30:18","date_gmt":"2026-05-11T14:30:18","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=2945"},"modified":"2026-05-18T07:19:08","modified_gmt":"2026-05-18T14:19:08","slug":"pattern-formation-via-curvature-driven-amplitude-relaxation-in-coherence-geometry","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/11\/pattern-formation-via-curvature-driven-amplitude-relaxation-in-coherence-geometry\/","title":{"rendered":"Pattern Formation via Curvature-Driven Amplitude Relaxation in Coherence Geometry"},"content":{"rendered":"\n

Internal ID:<\/strong> CGI-RSR-000015
Author(s):<\/strong> Barry L. Petersen
Document Type:<\/strong> Research Paper
Status:<\/strong> Public
Domains:<\/strong> Pattern Formation, Chemistry, Physics
Research Topics:<\/strong> Diffusion-Reaction, coherence-driven amplitude relaxation (CDAR)<\/p>\n\n\n\n

Abstract<\/h3>\n\n\n\n


We present a coherence\u2013geometric formulation of diffusion\u2013reaction\u2013like pattern formation based on curvature-driven amplitude relaxation<\/em> (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\u2013phase field governed by geometric coherence couplings. In its minimal single-phase instantiation (N<\/em> = 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\u2013kill 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 \u2265<\/em> 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\u2013reaction behavior can be understood as a projected or limiting description of a more structured coherence\u2013geometric substrate, providing a compact and extensible foundation for pattern formation across physical, biological, and computational contexts.<\/p>\n\n\n\n

Available Document<\/h3>\n\n\n\n

DOI:<\/strong> 10.5281\/zenodo.20121185<\/code><\/p>\n\n\n\n

Citation:<\/strong>
Petersen, B. L. (2026). Pattern Formation via Curvature-Driven Amplitude Relaxation in Coherence Geometry. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.20121185<\/p>\n\n\n\n

Source Code and Supporting Materials<\/h3>\n\n\n\n

Not yet included.<\/p>\n\n\n\n

Summary and Notes<\/h3>\n\n\n\n

This paper is listed in the Foundation Papers section because it serves as a source document for CDR-05.<\/p>\n\n\n\n

Related Work<\/h3>\n\n\n\n

N\/A<\/p>\n","protected":false},"excerpt":{"rendered":"

CGI-RSR-000015 | We present a coherence\u2013geometric formulation of diffusion\u2013reaction\u2013like 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\u2013phase field governed by geometric coherence couplings.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","footnotes":""},"categories":[72,32,59,73,31,56],"tags":[],"class_list":["post-2945","post","type-post","status-publish","format-standard","hentry","category-diffusion-reaction-pattern-formation","category-chemistry","category-pattern-formation","category-pattern-formation-physics","category-physics","category-research-papers"],"_links":{"self":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2945","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/comments?post=2945"}],"version-history":[{"count":5,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2945\/revisions"}],"predecessor-version":[{"id":3339,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2945\/revisions\/3339"}],"wp:attachment":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/media?parent=2945"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/categories?post=2945"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/tags?post=2945"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}