{"id":2588,"date":"2026-05-03T14:18:10","date_gmt":"2026-05-03T14:18:10","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=2588"},"modified":"2026-06-15T04:47:50","modified_gmt":"2026-06-15T11:47:50","slug":"local-coherence-hessians-a-structural-classification-of-spectral-gaps","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/03\/local-coherence-hessians-a-structural-classification-of-spectral-gaps\/","title":{"rendered":"Local Coherence Hessians: A Structural Classification of Spectral Gaps"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong>Internal ID:<\/strong> CGI-RSR-000003<br><strong>Author(s):<\/strong> Barry L. Petersen<br><strong>Document Type:<\/strong> Research Paper<br><strong>Publication Date: <\/strong>May 2026<br><strong>Original Creation Date: <\/strong>February 17, 2026<br><strong>Status:<\/strong> Public<br><strong>Domains:<\/strong> Mathematics, Physics<br><strong>Research Topics:<\/strong> Mathematical physics, Clay Millennium Problems, Yang-Mills<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Abstract<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><br><em>We provide a structural classification of spectral gaps for local coherence functionals on periodic lattices. For nearest-neighbor phase interactions, the quadratic Hessian at aligned configurations is a weighted graph Laplacian. In infinite volume this operator is gapless, with the lowest nonzero eigenvalue scaling like L^\u22122. A uniform spectral gap requires strictly positive onsite quadratic curvature. We show that multi-channel phase locking provides a canonical intrinsic source of such curvature on relative channel modes: when the channel-coupling graph is connected, all relative-channel fluctuations are uniformly gapped, independent of lattice size. After locking, the remaining degree of freedom is the synchronized phase. We prove a sharp dichotomy: if the restricted functional on the synchronized manifold retains global shift symmetry, the model is structurally gapless; if it exhibits positive onsite curvature, the full multi-channel Hessian admits a uniform spectral gap. This isolates the precise structural origin of mass scales in local coherence models: locality alone is insufficient; quadratic curvature in the final soft sector is necessary and sufficient.<\/em><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Available Document<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>DOI:<\/strong> <code><a href=\"https:\/\/doi.org\/10.5281\/zenodo.19969141\" rel=\"nofollow noopener\" target=\"_blank\">10.5281\/zenodo.19969141<\/a><\/code><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Citation:<\/strong><br>Petersen, B. L. (2026). Local Coherence Hessians: A Structural Classification of Spectral Gaps. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.19969141<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Source Code and Supporting Materials<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">None.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary and Notes<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This document is written in standard mathematical-physics language. It studies local coherence functionals, Hessians, graph Laplacians, spectral gaps, periodic lattices, and multi-channel phase locking without requiring the reader to adopt the full Coherence Geometry framework.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The CG role is organizational rather than linguistic: it motivates the focus on coherence channels, phase locking, compatibility constraints, and curvature as structural sources of effective excitation scales.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rather than beginning with fields as primitive objects, the approach studies how collective phase-locking, local coupling, and onsite curvature generate or fail to generate uniform spectral gaps.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From this viewpoint, a mass gap is interpreted structurally as robust quadratic curvature that remains after internal modes organize into stable locked sectors.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Scope Notes<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This document is written in standard mathematical-physics language. It studies local coherence functionals, Hessians, graph Laplacians, spectral gaps, periodic lattices, and multi-channel phase locking without requiring the reader to adopt the full Coherence Geometry framework.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The CG role is organizational rather than linguistic: it motivates the focus on coherence channels, phase locking, compatibility constraints, and curvature as structural sources of effective excitation scales.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rather than beginning with fields as primitive objects, the approach studies how collective phase-locking, local coupling, and onsite curvature generate or fail to generate uniform spectral gaps.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From this viewpoint, a mass gap is interpreted structurally as robust quadratic curvature that remains after internal modes organize into stable locked sectors.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Related Work<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This paper is part of the Coherence Geometry research sequence on spectral gaps, local stability, and projection of CG mechanisms into mathematical physics. Additional Yang\u2013Mills-specific translation work has not yet been released.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Internal ID: CGI-RSR-000003 | We provide a structural classification of spectral gaps for local coherence functionals on periodic lattices.<\/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":[30,36,31,56],"tags":[],"class_list":["post-2588","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-clay-millennium-problems","category-physics","category-research-papers"],"_links":{"self":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2588","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=2588"}],"version-history":[{"count":11,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2588\/revisions"}],"predecessor-version":[{"id":4235,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2588\/revisions\/4235"}],"wp:attachment":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/media?parent=2588"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/categories?post=2588"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/tags?post=2588"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}