{"id":2597,"date":"2026-05-03T14:55:37","date_gmt":"2026-05-03T14:55:37","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=2597"},"modified":"2026-05-09T14:06:26","modified_gmt":"2026-05-09T14:06:26","slug":"rank-one-coherence-obstructions-in-high-high-navier-stokes-interactions","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/03\/rank-one-coherence-obstructions-in-high-high-navier-stokes-interactions\/","title":{"rendered":"Rank-One Coherence Obstructions in High\u2013High Navier\u2013Stokes Interactions"},"content":{"rendered":"\n

Internal ID:<\/strong> CGI-RSR-000005
Document Type:<\/strong> Research Paper
Publication Date:<\/strong> May 2026
Status:<\/strong> Public
Domains:<\/strong> Mathematics, Physics
Research Topics:<\/strong> Partial differential equations \/ fluid dynamics, Clay Millennium Problems, Navier-Stokes<\/p>\n\n\n\n

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


We study comparable high\u2013high interactions in the three-dimensional incompressible Navier\u2013Stokes nonlinearity. For each dyadic output cell, we refine the interaction by angular sector, helicity, radial sub-shell, and output polarization, and form the Gram matrix of the corresponding output contributions. We call this Gram matrix the extended output coherence matrix. Its top eigenvalue measures rank-one coherent output summation, while the remaining spectrum measures secondary independent output channels. The main result is a high\u2013high coherence closure theorem. If a positive fraction of the Gram spectrum lies away from the top eigenvalue, then a spectral off-diagonal bilinear coherence inequality gives a strict gap from rank-one coherent saturation. Near the rank-one Gram locus, the Navier\u2013Stokes symbol forces a structural degeneracy: same-helicity same-radius coherence is Beltrami-depleted through the Lamb-vector identity, mixed-helicity and radially separated interactions split into orthogonal extended channels, and the only remaining finite radially separated beat is parabolically damped unless replenished by additional channels. Consequently, nondepleted comparable high\u2013high Navier\u2013Stokes interactions cannot sustain rank-one coherent output saturation in this extended Gram-matrix framework. The result is intended as a high\u2013high module for later integration with paraproduct estimates, critical continuation criteria, or profile-decomposition methods.<\/em><\/p>\n\n\n\n

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

DOI:<\/strong> 10.5281\/zenodo.19970064<\/a><\/code><\/p>\n\n\n\n

Citation:<\/strong>
Petersen, B. L. (2026). Rank-One Coherence Obstructions in High\u2013High Navier\u2013Stokes Interactions. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.19970064<\/p>\n\n\n\n

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

N\/A<\/p>\n\n\n\n

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

The CG viewpoint interprets dangerous nonlinear growth as a problem of excessive directional alignment among The CG viewpoint interprets dangerous nonlinear growth as a problem of excessive coherence among interacting modes.<\/p>\n\n\n\n

Rather than asking only how large the velocity field becomes, this approach asks how organized the nonlinear output becomes. In particular, it studies whether high-frequency interactions can collapse into a single dominant output channel capable of sustaining critical saturation.<\/p>\n\n\n\n

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

CGI-RSR-000006 : Projective Rank-one Closure for Terminal Navier\u2013Stokes Saturation<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

Internal ID: CGI-RSR-000005 | We study comparable high\u2013high interactions in the three-dimensional incompressible Navier\u2013Stokes nonlinearity.<\/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-2597","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\/2597","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=2597"}],"version-history":[{"count":1,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2597\/revisions"}],"predecessor-version":[{"id":2598,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/posts\/2597\/revisions\/2598"}],"wp:attachment":[{"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/media?parent=2597"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/categories?post=2597"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/coherencegeometry.com\/index.php\/wp-json\/wp\/v2\/tags?post=2597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}