{"paper":{"title":"A unified quaternion-complex framework for Navier-Stokes equations: new insights and implications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"physics.flu-dyn","authors_text":"Farrukh A. Chishtie","submitted_at":"2025-05-28T20:37:33Z","abstract_excerpt":"We present a novel, unified quaternion-complex framework for formulating the incompressible Navier-Stokes equations that reveals the geometric structure underlying viscous fluid motion and resolves the Clay Institute's Millennium Prize problem. By introducing complex coordinates $z = x + iy$ and expressing the velocity field as $F = u + iv$, we demonstrate that the nonlinear convection terms decompose as $(u \\cdot \\nabla)F = F \\cdot \\frac{\\partial F}{\\partial z} + F^* \\cdot \\frac{\\partial F}{\\partial \\bar{z}}$, separating inviscid convection from viscous coupling effects. We extend this framew"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.22853","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.22853/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}