{"paper":{"title":"Vector Invariance and Structural Closure of Julia-Type Iterations in Clifford Algebra","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Orgest Zaka","submitted_at":"2026-05-18T10:28:13Z","abstract_excerpt":"In this paper, we introduce a Clifford algebra framework for Julia-type dynamics driven by the geometric product. The nonlinear iteration \\[ f(\\vec{x}) = (\\vec{x}\\diamond \\vec{n})^p \\diamond \\vec{n} + \\vec{c}, \\qquad p \\ge 2, \\] is studied in a real $n$-dimensional inner-product space $V$, where $\\vec{x}, \\vec{n}, \\vec{c} \\in V$ and $\\vec{n}$ is a unit vector.\n  The main result reveals a previously unreported invariance phenomenon: although the geometric product generates higher-grade multivector components at intermediate stages, a built-in grade-reduction mechanism ensures complete collapse "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20252","kind":"arxiv","version":1},"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/2605.20252/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"}