{"paper":{"title":"The Four-Point Picard Theorem for Quaternionic Slice Regular Functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Guangbin Ren, Xin Zhao","submitted_at":"2026-06-07T14:37:13Z","abstract_excerpt":"Let $a_0,a_1,a_2,a_3\\in\\mathbb H$. We prove that a nonconstant entire slice regular function on $\\mathbb H$ can omit these four values if and only if they are affinely dependent. Thus the affinely independent case, the four-point borderline left open by the Bisi--Winkelmann Picard theorem, cannot occur. The proof converts omission into four zero-free entire functions $Q_j$ attached to the stem function. For affinely independent target points, the coordinate normal to their affine span is governed by a square-discriminant identity $T^2=\\Delta_A(Q_0,Q_1,Q_2,Q_3)$. Finite-order $Q$-data are exclu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08651","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/2606.08651/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"}