{"paper":{"title":"Structure and realizability for rational maps","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT","math.GT"],"primary_cat":"math.DG","authors_text":"Zhiqiang Wei","submitted_at":"2025-11-10T07:15:53Z","abstract_excerpt":"We establish a structure theorem for rational maps $f:\\overline{\\mathbb{C}}\\to\\overline{\\mathbb{C}}$: the pullback metric $f^{*}{\\rm d}s_{0}^{2}$ of the standard metric ${\\rm d}s_{0}^{2}$ admits a canonical decomposition into finitely many footballs -- Riemann spheres with two antipodal conical singularities of equal angle -- by cutting along a finite set of geodesics. This geometric decomposition provides a new framework for the Hurwitz existence problem. As an application, we prove that a collection $\\mathcal{D}$ of $k$ nontrivial partitions of a positive integer $d$ satisfying the Riemann--"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.06784","kind":"arxiv","version":2},"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/2511.06784/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"}