{"paper":{"title":"The Non-Orientable Topology of Condorcet's Paradox","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Condorcet's Paradox corresponds to the non-orientability of a Klein bottle or real projective plane.","cross_cats":["cs.GT","econ.TH"],"primary_cat":"math.AT","authors_text":"Mikhail Prokopenko, Ori Livson, Siddharth Pritam","submitted_at":"2026-01-12T07:38:25Z","abstract_excerpt":"Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of social choice theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory preference cycles. Topological methods have since broadened social choice theory and elucidated existing results. However, characterisations of preference cycles in topological social choice theory are lacking. In this paper, we address this gap by introducing a fr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein Bottle or Real Projective Plane, depending on how preference cycles are represented","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"that the chosen topological representation of preference cycles (generalizing Baryshnikov's model for strict ordinal preferences on three alternatives) faithfully captures the logical contradiction without introducing extraneous structure","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Condorcet's paradox corresponds to non-orientability of a surface homeomorphic to the Klein bottle or real projective plane in a generalized topological model of strict ordinal preferences.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Condorcet's Paradox corresponds to the non-orientability of a Klein bottle or real projective plane.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b78eb5c9c0b188ad54610812d9b859fdb08b97010735ce2112cc8f983a0b352b"},"source":{"id":"2601.07283","kind":"arxiv","version":4},"verdict":{"id":"66f7d342-2fba-4ed4-8748-985630776014","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T15:33:27.313337Z","strongest_claim":"the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein Bottle or Real Projective Plane, depending on how preference cycles are represented","one_line_summary":"Condorcet's paradox corresponds to non-orientability of a surface homeomorphic to the Klein bottle or real projective plane in a generalized topological model of strict ordinal preferences.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"that the chosen topological representation of preference cycles (generalizing Baryshnikov's model for strict ordinal preferences on three alternatives) faithfully captures the logical contradiction without introducing extraneous structure","pith_extraction_headline":"Condorcet's Paradox corresponds to the non-orientability of a Klein bottle or real projective plane."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.07283/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":2,"snapshot_sha256":"21caa4e7d7b72f86cbeecc2ebe408f3df17dc81de567bd53088a29714d28a243"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}