{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BZKHGD7WP35HDVEDJHTFAA63FN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"30d537d71cd471572e9b8f52d5899c0977b063c239e332df64d827a4dc46013b","cross_cats_sorted":["cs.CG","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-06-22T01:52:45Z","title_canon_sha256":"c5b92dbc4f1ee7791cc44da6d9959075463dc940dce3b9cc03c97ecab39bda81"},"schema_version":"1.0","source":{"id":"1506.06440","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06440","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06440v1","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06440","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"pith_short_12","alias_value":"BZKHGD7WP35H","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BZKHGD7WP35HDVED","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BZKHGD7W","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:b0f003d39a758bd3905e9ce0558f98b822f081e5b002160d8cc11acc9e3fafdf","target":"graph","created_at":"2026-05-18T01:41:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic definitions are due to Evako: the unit sphere of a vertex x of a graph G=(V,E) is the graph generated by {y | (x,y) in E} Inductively, a finite simple graph is called contractible if there is a vertex x such that both its unit sphere S(x) as well as the graph generated by V-{x} are","authors_text":"Oliver Knill","cross_cats":["cs.CG","math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-06-22T01:52:45Z","title":"The Jordan-Brouwer theorem for graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06440","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a5db144de626039e2fe20b200d447aa7e0f846dafa2ed12e60fbe76567c8a85e","target":"record","created_at":"2026-05-18T01:41:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"30d537d71cd471572e9b8f52d5899c0977b063c239e332df64d827a4dc46013b","cross_cats_sorted":["cs.CG","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-06-22T01:52:45Z","title_canon_sha256":"c5b92dbc4f1ee7791cc44da6d9959075463dc940dce3b9cc03c97ecab39bda81"},"schema_version":"1.0","source":{"id":"1506.06440","kind":"arxiv","version":1}},"canonical_sha256":"0e54730ff67efa71d48349e65003db2b721d659f10b69beb08aee046a09e3e08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e54730ff67efa71d48349e65003db2b721d659f10b69beb08aee046a09e3e08","first_computed_at":"2026-05-18T01:41:45.226490Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:41:45.226490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U9nsSZhNUXZIo9GHWdYiXlC7w+Gcd1uoNxVWrX8c9NuQzUuOcopWQac+o2xqYUN/Mkw9MJVVMZ8iih67FPPeCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:41:45.227173Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.06440","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a5db144de626039e2fe20b200d447aa7e0f846dafa2ed12e60fbe76567c8a85e","sha256:b0f003d39a758bd3905e9ce0558f98b822f081e5b002160d8cc11acc9e3fafdf"],"state_sha256":"625c980238c3e038325a061e16faff633de48d5f93c82010c12531b7e0226d36"}