{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:KXN4U6CMKJQ5CRIE4XYHWPGWOB","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":"41f0bb66e35857a726aa351fbadee201c458dbd08bae3d9eb2b1dabe728ef688","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-09-26T11:26:18Z","title_canon_sha256":"ddb0e296f4b5e4c892e7bd99416b3dfe0f3b4ee338f1831e4ba6fc5ff85bc67c"},"schema_version":"1.0","source":{"id":"2309.14848","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.14848","created_at":"2026-07-05T06:54:30Z"},{"alias_kind":"arxiv_version","alias_value":"2309.14848v1","created_at":"2026-07-05T06:54:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.14848","created_at":"2026-07-05T06:54:30Z"},{"alias_kind":"pith_short_12","alias_value":"KXN4U6CMKJQ5","created_at":"2026-07-05T06:54:30Z"},{"alias_kind":"pith_short_16","alias_value":"KXN4U6CMKJQ5CRIE","created_at":"2026-07-05T06:54:30Z"},{"alias_kind":"pith_short_8","alias_value":"KXN4U6CM","created_at":"2026-07-05T06:54:30Z"}],"graph_snapshots":[{"event_id":"sha256:455a1b938aac6d9d0b2c13db17d8800a39cdfaad776624cefe3e9f0356eff370","target":"graph","created_at":"2026-07-05T06:54:30Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2309.14848/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Bier sphere $Bier(\\mathcal{G}) = Bier(K) = K\\ast_\\Delta K^\\circ$ and the canonical fan $Fan(\\Gamma) = Fan(K)$ are combinatorial/geometric companions of a simple game $\\mathcal{G} = (P,\\Gamma)$ (equivalently the associated simplicial complex $K$), where $P$ is the set of players, $\\Gamma\\subseteq 2^P$ is the set of wining coalitions, and $K = 2^P\\setminus \\Gamma$ is the simplicial complex of losing coalitions. We characterize roughly weighted majority games as the games $\\Gamma$ such that $Bier(\\mathcal{G})$ (respectively $Fan(\\Gamma)$) is canonically polytopal (canonically pseudo-polytopal","authors_text":"Filip D. Jevti\\'c, Marinko Timotijevi\\'c, Rade T. \\v{Z}ivaljevi\\'c","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-09-26T11:26:18Z","title":"Polytopality of simple games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.14848","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:b766c4e276d7a4be141a65fdf74e2f21e751ab857f904e63949ec4bef2b8223e","target":"record","created_at":"2026-07-05T06:54:30Z","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":"41f0bb66e35857a726aa351fbadee201c458dbd08bae3d9eb2b1dabe728ef688","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-09-26T11:26:18Z","title_canon_sha256":"ddb0e296f4b5e4c892e7bd99416b3dfe0f3b4ee338f1831e4ba6fc5ff85bc67c"},"schema_version":"1.0","source":{"id":"2309.14848","kind":"arxiv","version":1}},"canonical_sha256":"55dbca784c5261d14504e5f07b3cd6706b7c26122d2dca00951a3b1e03563921","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"55dbca784c5261d14504e5f07b3cd6706b7c26122d2dca00951a3b1e03563921","first_computed_at":"2026-07-05T06:54:30.598379Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:54:30.598379Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VTxOKfqsdh0ifpfYsRIGQkBofHlZSNwmBxc9xVPbKhKyYWe4mSiYLLFEljaRDWTTgH0Z5JTEfNmy9yh2NFthBg==","signature_status":"signed_v1","signed_at":"2026-07-05T06:54:30.598783Z","signed_message":"canonical_sha256_bytes"},"source_id":"2309.14848","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b766c4e276d7a4be141a65fdf74e2f21e751ab857f904e63949ec4bef2b8223e","sha256:455a1b938aac6d9d0b2c13db17d8800a39cdfaad776624cefe3e9f0356eff370"],"state_sha256":"9b5dbfd8e32c3ac7d74c75cca9fedfd1963d53e1635d2c7e83bb4663218768d3"}