{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:KCRI5DP5NMVCW2YBL7UDJ2YOMJ","short_pith_number":"pith:KCRI5DP5","canonical_record":{"source":{"id":"1604.02395","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-08T16:41:54Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"eb26cc809c6e7183b0a68e2f90a9e8cf3445ac5cfe295cbfc8d9f613d2b56e8f","abstract_canon_sha256":"a9df4364511ce9e387b0a71246786d16cca979fdffd4a4ba8d28ae367f7ebbfe"},"schema_version":"1.0"},"canonical_sha256":"50a28e8dfd6b2a2b6b015fe834eb0e624c649f82f89721af95917191e674ae3d","source":{"kind":"arxiv","id":"1604.02395","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02395","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02395v1","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02395","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"pith_short_12","alias_value":"KCRI5DP5NMVC","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KCRI5DP5NMVCW2YB","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KCRI5DP5","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:KCRI5DP5NMVCW2YBL7UDJ2YOMJ","target":"record","payload":{"canonical_record":{"source":{"id":"1604.02395","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-08T16:41:54Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"eb26cc809c6e7183b0a68e2f90a9e8cf3445ac5cfe295cbfc8d9f613d2b56e8f","abstract_canon_sha256":"a9df4364511ce9e387b0a71246786d16cca979fdffd4a4ba8d28ae367f7ebbfe"},"schema_version":"1.0"},"canonical_sha256":"50a28e8dfd6b2a2b6b015fe834eb0e624c649f82f89721af95917191e674ae3d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:26.654904Z","signature_b64":"xja5Ram7t8IL0/dzOtyqL7YmtHl1MwzzodOupMSkGvFkzliXurJtC66N/9j/3Qp4IcDSUEzHkM5JxSW2Z9bMBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50a28e8dfd6b2a2b6b015fe834eb0e624c649f82f89721af95917191e674ae3d","last_reissued_at":"2026-05-18T01:17:26.654360Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:26.654360Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.02395","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sTdtu5UZSZgRK5vBr2Ngk2ofQyBWNBREK4ZYxLmxqkxCIWFbLVvwGnT0nMIFdAWUD76+MWjpLHuDsxfrnEp3Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T19:24:22.912284Z"},"content_sha256":"9114e0fc05d15e4049a3146fd685440a5b97ca8c94c0b0150757da78d2b19bae","schema_version":"1.0","event_id":"sha256:9114e0fc05d15e4049a3146fd685440a5b97ca8c94c0b0150757da78d2b19bae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:KCRI5DP5NMVCW2YBL7UDJ2YOMJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proving Tucker's Lemma with a Volume Argument","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Beauttie Kuture, Christopher Loa, Francis Edward Su, Mutiara Sondjaja, Oscar Leong","submitted_at":"2016-04-08T16:41:54Z","abstract_excerpt":"Sperner's lemma is a statement about labeled triangulations of a simplex. McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma by examining volumes of simplices in a triangulation under time-linear simplex-linear deformation. We adapt a similar argument to prove Tucker's Lemma on a triangulated cross-polytope $P$. The McLennan-Tourky technique does not directly apply because this deformation may distort the volume of $P$. We remedy this by inscribing $P$ in its dual polytope, triangulating it, and considering how the volumes of deformed simplices behave."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02395","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uzwvsfcYXed6VqakZlqaN1MMFTbiti358ad4ukyx4GXPdB6Wxr9dMwznsESgOPV0oSqe55TxWkCqDNL+MV3wBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T19:24:22.912653Z"},"content_sha256":"7bfb80827201b45a5cd852fad3737a5a30e2c7d9280b3a37d3b17c4e82694b1e","schema_version":"1.0","event_id":"sha256:7bfb80827201b45a5cd852fad3737a5a30e2c7d9280b3a37d3b17c4e82694b1e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KCRI5DP5NMVCW2YBL7UDJ2YOMJ/bundle.json","state_url":"https://pith.science/pith/KCRI5DP5NMVCW2YBL7UDJ2YOMJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KCRI5DP5NMVCW2YBL7UDJ2YOMJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T19:24:22Z","links":{"resolver":"https://pith.science/pith/KCRI5DP5NMVCW2YBL7UDJ2YOMJ","bundle":"https://pith.science/pith/KCRI5DP5NMVCW2YBL7UDJ2YOMJ/bundle.json","state":"https://pith.science/pith/KCRI5DP5NMVCW2YBL7UDJ2YOMJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KCRI5DP5NMVCW2YBL7UDJ2YOMJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KCRI5DP5NMVCW2YBL7UDJ2YOMJ","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":"a9df4364511ce9e387b0a71246786d16cca979fdffd4a4ba8d28ae367f7ebbfe","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-08T16:41:54Z","title_canon_sha256":"eb26cc809c6e7183b0a68e2f90a9e8cf3445ac5cfe295cbfc8d9f613d2b56e8f"},"schema_version":"1.0","source":{"id":"1604.02395","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02395","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02395v1","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02395","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"pith_short_12","alias_value":"KCRI5DP5NMVC","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KCRI5DP5NMVCW2YB","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KCRI5DP5","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:7bfb80827201b45a5cd852fad3737a5a30e2c7d9280b3a37d3b17c4e82694b1e","target":"graph","created_at":"2026-05-18T01:17:26Z","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":"Sperner's lemma is a statement about labeled triangulations of a simplex. McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma by examining volumes of simplices in a triangulation under time-linear simplex-linear deformation. We adapt a similar argument to prove Tucker's Lemma on a triangulated cross-polytope $P$. The McLennan-Tourky technique does not directly apply because this deformation may distort the volume of $P$. We remedy this by inscribing $P$ in its dual polytope, triangulating it, and considering how the volumes of deformed simplices behave.","authors_text":"Beauttie Kuture, Christopher Loa, Francis Edward Su, Mutiara Sondjaja, Oscar Leong","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-08T16:41:54Z","title":"Proving Tucker's Lemma with a Volume Argument"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02395","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":"sha256:9114e0fc05d15e4049a3146fd685440a5b97ca8c94c0b0150757da78d2b19bae","target":"record","created_at":"2026-05-18T01:17:26Z","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":"a9df4364511ce9e387b0a71246786d16cca979fdffd4a4ba8d28ae367f7ebbfe","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-08T16:41:54Z","title_canon_sha256":"eb26cc809c6e7183b0a68e2f90a9e8cf3445ac5cfe295cbfc8d9f613d2b56e8f"},"schema_version":"1.0","source":{"id":"1604.02395","kind":"arxiv","version":1}},"canonical_sha256":"50a28e8dfd6b2a2b6b015fe834eb0e624c649f82f89721af95917191e674ae3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50a28e8dfd6b2a2b6b015fe834eb0e624c649f82f89721af95917191e674ae3d","first_computed_at":"2026-05-18T01:17:26.654360Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:26.654360Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xja5Ram7t8IL0/dzOtyqL7YmtHl1MwzzodOupMSkGvFkzliXurJtC66N/9j/3Qp4IcDSUEzHkM5JxSW2Z9bMBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:26.654904Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.02395","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9114e0fc05d15e4049a3146fd685440a5b97ca8c94c0b0150757da78d2b19bae","sha256:7bfb80827201b45a5cd852fad3737a5a30e2c7d9280b3a37d3b17c4e82694b1e"],"state_sha256":"2799973265e19736e94b967556ab2b27ea5d8287790df85def1f5c8d87774c2e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xHds/5EddFDSHEsd/yXajqUfxZvJJRFU2goIAuGfKRHbSYa/tEL0X9fAEoioJinyCu8dBUaS90LzqmaUs7dkBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T19:24:22.914714Z","bundle_sha256":"32e5d818467188a2d927ee80774e7db4bc64948d9a98be35c991d01656f4395f"}}