{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:M6RPJRCQE72WAXALFNUVN7UAND","short_pith_number":"pith:M6RPJRCQ","canonical_record":{"source":{"id":"math/0609557","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-09-20T13:40:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6be791731de0c0189ea8b20fa6471a6ada62e6c68e4f2f2ae4f14cfd3cb77272","abstract_canon_sha256":"d05fe13f9912090797a632152b24f45852ed56d326490780e937c1e7d3fba3f4"},"schema_version":"1.0"},"canonical_sha256":"67a2f4c45027f5605c0b2b6956fe8068c2ae35089a375616ec762bef38c48b71","source":{"kind":"arxiv","id":"math/0609557","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609557","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609557v4","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609557","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"pith_short_12","alias_value":"M6RPJRCQE72W","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"M6RPJRCQE72WAXAL","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"M6RPJRCQ","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:M6RPJRCQE72WAXALFNUVN7UAND","target":"record","payload":{"canonical_record":{"source":{"id":"math/0609557","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-09-20T13:40:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6be791731de0c0189ea8b20fa6471a6ada62e6c68e4f2f2ae4f14cfd3cb77272","abstract_canon_sha256":"d05fe13f9912090797a632152b24f45852ed56d326490780e937c1e7d3fba3f4"},"schema_version":"1.0"},"canonical_sha256":"67a2f4c45027f5605c0b2b6956fe8068c2ae35089a375616ec762bef38c48b71","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:23.207072Z","signature_b64":"NfA+3av4OqiJXqzjynW5yzvF0Inch90kgS+SzPYzE+N9mChfd/QiIdq+yMxZxRGhaJJv//SL3xi8394HMB+ODw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67a2f4c45027f5605c0b2b6956fe8068c2ae35089a375616ec762bef38c48b71","last_reissued_at":"2026-05-18T03:50:23.206299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:23.206299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0609557","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"weGOpSg0IaMimzZ8pMVGKOotoxfGQm/YStqippMoezfZFHDXU1eX0hd6osdv77duMzS4/ZTmnOFjGx2w/c9QDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T00:43:17.107898Z"},"content_sha256":"9ff5cf6e03d221c6eff2eb9ecb5563820f3680f54d024657e9fd3aa76e38ede2","schema_version":"1.0","event_id":"sha256:9ff5cf6e03d221c6eff2eb9ecb5563820f3680f54d024657e9fd3aa76e38ede2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:M6RPJRCQE72WAXALFNUVN7UAND","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Manifolds associated with $(Z_2)^n$-colored regular graphs","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GT","authors_text":"Zhi L\\\"u, Zhiqiang Bao","submitted_at":"2006-09-20T13:40:08Z","abstract_excerpt":"In this article we describe a canonical way to expand a certain kind of $(\\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial $n$-manifold can be obtained in this way. When $n\\leq 3$, we give simple equivalent conditions for a colored graph to admit an expansion. In addition, we show that if a $(\\mathbb Z_2)^{n+1}$-colored regular graph admits an $n$-skeletal expansion, then it is realizable as the moment graph of an $(n+1)$-dimensional closed $(\\mathbb Z_2)^{n+1}$-manifold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609557","kind":"arxiv","version":4},"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":"2026-05-18T03:50:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jqV7JOEVkaUD0XxxpGhiQ/IK4vZS07tklIl+o9mPZQ0K3hIQsffS4jHDJtcPGCCTcVRV5pUEvgt9OcangqktCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T00:43:17.108523Z"},"content_sha256":"e4e407ffdd1f59b8474a8b369dac44209283b3be22ece5b90e50b68609e933ca","schema_version":"1.0","event_id":"sha256:e4e407ffdd1f59b8474a8b369dac44209283b3be22ece5b90e50b68609e933ca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M6RPJRCQE72WAXALFNUVN7UAND/bundle.json","state_url":"https://pith.science/pith/M6RPJRCQE72WAXALFNUVN7UAND/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M6RPJRCQE72WAXALFNUVN7UAND/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":"2026-05-20T00:43:17Z","links":{"resolver":"https://pith.science/pith/M6RPJRCQE72WAXALFNUVN7UAND","bundle":"https://pith.science/pith/M6RPJRCQE72WAXALFNUVN7UAND/bundle.json","state":"https://pith.science/pith/M6RPJRCQE72WAXALFNUVN7UAND/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M6RPJRCQE72WAXALFNUVN7UAND/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:M6RPJRCQE72WAXALFNUVN7UAND","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":"d05fe13f9912090797a632152b24f45852ed56d326490780e937c1e7d3fba3f4","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.GT","submitted_at":"2006-09-20T13:40:08Z","title_canon_sha256":"6be791731de0c0189ea8b20fa6471a6ada62e6c68e4f2f2ae4f14cfd3cb77272"},"schema_version":"1.0","source":{"id":"math/0609557","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609557","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609557v4","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609557","created_at":"2026-05-18T03:50:23Z"},{"alias_kind":"pith_short_12","alias_value":"M6RPJRCQE72W","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"M6RPJRCQE72WAXAL","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"M6RPJRCQ","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:e4e407ffdd1f59b8474a8b369dac44209283b3be22ece5b90e50b68609e933ca","target":"graph","created_at":"2026-05-18T03:50:23Z","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":"In this article we describe a canonical way to expand a certain kind of $(\\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial $n$-manifold can be obtained in this way. When $n\\leq 3$, we give simple equivalent conditions for a colored graph to admit an expansion. In addition, we show that if a $(\\mathbb Z_2)^{n+1}$-colored regular graph admits an $n$-skeletal expansion, then it is realizable as the moment graph of an $(n+1)$-dimensional closed $(\\mathbb Z_2)^{n+1}$-manifold.","authors_text":"Zhi L\\\"u, Zhiqiang Bao","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2006-09-20T13:40:08Z","title":"Manifolds associated with $(Z_2)^n$-colored regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609557","kind":"arxiv","version":4},"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":"sha256:9ff5cf6e03d221c6eff2eb9ecb5563820f3680f54d024657e9fd3aa76e38ede2","target":"record","created_at":"2026-05-18T03:50:23Z","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":"d05fe13f9912090797a632152b24f45852ed56d326490780e937c1e7d3fba3f4","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.GT","submitted_at":"2006-09-20T13:40:08Z","title_canon_sha256":"6be791731de0c0189ea8b20fa6471a6ada62e6c68e4f2f2ae4f14cfd3cb77272"},"schema_version":"1.0","source":{"id":"math/0609557","kind":"arxiv","version":4}},"canonical_sha256":"67a2f4c45027f5605c0b2b6956fe8068c2ae35089a375616ec762bef38c48b71","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"67a2f4c45027f5605c0b2b6956fe8068c2ae35089a375616ec762bef38c48b71","first_computed_at":"2026-05-18T03:50:23.206299Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:23.206299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NfA+3av4OqiJXqzjynW5yzvF0Inch90kgS+SzPYzE+N9mChfd/QiIdq+yMxZxRGhaJJv//SL3xi8394HMB+ODw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:23.207072Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0609557","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ff5cf6e03d221c6eff2eb9ecb5563820f3680f54d024657e9fd3aa76e38ede2","sha256:e4e407ffdd1f59b8474a8b369dac44209283b3be22ece5b90e50b68609e933ca"],"state_sha256":"a6b792740602bcbe7908ceabe7ce7716721564568a807a93583960370c204809"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nx78DMMk3EfJ+lN267HaGETLbFXo6B0EVCaBmrkrRd3Ze5EP+LVit/X2ERQGSaouFTrnWkXQ8aKAlIZbinjfAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T00:43:17.112202Z","bundle_sha256":"16f48bc8f91a5bcc0d497a484abea29bf8129cf7a6d4ff509f9ef57f17f54b0c"}}