{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:23IAFEMJCML6TT5V2UXZSHGNJS","short_pith_number":"pith:23IAFEMJ","canonical_record":{"source":{"id":"0710.1172","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-10-05T10:28:55Z","cross_cats_sorted":[],"title_canon_sha256":"0025412d14b9992a2ead61e4519382a3d6bcedf5b3c42bbbc6cb1b15e3db3e15","abstract_canon_sha256":"641fb66d0ca0c912558fe3e7a4a209dd679231c91298e155f5a24ba41e3c4583"},"schema_version":"1.0"},"canonical_sha256":"d6d00291891317e9cfb5d52f991ccd4c9c61627ccd9edc67531c92c729ea5bb6","source":{"kind":"arxiv","id":"0710.1172","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.1172","created_at":"2026-05-18T04:10:34Z"},{"alias_kind":"arxiv_version","alias_value":"0710.1172v3","created_at":"2026-05-18T04:10:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.1172","created_at":"2026-05-18T04:10:34Z"},{"alias_kind":"pith_short_12","alias_value":"23IAFEMJCML6","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"23IAFEMJCML6TT5V","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"23IAFEMJ","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:23IAFEMJCML6TT5V2UXZSHGNJS","target":"record","payload":{"canonical_record":{"source":{"id":"0710.1172","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-10-05T10:28:55Z","cross_cats_sorted":[],"title_canon_sha256":"0025412d14b9992a2ead61e4519382a3d6bcedf5b3c42bbbc6cb1b15e3db3e15","abstract_canon_sha256":"641fb66d0ca0c912558fe3e7a4a209dd679231c91298e155f5a24ba41e3c4583"},"schema_version":"1.0"},"canonical_sha256":"d6d00291891317e9cfb5d52f991ccd4c9c61627ccd9edc67531c92c729ea5bb6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:34.750761Z","signature_b64":"fnB1c6FPW/b0Ow0wb+b2tFw8wKxQxxzH9kwem+6u/9Cc2agZ0RevQ2icYa8Gb1/96HBF+aMstcDAi4QDNlwhDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6d00291891317e9cfb5d52f991ccd4c9c61627ccd9edc67531c92c729ea5bb6","last_reissued_at":"2026-05-18T04:10:34.749992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:34.749992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0710.1172","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:10:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SPxXRgWkeV58cN+b160WuIep3RONszyKxDppIWkWnLy+6cBxMvg8A4A2UnA4Gs3XPYTee1yJ8lQmZVuJd15uAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:41:40.202693Z"},"content_sha256":"df80596cd96d99d45408cc7da1a7c01bdf0b11d86f4b783730e9b1173e83d406","schema_version":"1.0","event_id":"sha256:df80596cd96d99d45408cc7da1a7c01bdf0b11d86f4b783730e9b1173e83d406"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:23IAFEMJCML6TT5V2UXZSHGNJS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combinatorial Alexander Duality -- a Short and Elementary Proof","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anders Bj\\\"orner, Martin Tancer","submitted_at":"2007-10-05T10:28:55Z","abstract_excerpt":"Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \\subset V: V \\setminus A \\notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.1172","kind":"arxiv","version":3},"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":"2026-05-18T04:10:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nte3kpyw3mIcI69gV5hqNQZcD5HSwb/q+gBpRmaEP+HEsT6MeoGU+9yHX8nJkdgi9Jj+Me+wni25zrAwdrpECw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:41:40.203391Z"},"content_sha256":"aa91fd74607012d79ad08182d473d8dc3f08d9b0676116cc7105e4688c81d923","schema_version":"1.0","event_id":"sha256:aa91fd74607012d79ad08182d473d8dc3f08d9b0676116cc7105e4688c81d923"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/23IAFEMJCML6TT5V2UXZSHGNJS/bundle.json","state_url":"https://pith.science/pith/23IAFEMJCML6TT5V2UXZSHGNJS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/23IAFEMJCML6TT5V2UXZSHGNJS/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":"2026-05-27T16:41:40Z","links":{"resolver":"https://pith.science/pith/23IAFEMJCML6TT5V2UXZSHGNJS","bundle":"https://pith.science/pith/23IAFEMJCML6TT5V2UXZSHGNJS/bundle.json","state":"https://pith.science/pith/23IAFEMJCML6TT5V2UXZSHGNJS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/23IAFEMJCML6TT5V2UXZSHGNJS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:23IAFEMJCML6TT5V2UXZSHGNJS","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":"641fb66d0ca0c912558fe3e7a4a209dd679231c91298e155f5a24ba41e3c4583","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-10-05T10:28:55Z","title_canon_sha256":"0025412d14b9992a2ead61e4519382a3d6bcedf5b3c42bbbc6cb1b15e3db3e15"},"schema_version":"1.0","source":{"id":"0710.1172","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.1172","created_at":"2026-05-18T04:10:34Z"},{"alias_kind":"arxiv_version","alias_value":"0710.1172v3","created_at":"2026-05-18T04:10:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.1172","created_at":"2026-05-18T04:10:34Z"},{"alias_kind":"pith_short_12","alias_value":"23IAFEMJCML6","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"23IAFEMJCML6TT5V","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"23IAFEMJ","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:aa91fd74607012d79ad08182d473d8dc3f08d9b0676116cc7105e4688c81d923","target":"graph","created_at":"2026-05-18T04:10:34Z","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":"Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \\subset V: V \\setminus A \\notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof.","authors_text":"Anders Bj\\\"orner, Martin Tancer","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2007-10-05T10:28:55Z","title":"Combinatorial Alexander Duality -- a Short and Elementary Proof"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.1172","kind":"arxiv","version":3},"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:df80596cd96d99d45408cc7da1a7c01bdf0b11d86f4b783730e9b1173e83d406","target":"record","created_at":"2026-05-18T04:10:34Z","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":"641fb66d0ca0c912558fe3e7a4a209dd679231c91298e155f5a24ba41e3c4583","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-10-05T10:28:55Z","title_canon_sha256":"0025412d14b9992a2ead61e4519382a3d6bcedf5b3c42bbbc6cb1b15e3db3e15"},"schema_version":"1.0","source":{"id":"0710.1172","kind":"arxiv","version":3}},"canonical_sha256":"d6d00291891317e9cfb5d52f991ccd4c9c61627ccd9edc67531c92c729ea5bb6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6d00291891317e9cfb5d52f991ccd4c9c61627ccd9edc67531c92c729ea5bb6","first_computed_at":"2026-05-18T04:10:34.749992Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:34.749992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fnB1c6FPW/b0Ow0wb+b2tFw8wKxQxxzH9kwem+6u/9Cc2agZ0RevQ2icYa8Gb1/96HBF+aMstcDAi4QDNlwhDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:34.750761Z","signed_message":"canonical_sha256_bytes"},"source_id":"0710.1172","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:df80596cd96d99d45408cc7da1a7c01bdf0b11d86f4b783730e9b1173e83d406","sha256:aa91fd74607012d79ad08182d473d8dc3f08d9b0676116cc7105e4688c81d923"],"state_sha256":"835b6bd9c9d87d3c919a0ade4efcba0c4085a949c6ecc59d20abb1ab09bb7b0e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wWbGAZXL839mMXDB9oD3/VkH23OWTA5pJC08805IPT8IlvOGe5cevYT+fID2gThzqMPOkhh8OgsDmjsnSq1iAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T16:41:40.207072Z","bundle_sha256":"70e3e8b6632ba2c29bebf2d307f7cd5a94070f60835c56497da9a0035037ce1d"}}