{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:DKH5HUCLKYNQ5MFWYKSMDK2IU4","short_pith_number":"pith:DKH5HUCL","canonical_record":{"source":{"id":"2101.07245","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-01-18T18:55:53Z","cross_cats_sorted":["math.AC","math.AG","math.AT"],"title_canon_sha256":"7f8002f66dd3f865c20cbf4280eba3d962ce8c1a777727887d30f537827f4b0e","abstract_canon_sha256":"f05cc7ecd6f797542454c0f3e744ea925808a99c5a678b9962acbc911fa7d985"},"schema_version":"1.0"},"canonical_sha256":"1a8fd3d04b561b0eb0b6c2a4c1ab48a713df2e33e6f441db83c62cfd7f6e8986","source":{"kind":"arxiv","id":"2101.07245","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.07245","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"arxiv_version","alias_value":"2101.07245v2","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.07245","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"pith_short_12","alias_value":"DKH5HUCLKYNQ","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"pith_short_16","alias_value":"DKH5HUCLKYNQ5MFW","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"pith_short_8","alias_value":"DKH5HUCL","created_at":"2026-07-05T02:42:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:DKH5HUCLKYNQ5MFWYKSMDK2IU4","target":"record","payload":{"canonical_record":{"source":{"id":"2101.07245","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-01-18T18:55:53Z","cross_cats_sorted":["math.AC","math.AG","math.AT"],"title_canon_sha256":"7f8002f66dd3f865c20cbf4280eba3d962ce8c1a777727887d30f537827f4b0e","abstract_canon_sha256":"f05cc7ecd6f797542454c0f3e744ea925808a99c5a678b9962acbc911fa7d985"},"schema_version":"1.0"},"canonical_sha256":"1a8fd3d04b561b0eb0b6c2a4c1ab48a713df2e33e6f441db83c62cfd7f6e8986","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:42:47.433324Z","signature_b64":"RACzVCUo7OdPWILTMmvugG8hTgeit5hX8O+BJUvD9FBUEGuSl8Cnw61EHsv1M3liy3hBcQuQ3Xthe85kCLpMDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a8fd3d04b561b0eb0b6c2a4c1ab48a713df2e33e6f441db83c62cfd7f6e8986","last_reissued_at":"2026-07-05T02:42:47.432825Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:42:47.432825Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2101.07245","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:42:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X89cNXsTmliobBcVR7UT0zA0HCPIeXa712z74ts5bVqWWStKTIcaXCi/2izpp3xqODZbigULuN9XSI7fZpNkBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T02:01:21.320172Z"},"content_sha256":"a84074c1b35d64751c3ecb964adf69bcbd106f5b37c1d96e0ee0c624d92e1c0f","schema_version":"1.0","event_id":"sha256:a84074c1b35d64751c3ecb964adf69bcbd106f5b37c1d96e0ee0c624d92e1c0f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:DKH5HUCLKYNQ5MFWYKSMDK2IU4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AC","math.AG","math.AT"],"primary_cat":"math.CO","authors_text":"Karim Adiprasito, Stavros Argyrios Papadakis, Vasiliki Petrotou","submitted_at":"2021-01-18T18:55:53Z","abstract_excerpt":"We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.07245","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.07245/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:42:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VGiVxmZpIZJ3MKzuwYkaFUvemtPuY2/oKjMzVvDbDdp7efA4PMA3KIpmt4+ElHPQzfp2+9uKSj2J2ldmVtnXDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T02:01:21.320663Z"},"content_sha256":"f8e874c27eea58d721b5b2c9b38c4c88afb773ecdd774e9a1ce2820bf53735b8","schema_version":"1.0","event_id":"sha256:f8e874c27eea58d721b5b2c9b38c4c88afb773ecdd774e9a1ce2820bf53735b8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DKH5HUCLKYNQ5MFWYKSMDK2IU4/bundle.json","state_url":"https://pith.science/pith/DKH5HUCLKYNQ5MFWYKSMDK2IU4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DKH5HUCLKYNQ5MFWYKSMDK2IU4/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":"2026-08-19T02:01:21Z","links":{"resolver":"https://pith.science/pith/DKH5HUCLKYNQ5MFWYKSMDK2IU4","bundle":"https://pith.science/pith/DKH5HUCLKYNQ5MFWYKSMDK2IU4/bundle.json","state":"https://pith.science/pith/DKH5HUCLKYNQ5MFWYKSMDK2IU4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DKH5HUCLKYNQ5MFWYKSMDK2IU4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:DKH5HUCLKYNQ5MFWYKSMDK2IU4","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":"f05cc7ecd6f797542454c0f3e744ea925808a99c5a678b9962acbc911fa7d985","cross_cats_sorted":["math.AC","math.AG","math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-01-18T18:55:53Z","title_canon_sha256":"7f8002f66dd3f865c20cbf4280eba3d962ce8c1a777727887d30f537827f4b0e"},"schema_version":"1.0","source":{"id":"2101.07245","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.07245","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"arxiv_version","alias_value":"2101.07245v2","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.07245","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"pith_short_12","alias_value":"DKH5HUCLKYNQ","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"pith_short_16","alias_value":"DKH5HUCLKYNQ5MFW","created_at":"2026-07-05T02:42:47Z"},{"alias_kind":"pith_short_8","alias_value":"DKH5HUCL","created_at":"2026-07-05T02:42:47Z"}],"graph_snapshots":[{"event_id":"sha256:f8e874c27eea58d721b5b2c9b38c4c88afb773ecdd774e9a1ce2820bf53735b8","target":"graph","created_at":"2026-07-05T02:42:47Z","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/2101.07245/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles.","authors_text":"Karim Adiprasito, Stavros Argyrios Papadakis, Vasiliki Petrotou","cross_cats":["math.AC","math.AG","math.AT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-01-18T18:55:53Z","title":"Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.07245","kind":"arxiv","version":2},"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:a84074c1b35d64751c3ecb964adf69bcbd106f5b37c1d96e0ee0c624d92e1c0f","target":"record","created_at":"2026-07-05T02:42:47Z","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":"f05cc7ecd6f797542454c0f3e744ea925808a99c5a678b9962acbc911fa7d985","cross_cats_sorted":["math.AC","math.AG","math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-01-18T18:55:53Z","title_canon_sha256":"7f8002f66dd3f865c20cbf4280eba3d962ce8c1a777727887d30f537827f4b0e"},"schema_version":"1.0","source":{"id":"2101.07245","kind":"arxiv","version":2}},"canonical_sha256":"1a8fd3d04b561b0eb0b6c2a4c1ab48a713df2e33e6f441db83c62cfd7f6e8986","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1a8fd3d04b561b0eb0b6c2a4c1ab48a713df2e33e6f441db83c62cfd7f6e8986","first_computed_at":"2026-07-05T02:42:47.432825Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:42:47.432825Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RACzVCUo7OdPWILTMmvugG8hTgeit5hX8O+BJUvD9FBUEGuSl8Cnw61EHsv1M3liy3hBcQuQ3Xthe85kCLpMDw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:42:47.433324Z","signed_message":"canonical_sha256_bytes"},"source_id":"2101.07245","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a84074c1b35d64751c3ecb964adf69bcbd106f5b37c1d96e0ee0c624d92e1c0f","sha256:f8e874c27eea58d721b5b2c9b38c4c88afb773ecdd774e9a1ce2820bf53735b8"],"state_sha256":"6aa4edc1f85beb3298702ff9c9ff48f52d817115b35cbb7179f1d472bc9c4ead"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8EtJUledD9bv7CB/xTGVHiFGeCy88exGBKLCfBRdt97V6paVjLY/wDpfoIdv/cYHJZqipmv7zqL00kY+9ElsDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T02:01:21.325118Z","bundle_sha256":"91f4469329db484a7814aa47fcb157d48846b18197169926dccc2925729f9e46"}}