{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:WNN2OOZZPPXDQSLLL56QA2YL4Z","short_pith_number":"pith:WNN2OOZZ","canonical_record":{"source":{"id":"2412.20548","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-12-29T18:38:16Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"5a318df3a04ec26839dd514ef046c346171c09361eccfb983594b889b411034e","abstract_canon_sha256":"7ea30e2f8d8e2d622fb508881512ed3e21ae401b4bb795e925e31538b2cabf68"},"schema_version":"1.0"},"canonical_sha256":"b35ba73b397bee38496b5f7d006b0be65d7a15a4d890e908eda39ae36d8d7d6a","source":{"kind":"arxiv","id":"2412.20548","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.20548","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"arxiv_version","alias_value":"2412.20548v2","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.20548","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"pith_short_12","alias_value":"WNN2OOZZPPXD","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"pith_short_16","alias_value":"WNN2OOZZPPXDQSLL","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"pith_short_8","alias_value":"WNN2OOZZ","created_at":"2026-07-05T10:07:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:WNN2OOZZPPXDQSLLL56QA2YL4Z","target":"record","payload":{"canonical_record":{"source":{"id":"2412.20548","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-12-29T18:38:16Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"5a318df3a04ec26839dd514ef046c346171c09361eccfb983594b889b411034e","abstract_canon_sha256":"7ea30e2f8d8e2d622fb508881512ed3e21ae401b4bb795e925e31538b2cabf68"},"schema_version":"1.0"},"canonical_sha256":"b35ba73b397bee38496b5f7d006b0be65d7a15a4d890e908eda39ae36d8d7d6a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:07:47.329631Z","signature_b64":"4+qqpoQCJiF8hkijXxNZZxsWIc5Hl0v4CTN1LnpMmQyI7Ig2VEJsqCmdSoEIMyVLpyRaCEFg4D9AeHJla17MAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b35ba73b397bee38496b5f7d006b0be65d7a15a4d890e908eda39ae36d8d7d6a","last_reissued_at":"2026-07-05T10:07:47.328928Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:07:47.328928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.20548","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":"2026-07-05T10:07:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1K8qwp6kVGldMjYusqUPcHwZrwOnN7CgKDX0gqkgnf/NHPJ7jluJlrRweBd2z1QTxhw6rMDMQ2G1BiqO8RwQDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T05:23:47.469301Z"},"content_sha256":"d6197f43423925ff9c9d840e90a145613d9b118e291e16f48348c19dfee9e59d","schema_version":"1.0","event_id":"sha256:d6197f43423925ff9c9d840e90a145613d9b118e291e16f48348c19dfee9e59d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:WNN2OOZZPPXDQSLLL56QA2YL4Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Six-Functor Formalisms III: The construction and extension of 6FFs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AG","authors_text":"Chirantan Chowdhury","submitted_at":"2024-12-29T18:38:16Z","abstract_excerpt":"This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding various functors altogether by taking adjoints along specific directions. Combined with the $\\infty$-categorical compactification theorem from the previous article, we can construct abstract six-functor formalisms in reasonable geometric setups of our interest. We also reprove the simplified versions of the DESCENT program due to Liu-Zheng, which allows us to ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.20548","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/2412.20548/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":"2026-07-05T10:07:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4O9bgdbYgYBkEMhZjuBplrPIdDvFRHIZo84jsih2PE8E5vIBNxT7LEGJH1Mnt4gkAEX5cMmJ2jjo6yFhoN3JCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T05:23:47.470102Z"},"content_sha256":"7f8fff75b6df413aabda13d404d415afa133b5d560b04e5bfc3d048dd7b4a33c","schema_version":"1.0","event_id":"sha256:7f8fff75b6df413aabda13d404d415afa133b5d560b04e5bfc3d048dd7b4a33c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WNN2OOZZPPXDQSLLL56QA2YL4Z/bundle.json","state_url":"https://pith.science/pith/WNN2OOZZPPXDQSLLL56QA2YL4Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WNN2OOZZPPXDQSLLL56QA2YL4Z/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":"2026-08-12T05:23:47Z","links":{"resolver":"https://pith.science/pith/WNN2OOZZPPXDQSLLL56QA2YL4Z","bundle":"https://pith.science/pith/WNN2OOZZPPXDQSLLL56QA2YL4Z/bundle.json","state":"https://pith.science/pith/WNN2OOZZPPXDQSLLL56QA2YL4Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WNN2OOZZPPXDQSLLL56QA2YL4Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:WNN2OOZZPPXDQSLLL56QA2YL4Z","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":"7ea30e2f8d8e2d622fb508881512ed3e21ae401b4bb795e925e31538b2cabf68","cross_cats_sorted":["math.KT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-12-29T18:38:16Z","title_canon_sha256":"5a318df3a04ec26839dd514ef046c346171c09361eccfb983594b889b411034e"},"schema_version":"1.0","source":{"id":"2412.20548","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.20548","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"arxiv_version","alias_value":"2412.20548v2","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.20548","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"pith_short_12","alias_value":"WNN2OOZZPPXD","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"pith_short_16","alias_value":"WNN2OOZZPPXDQSLL","created_at":"2026-07-05T10:07:47Z"},{"alias_kind":"pith_short_8","alias_value":"WNN2OOZZ","created_at":"2026-07-05T10:07:47Z"}],"graph_snapshots":[{"event_id":"sha256:7f8fff75b6df413aabda13d404d415afa133b5d560b04e5bfc3d048dd7b4a33c","target":"graph","created_at":"2026-07-05T10:07:47Z","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/2412.20548/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding various functors altogether by taking adjoints along specific directions. Combined with the $\\infty$-categorical compactification theorem from the previous article, we can construct abstract six-functor formalisms in reasonable geometric setups of our interest. We also reprove the simplified versions of the DESCENT program due to Liu-Zheng, which allows us to ex","authors_text":"Chirantan Chowdhury","cross_cats":["math.KT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-12-29T18:38:16Z","title":"Six-Functor Formalisms III: The construction and extension of 6FFs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.20548","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:d6197f43423925ff9c9d840e90a145613d9b118e291e16f48348c19dfee9e59d","target":"record","created_at":"2026-07-05T10:07:47Z","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":"7ea30e2f8d8e2d622fb508881512ed3e21ae401b4bb795e925e31538b2cabf68","cross_cats_sorted":["math.KT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-12-29T18:38:16Z","title_canon_sha256":"5a318df3a04ec26839dd514ef046c346171c09361eccfb983594b889b411034e"},"schema_version":"1.0","source":{"id":"2412.20548","kind":"arxiv","version":2}},"canonical_sha256":"b35ba73b397bee38496b5f7d006b0be65d7a15a4d890e908eda39ae36d8d7d6a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b35ba73b397bee38496b5f7d006b0be65d7a15a4d890e908eda39ae36d8d7d6a","first_computed_at":"2026-07-05T10:07:47.328928Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:07:47.328928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4+qqpoQCJiF8hkijXxNZZxsWIc5Hl0v4CTN1LnpMmQyI7Ig2VEJsqCmdSoEIMyVLpyRaCEFg4D9AeHJla17MAA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:07:47.329631Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.20548","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d6197f43423925ff9c9d840e90a145613d9b118e291e16f48348c19dfee9e59d","sha256:7f8fff75b6df413aabda13d404d415afa133b5d560b04e5bfc3d048dd7b4a33c"],"state_sha256":"a6783defb57fd65d9d6d5bb9798f4f486e01d6db44787dc0fd17c9ca453f39d9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WHF6yKKJEv1dQFun8xHJ/9vcXlR3XTog+tJWnaBmWkJyfxa4V60RVwNGptd7mCVsDHAh6uYPy7+uTvzoqK1gCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T05:23:47.480019Z","bundle_sha256":"8f8e2ab9f2bdb16a59d16b1a1f8b4d8e5d77a49398b93b8178abd9d60f13af99"}}