{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:GG74BRVATVAHYGWMKQ2VZLKHR3","short_pith_number":"pith:GG74BRVA","canonical_record":{"source":{"id":"math/0602489","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2006-02-22T09:55:52Z","cross_cats_sorted":[],"title_canon_sha256":"c8178138167bac5636abd1be8433eed5c4714473aa7ccb507b13b7fe3cde51fc","abstract_canon_sha256":"ea68e387b40529a805dee0b10e5a9c2252ab8db5fa7be4c4e813014fe2d441c7"},"schema_version":"1.0"},"canonical_sha256":"31bfc0c6a09d407c1acc54355cad478ef86f915001377f11eb3925d08e207d1a","source":{"kind":"arxiv","id":"math/0602489","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0602489","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"math/0602489v1","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602489","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"GG74BRVATVAH","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"GG74BRVATVAHYGWM","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"GG74BRVA","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:GG74BRVATVAHYGWMKQ2VZLKHR3","target":"record","payload":{"canonical_record":{"source":{"id":"math/0602489","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2006-02-22T09:55:52Z","cross_cats_sorted":[],"title_canon_sha256":"c8178138167bac5636abd1be8433eed5c4714473aa7ccb507b13b7fe3cde51fc","abstract_canon_sha256":"ea68e387b40529a805dee0b10e5a9c2252ab8db5fa7be4c4e813014fe2d441c7"},"schema_version":"1.0"},"canonical_sha256":"31bfc0c6a09d407c1acc54355cad478ef86f915001377f11eb3925d08e207d1a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:24.377972Z","signature_b64":"U2PTHO1co5LRw48ROurQ6dE3aRXplSH2nWK9V/tgy50N54YcXXUZAf7qlsFUILweTrP7CG4nRZpFHsYab+4NCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31bfc0c6a09d407c1acc54355cad478ef86f915001377f11eb3925d08e207d1a","last_reissued_at":"2026-05-18T01:38:24.377257Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:24.377257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0602489","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":"2026-05-18T01:38:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"biQfIaLYxC3WkImozKVP8JiAgh6IGt0Cq/8cRUy/DGihDRDt7KrbupcBZTsXhwfUB2YM2uoT10lCbg6VD1QtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T02:39:52.449383Z"},"content_sha256":"5590276b18cd9a15639eb3842e37e304d5013fb36c551030f0faaec76244c72a","schema_version":"1.0","event_id":"sha256:5590276b18cd9a15639eb3842e37e304d5013fb36c551030f0faaec76244c72a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:GG74BRVATVAHYGWMKQ2VZLKHR3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cohomology for a group of diffeomorphisms of a manifold preserving an exact form","license":"","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Mark Losik, Peter W. Michor","submitted_at":"2006-02-22T09:55:52Z","abstract_excerpt":"Let $M$ be a $G$-manifold and $\\om$ a $G$-invariant exact $m$-form on $M$. We indicate when these data allow us to constract a cocycle on a group $G$ with values in the trivial $G$-module $\\mathbb R$ and when this cocycle is nontrivial."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602489","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":"2026-05-18T01:38:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I2QgNvCZN3Kh6OuWuwiniX+wHhnqZeij2fyXzMTFc9pTCKY4Ngu4A/aS/BHHVOUWS2F+JH1UfKGF9y2Jdm3cBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T02:39:52.449847Z"},"content_sha256":"f0cbac7a9a86ce91b987d9ee433b7d08b4f5498312ec359972fda1ef2a876cd6","schema_version":"1.0","event_id":"sha256:f0cbac7a9a86ce91b987d9ee433b7d08b4f5498312ec359972fda1ef2a876cd6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GG74BRVATVAHYGWMKQ2VZLKHR3/bundle.json","state_url":"https://pith.science/pith/GG74BRVATVAHYGWMKQ2VZLKHR3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GG74BRVATVAHYGWMKQ2VZLKHR3/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":"2026-06-26T02:39:52Z","links":{"resolver":"https://pith.science/pith/GG74BRVATVAHYGWMKQ2VZLKHR3","bundle":"https://pith.science/pith/GG74BRVATVAHYGWMKQ2VZLKHR3/bundle.json","state":"https://pith.science/pith/GG74BRVATVAHYGWMKQ2VZLKHR3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GG74BRVATVAHYGWMKQ2VZLKHR3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:GG74BRVATVAHYGWMKQ2VZLKHR3","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":"ea68e387b40529a805dee0b10e5a9c2252ab8db5fa7be4c4e813014fe2d441c7","cross_cats_sorted":[],"license":"","primary_cat":"math.DG","submitted_at":"2006-02-22T09:55:52Z","title_canon_sha256":"c8178138167bac5636abd1be8433eed5c4714473aa7ccb507b13b7fe3cde51fc"},"schema_version":"1.0","source":{"id":"math/0602489","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0602489","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"math/0602489v1","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602489","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"GG74BRVATVAH","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"GG74BRVATVAHYGWM","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"GG74BRVA","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:f0cbac7a9a86ce91b987d9ee433b7d08b4f5498312ec359972fda1ef2a876cd6","target":"graph","created_at":"2026-05-18T01:38:24Z","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 $M$ be a $G$-manifold and $\\om$ a $G$-invariant exact $m$-form on $M$. We indicate when these data allow us to constract a cocycle on a group $G$ with values in the trivial $G$-module $\\mathbb R$ and when this cocycle is nontrivial.","authors_text":"Mark Losik, Peter W. Michor","cross_cats":[],"headline":"","license":"","primary_cat":"math.DG","submitted_at":"2006-02-22T09:55:52Z","title":"Cohomology for a group of diffeomorphisms of a manifold preserving an exact form"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602489","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":"sha256:5590276b18cd9a15639eb3842e37e304d5013fb36c551030f0faaec76244c72a","target":"record","created_at":"2026-05-18T01:38:24Z","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":"ea68e387b40529a805dee0b10e5a9c2252ab8db5fa7be4c4e813014fe2d441c7","cross_cats_sorted":[],"license":"","primary_cat":"math.DG","submitted_at":"2006-02-22T09:55:52Z","title_canon_sha256":"c8178138167bac5636abd1be8433eed5c4714473aa7ccb507b13b7fe3cde51fc"},"schema_version":"1.0","source":{"id":"math/0602489","kind":"arxiv","version":1}},"canonical_sha256":"31bfc0c6a09d407c1acc54355cad478ef86f915001377f11eb3925d08e207d1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31bfc0c6a09d407c1acc54355cad478ef86f915001377f11eb3925d08e207d1a","first_computed_at":"2026-05-18T01:38:24.377257Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:24.377257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U2PTHO1co5LRw48ROurQ6dE3aRXplSH2nWK9V/tgy50N54YcXXUZAf7qlsFUILweTrP7CG4nRZpFHsYab+4NCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:24.377972Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0602489","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5590276b18cd9a15639eb3842e37e304d5013fb36c551030f0faaec76244c72a","sha256:f0cbac7a9a86ce91b987d9ee433b7d08b4f5498312ec359972fda1ef2a876cd6"],"state_sha256":"ccce66f2e61aaed31691740fda0d7bfc28d72ffd21f493f09e72f4808d3b6bc6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T4swqcRa40NunFc4vnTz+LMLlKnUXIahhpI5NvmNl0ZGL9O5UfycRPfo47Ys39oOVkBryPsMU5OVfRcgsgOQDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T02:39:52.452800Z","bundle_sha256":"3ada3ac06e037f418f12335967876f7423a9f19e4428479e04b6b60207394102"}}