{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:NK7A6DHCQN2FGRZCZPJXBOYWFN","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":"349f977ca454567bcd319d213e4409fae713ec480b280974674be42bb24c928a","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-11-23T15:33:07Z","title_canon_sha256":"01a01e8426f37f002e7b1797bc41408ac12d9c2513e14a7a8e2aa6e9edfa33ec"},"schema_version":"1.0","source":{"id":"0911.4428","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.4428","created_at":"2026-05-18T04:39:18Z"},{"alias_kind":"arxiv_version","alias_value":"0911.4428v2","created_at":"2026-05-18T04:39:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4428","created_at":"2026-05-18T04:39:18Z"},{"alias_kind":"pith_short_12","alias_value":"NK7A6DHCQN2F","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"NK7A6DHCQN2FGRZC","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"NK7A6DHC","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:a3f1f422b359dc3d0375f32c2922f5463589e5f927b100cec77ae4624deb7fa5","target":"graph","created_at":"2026-05-18T04:39:18Z","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":"We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the operadic semidirect product with the circle group. The idea of our proof is to show that Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions and so one can essentially take the operadic semidirect product with the homology of S^1 everywhere to obtain a chain of quasi-isomorphi","authors_text":"Jeffrey Giansiracusa, Paolo Salvatore","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-11-23T15:33:07Z","title":"Formality of the framed little 2-discs operad and semidirect products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4428","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":"sha256:65ea9f28f11033977a0b6406a362c0597fbceb8c8d05c076b5debd94417a50f3","target":"record","created_at":"2026-05-18T04:39:18Z","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":"349f977ca454567bcd319d213e4409fae713ec480b280974674be42bb24c928a","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-11-23T15:33:07Z","title_canon_sha256":"01a01e8426f37f002e7b1797bc41408ac12d9c2513e14a7a8e2aa6e9edfa33ec"},"schema_version":"1.0","source":{"id":"0911.4428","kind":"arxiv","version":2}},"canonical_sha256":"6abe0f0ce28374534722cbd370bb162b6f0937fd83ca74bac2d5920ad0e834de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6abe0f0ce28374534722cbd370bb162b6f0937fd83ca74bac2d5920ad0e834de","first_computed_at":"2026-05-18T04:39:18.353460Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:18.353460Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mYDg8Qu1Xm26HuJl+5yhRIgiM8ybAxmZNdtCilUde6IpPNLrmqugA0MT9V5i/JbhAGcmb6WmbetiWMl8MCMuDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:18.354063Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.4428","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65ea9f28f11033977a0b6406a362c0597fbceb8c8d05c076b5debd94417a50f3","sha256:a3f1f422b359dc3d0375f32c2922f5463589e5f927b100cec77ae4624deb7fa5"],"state_sha256":"f732ed7e80a9c952c78df58235549a45efe74dab35ce35e20279e3db00d2d3c5"}