{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:XPMRRAI63CACQ7UHZPDA34RVUT","short_pith_number":"pith:XPMRRAI6","canonical_record":{"source":{"id":"1012.0318","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-01T21:03:31Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"d6ba79c11dd0f344b998757b392b1fd3adde583b578e11ff32d90133553007ff","abstract_canon_sha256":"c4a0eba25c5162d22bfb9d22417ab550025e8e4621c9aff7953d6ffb2a8f16dc"},"schema_version":"1.0"},"canonical_sha256":"bbd918811ed880287e87cbc60df235a4eb1b3f46eff13ec2fec380f636650d15","source":{"kind":"arxiv","id":"1012.0318","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0318","created_at":"2026-05-18T04:34:17Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0318v1","created_at":"2026-05-18T04:34:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0318","created_at":"2026-05-18T04:34:17Z"},{"alias_kind":"pith_short_12","alias_value":"XPMRRAI63CAC","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XPMRRAI63CACQ7UH","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XPMRRAI6","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:XPMRRAI63CACQ7UHZPDA34RVUT","target":"record","payload":{"canonical_record":{"source":{"id":"1012.0318","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-01T21:03:31Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"d6ba79c11dd0f344b998757b392b1fd3adde583b578e11ff32d90133553007ff","abstract_canon_sha256":"c4a0eba25c5162d22bfb9d22417ab550025e8e4621c9aff7953d6ffb2a8f16dc"},"schema_version":"1.0"},"canonical_sha256":"bbd918811ed880287e87cbc60df235a4eb1b3f46eff13ec2fec380f636650d15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:17.993024Z","signature_b64":"nazdfN7VlgKYQJU0fA72OTsOyJ51K/kyF0UyuDVglZ3EJsUhdsmPzWSYgOsqemReCLu8HxlGjGi3RuO5O/W9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbd918811ed880287e87cbc60df235a4eb1b3f46eff13ec2fec380f636650d15","last_reissued_at":"2026-05-18T04:34:17.992475Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:17.992475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.0318","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":"2026-05-18T04:34:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5mEPcpJ2VPd8FB/5OvfWHDNb1L/Tx3EBqzue7xufCHb1Y4HzcepiRZDPW1ltzKbRAvtrFqzwxM+K226q5afCAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T21:05:24.014578Z"},"content_sha256":"dab55d7e7819b2a7f111acae335b0ce55be2b9f75c81b5e071d35e13893a568e","schema_version":"1.0","event_id":"sha256:dab55d7e7819b2a7f111acae335b0ce55be2b9f75c81b5e071d35e13893a568e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:XPMRRAI63CACQ7UHZPDA34RVUT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The translation operator for self-projective coalgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"William Chin","submitted_at":"2010-12-01T21:03:31Z","abstract_excerpt":"We describe the transpose operator for self-projective and symmetric coalgebras in terms of the syzygy and Nakayama functors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0318","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":"2026-05-18T04:34:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FzrvNghzihAwF/Lixt2/QuN4n2A1L/TZvS9vMQfdlTBDSkBTVTtYwEZoipvm69vTFnuAKdoTUEYjsw+JubYPAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T21:05:24.015290Z"},"content_sha256":"3efcf6313c98c180cb9bfae68564cc5907ebf8fdca98b1b1aa215c993a90feaf","schema_version":"1.0","event_id":"sha256:3efcf6313c98c180cb9bfae68564cc5907ebf8fdca98b1b1aa215c993a90feaf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XPMRRAI63CACQ7UHZPDA34RVUT/bundle.json","state_url":"https://pith.science/pith/XPMRRAI63CACQ7UHZPDA34RVUT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XPMRRAI63CACQ7UHZPDA34RVUT/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":"2026-06-07T21:05:24Z","links":{"resolver":"https://pith.science/pith/XPMRRAI63CACQ7UHZPDA34RVUT","bundle":"https://pith.science/pith/XPMRRAI63CACQ7UHZPDA34RVUT/bundle.json","state":"https://pith.science/pith/XPMRRAI63CACQ7UHZPDA34RVUT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XPMRRAI63CACQ7UHZPDA34RVUT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XPMRRAI63CACQ7UHZPDA34RVUT","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":"c4a0eba25c5162d22bfb9d22417ab550025e8e4621c9aff7953d6ffb2a8f16dc","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-01T21:03:31Z","title_canon_sha256":"d6ba79c11dd0f344b998757b392b1fd3adde583b578e11ff32d90133553007ff"},"schema_version":"1.0","source":{"id":"1012.0318","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0318","created_at":"2026-05-18T04:34:17Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0318v1","created_at":"2026-05-18T04:34:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0318","created_at":"2026-05-18T04:34:17Z"},{"alias_kind":"pith_short_12","alias_value":"XPMRRAI63CAC","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XPMRRAI63CACQ7UH","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XPMRRAI6","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:3efcf6313c98c180cb9bfae68564cc5907ebf8fdca98b1b1aa215c993a90feaf","target":"graph","created_at":"2026-05-18T04:34:17Z","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 describe the transpose operator for self-projective and symmetric coalgebras in terms of the syzygy and Nakayama functors.","authors_text":"William Chin","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-01T21:03:31Z","title":"The translation operator for self-projective coalgebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0318","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:dab55d7e7819b2a7f111acae335b0ce55be2b9f75c81b5e071d35e13893a568e","target":"record","created_at":"2026-05-18T04:34:17Z","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":"c4a0eba25c5162d22bfb9d22417ab550025e8e4621c9aff7953d6ffb2a8f16dc","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-01T21:03:31Z","title_canon_sha256":"d6ba79c11dd0f344b998757b392b1fd3adde583b578e11ff32d90133553007ff"},"schema_version":"1.0","source":{"id":"1012.0318","kind":"arxiv","version":1}},"canonical_sha256":"bbd918811ed880287e87cbc60df235a4eb1b3f46eff13ec2fec380f636650d15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bbd918811ed880287e87cbc60df235a4eb1b3f46eff13ec2fec380f636650d15","first_computed_at":"2026-05-18T04:34:17.992475Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:34:17.992475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nazdfN7VlgKYQJU0fA72OTsOyJ51K/kyF0UyuDVglZ3EJsUhdsmPzWSYgOsqemReCLu8HxlGjGi3RuO5O/W9AA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:34:17.993024Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.0318","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dab55d7e7819b2a7f111acae335b0ce55be2b9f75c81b5e071d35e13893a568e","sha256:3efcf6313c98c180cb9bfae68564cc5907ebf8fdca98b1b1aa215c993a90feaf"],"state_sha256":"92e03b80eadbbfe499b2c1d3187afe1df59a6372ce38e3d1c4fa2dc1bfc0d36c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e9tE/kvC3GvYHVJKbMddt4lhaynxvppVjEI3UMm6HHbe37FXjHG34a8QsA5uvm/eT0vSW2qW3j7HeWJv3oJPCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T21:05:24.019334Z","bundle_sha256":"24ea1d6b43a81571aa32d7b631a7bb162bafa54c21c0bbf4aad93de87d534ec0"}}