{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:3QAENFU3MFHBQLFDBZNGGFMA2O","short_pith_number":"pith:3QAENFU3","canonical_record":{"source":{"id":"1006.5122","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T08:35:46Z","cross_cats_sorted":["math.CT","math.GN"],"title_canon_sha256":"b03bb8f571a29f9e35d4e467611461e11cae5c8912e65c4d5a4e077cbdee98c5","abstract_canon_sha256":"8f2b5fd72bbae84b1ca8a485d5cc96bdf0ddd8c85421ec61a6ceca143c9ff22e"},"schema_version":"1.0"},"canonical_sha256":"dc0046969b614e182ca30e5a631580d3802f324039a62d746ee1c12fb885c98b","source":{"kind":"arxiv","id":"1006.5122","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.5122","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"arxiv_version","alias_value":"1006.5122v2","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.5122","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"pith_short_12","alias_value":"3QAENFU3MFHB","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3QAENFU3MFHBQLFD","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3QAENFU3","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:3QAENFU3MFHBQLFDBZNGGFMA2O","target":"record","payload":{"canonical_record":{"source":{"id":"1006.5122","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T08:35:46Z","cross_cats_sorted":["math.CT","math.GN"],"title_canon_sha256":"b03bb8f571a29f9e35d4e467611461e11cae5c8912e65c4d5a4e077cbdee98c5","abstract_canon_sha256":"8f2b5fd72bbae84b1ca8a485d5cc96bdf0ddd8c85421ec61a6ceca143c9ff22e"},"schema_version":"1.0"},"canonical_sha256":"dc0046969b614e182ca30e5a631580d3802f324039a62d746ee1c12fb885c98b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:47.596946Z","signature_b64":"/sBfO8XdVFFXemUsICK3/H4jwHWxsSSv2eHGaM+VErXmQmuSNu+daHRwufE5Y2pKqLHYd1YYDe9StfwUMlL/Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc0046969b614e182ca30e5a631580d3802f324039a62d746ee1c12fb885c98b","last_reissued_at":"2026-05-18T04:27:47.596411Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:47.596411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1006.5122","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":"2026-05-18T04:27:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e6Uhz+GLwaOh78HIs9nv05HOfHG5cS83er9jYMQ8x7WwPFIgEGzOsFKbNPodG3lISU3ouHuR5lxWQkscdgk7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:37:45.684003Z"},"content_sha256":"1556cdbba1069563a8162bd0c422191e983b8401d3bd51be5617dae97507305f","schema_version":"1.0","event_id":"sha256:1556cdbba1069563a8162bd0c422191e983b8401d3bd51be5617dae97507305f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:3QAENFU3MFHBQLFDBZNGGFMA2O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropy in a category","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.GN"],"primary_cat":"math.GR","authors_text":"Anna Giordano Bruno, Dikran Dikranjan","submitted_at":"2010-06-26T08:35:46Z","abstract_excerpt":"The Pinsker subgroup of an abelian group with respect to an endomorphism was introduced in the context of algebraic entropy. Motivated by the nice properties and characterizations of the Pinsker subgroup, we generalize its construction in two directions. We introduce the concept of entropy function h of an abelian category and define the Pinsker radical with respect to h, so that the class of all objects with trivial Pinsker radical is the torsion class of a torsion theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5122","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":""},"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":"2026-05-18T04:27:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P4Iq++FEI/25U3ZhaJ8fjGtnFZ+DqE6pwyPURmGbvT2t/OBKD84lWlPKyrEcVPpRQtTYXFf9U1NXOgFDDWe2DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:37:45.684526Z"},"content_sha256":"c0b116eebdfb0488713b6417fffae0ac29d6b312c41edeedaf7c0f4659f32926","schema_version":"1.0","event_id":"sha256:c0b116eebdfb0488713b6417fffae0ac29d6b312c41edeedaf7c0f4659f32926"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3QAENFU3MFHBQLFDBZNGGFMA2O/bundle.json","state_url":"https://pith.science/pith/3QAENFU3MFHBQLFDBZNGGFMA2O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3QAENFU3MFHBQLFDBZNGGFMA2O/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":"2026-06-01T14:37:45Z","links":{"resolver":"https://pith.science/pith/3QAENFU3MFHBQLFDBZNGGFMA2O","bundle":"https://pith.science/pith/3QAENFU3MFHBQLFDBZNGGFMA2O/bundle.json","state":"https://pith.science/pith/3QAENFU3MFHBQLFDBZNGGFMA2O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3QAENFU3MFHBQLFDBZNGGFMA2O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:3QAENFU3MFHBQLFDBZNGGFMA2O","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":"8f2b5fd72bbae84b1ca8a485d5cc96bdf0ddd8c85421ec61a6ceca143c9ff22e","cross_cats_sorted":["math.CT","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T08:35:46Z","title_canon_sha256":"b03bb8f571a29f9e35d4e467611461e11cae5c8912e65c4d5a4e077cbdee98c5"},"schema_version":"1.0","source":{"id":"1006.5122","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.5122","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"arxiv_version","alias_value":"1006.5122v2","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.5122","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"pith_short_12","alias_value":"3QAENFU3MFHB","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3QAENFU3MFHBQLFD","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3QAENFU3","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:c0b116eebdfb0488713b6417fffae0ac29d6b312c41edeedaf7c0f4659f32926","target":"graph","created_at":"2026-05-18T04:27:47Z","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"},"paper":{"abstract_excerpt":"The Pinsker subgroup of an abelian group with respect to an endomorphism was introduced in the context of algebraic entropy. Motivated by the nice properties and characterizations of the Pinsker subgroup, we generalize its construction in two directions. We introduce the concept of entropy function h of an abelian category and define the Pinsker radical with respect to h, so that the class of all objects with trivial Pinsker radical is the torsion class of a torsion theory.","authors_text":"Anna Giordano Bruno, Dikran Dikranjan","cross_cats":["math.CT","math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T08:35:46Z","title":"Entropy in a category"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5122","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":"sha256:1556cdbba1069563a8162bd0c422191e983b8401d3bd51be5617dae97507305f","target":"record","created_at":"2026-05-18T04:27:47Z","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":"8f2b5fd72bbae84b1ca8a485d5cc96bdf0ddd8c85421ec61a6ceca143c9ff22e","cross_cats_sorted":["math.CT","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T08:35:46Z","title_canon_sha256":"b03bb8f571a29f9e35d4e467611461e11cae5c8912e65c4d5a4e077cbdee98c5"},"schema_version":"1.0","source":{"id":"1006.5122","kind":"arxiv","version":2}},"canonical_sha256":"dc0046969b614e182ca30e5a631580d3802f324039a62d746ee1c12fb885c98b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dc0046969b614e182ca30e5a631580d3802f324039a62d746ee1c12fb885c98b","first_computed_at":"2026-05-18T04:27:47.596411Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:47.596411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/sBfO8XdVFFXemUsICK3/H4jwHWxsSSv2eHGaM+VErXmQmuSNu+daHRwufE5Y2pKqLHYd1YYDe9StfwUMlL/Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:47.596946Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.5122","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1556cdbba1069563a8162bd0c422191e983b8401d3bd51be5617dae97507305f","sha256:c0b116eebdfb0488713b6417fffae0ac29d6b312c41edeedaf7c0f4659f32926"],"state_sha256":"97067a1d76bef46c843f99f4e8bbf028af63a30d84042bb779682d92ff547e70"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"su/vYwmRt2EoZqR0tXusUfB9pCcPrJ/5g/z+4MaQesE5rGAhW8RkQyL6IZVCIlr68TCYJqCO2AGtyXN4s7GIBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T14:37:45.687133Z","bundle_sha256":"ba065381c8998a0aa804d99561922e594b5445685afaee0d74860e36e5203c9b"}}